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we develop a method for out-of-sample testing to assess the performance of our model under
new price scenario realizations.
3.2 Introduction
The costs incurred by many industries in today’s economy, such as oil and gas,
farming, electrical utilities, and airlines, depend heavily on commodity prices. Resource and
fuel purchases often constitute a substantial portion of operating costs for such organizations
and are therefore at risk to price fluctuations. For example, over half of residential and
industrial customers’ electricity bills [38] and about 35% of airline costs [39] relate to coal,
gas, andoilprices; asthesepricesrise, consumerburdenandairlineticketpricesalsoincrease,
while a drop in prices likely yields a corresponding decrease. As such, companies with a large
amount of price exposure can greatly benefit from making decisions to hedge against price
fluctuationsandminimizeexpectedcostsforresourceswhentheunderlyingpricesaresubject
to significant uncertainty.
Resource purchases are often long-term decisions; for example, a soy farm may con-
tractacropforpurchasebeforeaseason, oranairlinemaypurchasefuelforyearsinadvance.
Similarly, an electrical utility may engage in long-term fuel contracts, with scheduled deliv-
eries in subsequent years. It is common practice for these long-term decisions to be made
before future prices are known. Companies engage in hedging strategies to both reduce the
effect of future price uncertainty and attempt to minimize present and future costs by ex-
ecuting contracts at a price that they believe to be most favorable. For a purchaser, this
means trying to secure contracts for future time periods at today’s price if there is evidence
that prices will likely increase. By contrast, sellers try to engage in sales for future deliv-
ery at today’s prices when prices are likely to decrease in the future. This behavior is an
attempt to hedge against future price volatility. Once contracts are established at known
prices, contract holders are no longer exposed to unfavorable price movements. At the same
time, hedging against unfavorable future price movements reduces upside exposure as well;
if a buyer locks in today’s price for a future delivery and the price continues to drop, the
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buyer ends up overpaying.
Traditionally, electric utilities in the United States have enjoyed relatively stable and
therefore predictable fuel prices. Because fuel prices have been consistent, planning for the
purchase and delivery of coal has been a straightforward decision in which a policy that
has been implemented historically is carried forward to future years with small adjustments.
Recently, coal prices have been more volatile in response to increases in volumes of renewable
resources such as wind and solar, technological advances in fracking that have made natural
gas cost-competitive with coal, and reductions in global coal demand. In response, electri-
cal utilities have been forced to reassess their coal purchasing strategy in order to protect
themselves against increasing fuel price volatility.
In this chapter, we build a multi-stage stochastic program that minimizes the con-
ditional value-at-risk of future purchases to help inform today’s decisions and to provide a
minimum cost solution that is robust to future price uncertainty. We use a utility’s coal
procurement decisions over the next five years, though this process is applicable to any se-
quential purchase decision to satisfy demand over time under price uncertainty. The utility
makes annual fuel purchase decisions and can procure coal for delivery in the current year or
for future years. We first build a stochastic model to generate future price scenarios, which
are realized weekly. Annual purchase decisions are made using the weekly price available at
the time of the purchase.
We next solve a facility location model to select and assign probabilities to a smaller
number of scenarios that are representative of the Monte Carlo samples generated for each
stageinwhichweneedtomakepurchasedecisions. Wethenformulateandsolvetwoversions
of a multi-stage stochastic program to determine how much coal to purchase in each year
for physical delivery in both current and future years with the goal of minimizing expected
cost and the conditional value-at-risk. Finally, we discuss our results and compare the
performance of the two models using out-of-sample testing. Through the work presented in
this paper, we contribute the following:
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1. An methodology to determine optimal purchase decisions over multiple time periods
in which future costs are subject to uncertainty.
2. A generalized and notationally compact multi-stage nested CVaR implementation.
3. A comparison of the performance of a nested CVaR model formulation to that of an
expected CVaR model.
4. A methodology for out-of-sample solution performance testing in a multi-stage envi-
ronment.
This chapter is organized as follows: we first describe the forecasting model for fu-
ture price scenarios in Section 3.3. In Section 3.4, we discuss our strategy for selecting
representative scenarios from a larger number of scenarios that we generate using Monte
Carlo simulation and our forecasting model. The result of the scenario-reduction scheme are
tractable multi-stage, nested and expected CVaR models, the formulations of which are de-
tailed in Sections 3.5 and 3.6. Finally, we discuss the data we use to construct these models
and present our results in Section 3.7.
3.3 Price model
We construct a stochastic model for future coal prices using regression and vector
autoregressive time series models. We first construct a regression model using historical
data to predict future coal prices. We build models for seven different coal types from which
the utility can choose to source its supply. These coal types are differentiated by quality
specifications involving heat, ash, sulfur, and moisture content. Coals that have high heat
contentandlowash, sulfur, andmoisturecontentarehigherqualitybecausetheyarecleaner-
burning and more efficient, and are therefore more expensive. All of the coal prices in our
model are derived from two underlying weekly index prices, denoted P , where i ∈ {1,2}
it
denotes the index and t ∈ {1,...,219} denotes the week. We build a regression model using
time and natural gas prices as dependent variables, then transform the resulting forecasted
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values to obtain predictions for the specific type of coal using the linear transformation in
Equation (3.1). We index coal types by θ ∈ {1,...,7}; these differ from the index prices by
the inclusion of transportation costs, premiums for purchases made in advance of delivery,
and quality adjustments. These additions are unique to each supply and demand node
location, purchase and delivery date, and coal type. We use Equation (3.1) to transform
index price, pk , to coal type price, ck ; the cost to procure coal type θ from supply node
it′ sdθt′t
s in the year beginning with week t′ for delivery in year beginning with week t at demand
node d under price realization k is given by
ck
sdθt′t
= I iθ(pk
it′
+q iθ)+r
sd
+a t′t, (3.1)
k∈K
X
where
• I is a binary indicator that is 1 if price index i is used to derive the cost of coal θ.
iθ
The price of each coal type depends on a single index price.
• pk is the price of index coal i in time t′ under realization k ($ per ton)
it′
• q is a quality adjustment made to price index i to yield the cost of coal θ if price
iθ
index i is used to derive the cost of coal θ. The magnitude and direction of the quality
adjustment is made based on the difference in coal quality specifications. ($ per ton)
• r is the transportation cost of sending coal from supply node s to demand node d ($
sd
per ton)
• a t′t is added to reflect the premium paid in time t′ for delivery in a future time period,
t ($ per ton)
To obtain the cost in Equation (3.1), we need realizations of the price indices, pk .
it′
We follow a multi-step approach in which we remove linear and seasonal trends in the price
indices and then use a vector autoregressive lag-one (VAR(1)) model for their residuals.
The methodology that we use for the linear and seasonal trends is detailed in [40], and the
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VAR lag-one model is discussed in [41]. The utility provides us four years of daily historical
coal price data that we use to build the forecast model. As price data are not available for
weekends or holidays, we average the daily data to obtain a weekly price, P = (P ,P )T
t 1t 2t
where P is the price for index i at week t. The regression method that we now describe
it
models P as a function of gas prices (g ), weekly time indices (t), and month into which
t t
time t falls (m ).
t
3.3.1 Linear trend
Wefirstremoveanytrendsfromthepricebyfittingamultiplelinearregressionmodel.
Because coal prices display a downward trend over time and are dependent on natural gas
prices, we fit the linear trend:
P = α+β t+β g +ǫ , (3.2)
t 1 2 t t
where
• t is the week of the observation
• P is the observed vector of the price indices at week t; P = (P ,P )′
t t 1t 2t
• β is a 2×1 vector of time coefficients for the linear trend; β = (β ,β )′
1 1 11 12
• β is a 2×1 vector of gas coefficients for the linear trend; β = (β ,β )′
2 2 21 22
• g is the price of natural gas at time t
t
• α = (α ,α )′ is the intercept for the linear trend
1 2
• ǫ t = (ǫ 1t,ǫ 2t)′ is an error term at time t, whose additional structure is discussed in
Section 3.3.2
We remove the effect of natural gas prices and the downward trend by forming resid-
uals, Pr, of the linear regression model:
t
Pr = Pˆ −P , (3.3)
t t t
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where Pˆ is the price predicted by Model (3.2) at time t and P is the observed price index
t t
at time t.
3.3.2 Seasonality
We next remove the effects of seasonality from the price index residuals by fitting a
periodic trend (3.4) to the residuals of the linear Model (3.2), Pr:
t
Pr = αr +βrsin(m˜ )+βrcos(m˜ )+ǫr, (3.4)
t 1 t 2 t t
where
• Pr: a vector of observed detrended price indices at week t; Pr = (Pr,Pr)′
t t 1t 2t
• αr = (αr,αr): the intercept of the periodic trend
1 2
• βr: a 2×1 vector of the sine coefficients of the periodic trend; βr = (βr ,βr )′
1 1 11 12
• βr: a 2×1 vector of the cosine coefficients of the periodic trend; βr = (βr ,βr )′
2 2 21 22
• m˜ : the month into which time t falls, transformed into radians; m˜ = mt · π
t t 360 180
• ǫr = (ǫr ,ǫr ): an error term at time t whose structure is discussed in Section 3.3.3
t 1t 2t
Weremovetheeffectofseasonalityfromourpriceobservationsbytakingtheresiduals,
Prr, of the periodic model:
t
Prr = Pˆr −Pr, (3.5)
t t t
where Pˆr is the detrended price predicted by the periodic trend at time t, and Pr is the
t t
detrended price observation at time t.
3.3.3 Vector autoregressive lag-one model
We then fit a vector autoregressive model of lag one (VAR(1)) in Equation (3.6) to
the residuals of the periodic Model (3.4) to account for autocorrelation and the interaction
between the two indices:
Prr = a +A Prr +ǫrr, (3.6)
t 0 1 t−1 t
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where
• Prr: a vector of detrended and deseasonalized price indices at week t
t
• a : a 2×1 vector of constants
0
• A : a 2×2 matrix of coefficients
1
• ǫrr = (ǫrr,ǫrr)T: an error term at time t. The ǫrr are independently and normally
t 1t 2t t
distributed, N(0,Σ), where Σ is the 2×2 covariance matrix
We confirm that the residuals from this model are approximately normal by producing
histograms in Figure 3.1. We ensure that we have removed all dependence and trends
from the price index data by plotting the auto-correlation function (ACF) and partial auto-
correclation function (PACF) plots of the periodic trend model and VAR(1) error terms in
Figure 3.2 and Figure 3.3. Visual inspection of the histograms suggests that the residuals
approximately follow a normal distribution. The reduction of strongly significant values at
all lags in both the ACF and PACF plots once the VAR(1) model has been applied to the
period trend model error terms suggests that the temporal dependence has been removed.
Figure 3.1: Histogram of VAR(1) residuals from Equation (3.6).
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The model fits are given in Table 3.1. The overall R2 values of 92.3% and 93.1%
demonstrate that the modeling approach explains a large percentage of the variance in the
observed prices. The majority of the variance for both index prices is explained by the linear
model and VAR(1) model; the periodic model explains a much smaller percentage of the
observed price variance. This is not unexpected as coal contracts are typically longer term
contracts, and coal can easily be stored in inventory, which reduces the effect of seasonality
on prices.
Table 3.1: Percentage of variation in price index explained by regression model.
R2
Model (Equation)
Price 1 Price 2
Linear trend (4.2) 62.4% 63.7%
Periodic trend (4.3) 3.4% 4.2%
VAR(1) (4.4) 26.5% 25.2%
Overall 92.3% 93.1%
Figure 3.4 shows observed versus predicted historical values for index prices one
and two. Visual inspection suggests our model is a good predictor of the historical price
movements.
3.3.4 Scenario sampling
We use Monte Carlo sampling from the stochastic model in Equations (3.2)-(3.6) to
generate price inputs to our stochastic programming model. We build future price scenarios
by first drawing n = 1000 samples from the bivariate normal distribution, N(0,Σˆ ), denoted
ǫrrk = (ǫrrk,ǫrrk), fork = 1,...,n, whereΣˆ istheestimatedvalueofΣbasedontheresiduals
t 1t 2t
of Equation (3.6). We in turn pass the realizations of ǫrrk through the autoregressive Model
t
(3.6), our periodic trend Model (3.4), and our linear trend Model (3.2) to include the effects
of seasonality, gas prices, and the linear trend. We do this for each year in our study horizon,
which results in n forecasted price index scenarios for each year. If the resulting index price
is below a given value, we replace it with a price floor, which is set to a level below which coal
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Figure 3.4: Historical observed and predicted prices for price index 1 (left) and 2 (right).
mines would not be able to produce. We then apply the linear transformation from Equation
(3.1), which yields the per ton cost for each coal type, at each supply-demand node pair,
both for delivery in the current time period and for delivery in future time periods.
We calculate the coefficient of variation for the historical prices as well as for our
generated prices to check how well our model captures the behavior of both price indices
1 and 2. Over the past four years, price index 1 has a coefficient of variation of about
8%, and price index 2 has a coefficient of variation of about 9%. For the generated price
distributions, the coefficients of variation are just over 6% for both price indices 1 and 2. The
coefficients of variation for our generated prices are slightly lower than those of the historic
price indices because we include a price floor below which we would expect producers to
stop production and historically, coal has not been cheap enough to see the effects of a price
floor. As coal prices have fallen to historical lows, it is necessary to include this floor in our
forecast to prevent unrealistically low coal prices. The next section discusses how we reduce
the n = 1000 scenarios per year to a representative set of smaller size, n¯ < n.
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3.4 Scenario reduction
Multi-stage stochastic programs can become intractable quickly due to exponential
growth in scenarios by time period. For this reason, we reduce the n sampled realizations
to a representative n¯ scenarios while preserving the characteristics of the original n samples.
The approach we use to reduce the set of scenarios at each stage from n to n¯ is based on
minimizing a distance between the two probability distributions (one with n samples and
one with n¯), while also accounting for first and second moments of each distribution. The
distance we use is known as the Wasserstein metric, and our approach is based on ideas in
Dupaˇcova´ et al. [42] and in Hochreiter and Pflug [43]. See Dupaˇcova´ et al. [44] for further
discussions on scenario generation.
We formulate and solve a facility location problem to select the representative n¯
samplesthatweuseasinputtoourstochasticoptimizationmodel. Weselectarepresentative
subset of size n¯ from the n = 1000 Monte Carlo samples by minimizing the probability-
weighted sum of Euclidean distances between each scenario in the n-sample and n¯-sample
distributions. We further incorporate distances between the first two moments from the two
distributions. Because we are minimizing CVaR, which focuses on the (1−α) most extreme
realizations, we require that the facility location model choose scenarios from the tails of the
distribution with n = 1,000 scenarios.
We generate realizations and choose representative scenarios at each point at which
a new purchase decision is to be made. In Section 3.3, we generate weekly prices while coal
purchase decisions are made annually using the prices available at the start of the purchase
year; we solve the facility location problem for each period of our multi-stage stochastic
program independently, with predicted prices from the first week of each year as input. The
representative scenarios and associated probabilities provide the predicted prices that we
use in our multi-stage stochastic optimization problem. We discuss this in greater detail in
Section 3.5.1. Figure 3.5 depicts the points at which we are selecting representative samples,
as well as our purchase and delivery timing. We produce a five-year purchase plan using
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• ǫrrk: VAR(1) residual for price index i under scenario k; see Model (3.6)
i
• n: number of realizations
• n¯: number of representative scenarios
• µˆ : mean of VAR(1) residuals for price index i
i
• σˆ2: variance of VAR(1) residuals for price index i
i
• σˆ : covariance of VAR(1) residuals for price indices i and j
ij
• α: probability level defining the tails of a distribution
1 if index price residual i in realization k is in the upper 1−α tail
• τ ik:
of the n = 1000 scenario distribution
0 otherwise
We c alculate the distance between scenarios k and k′ as a Euclidean distance:
dkk′ = (ǫrrk −ǫrrk′ )2. (3.7)
t i i
si∈I
X
Variables
1 if realization k is chosen as a representative scenario
• yk:
0 otherwise
1 if realization k′ is assigned to representative scenario k
•
xk′k:
0 otherwise
• zk: nu mber of realizations assigned to representative scenario k
• eµ: difference between the mean of VAR(1) residuals for price index i from the
i
distribution with n¯ scenarios and the mean of the VAR(1) residuals for price index i
from the distribution with n scenarios
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• eσ2 : difference between the variance of VAR(1) residuals for price index i from the
i
distribution with
• n¯: scenarios and the variance of the VAR(1) residuals for price index i from the dis-
tribution with n scenarios
• eσ: difference between the covariance of the VAR(1) residuals for price indices i and
ij
j from the distribution with n¯ scenarios and the covariance of the VAR(1) residuals
for price indices i and j from the distribution with n scenarios
We solve the following facility location model to select the representative scenarios for
each stage, again with the t-index suppressed. We only consider the prices for the week we
makepurchasedecisionsinourfacilitylocationmodel; however, ourregressionmodelinforms
how the prices evolve over the year between decision points. While purchase decisions are
currently made once a year, including weekly prices allows the decision maker the capability
tousethismodeltomakemorefrequentpurchases. Usingthismodel,wereducethen = 1000
generated realizations for each decision point to n¯ = 6 representative samples.
minimize
dkk′ xk′k + eµ +eσ2 + eσ (3.8a)
t i i ij
k ω, Xk 6=′∈ ωK ′: Xi∈I (cid:16) (cid:17) i X, ij <∈ jI:
subject to
xk′k ≤ yk ∀k,k′ ∈ K : k 6= k′ (3.8b)
yk = n¯ (3.8c)
k∈Kt
X
xk′k = 1 ∀k,k′ ∈ K : k 6= k′ (3.8d)
k∈K
X
xk′k = zk ∀k,k′ ∈ K : k 6= k′ (3.8e)
k′∈K
X
1
Pkzk −µˆ ≤ eµ ∀i ∈ I (3.8f)
n i i i
(cid:12) (cid:12)
(cid:12) Xk∈K (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
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zk
(Pk −µˆ )2 −σˆ2 ≤ eσ2 ∀i ∈ I (3.8g)
n i i i i
(cid:12) ! (cid:12)
(cid:12) Xk∈K (cid:12)
(cid:12) (cid:12)
(cid:12) zk (cid:12)
(cid:12) (Pk −µˆ ) −σˆ ≤(cid:12) eσ ∀i,j ∈ I : i 6= j (3.8h)
n i i ij ij
(cid:12) ! (cid:12)
(cid:12) Xk∈K Yi∈I (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) τkyk ≥ 1 (cid:12) (cid:12) ∀i ∈ I (3.8i)
i
k∈K
X
zk ≥ 0; yk,xk′k ∈ {0,1} ∀k,k′ ∈ K
Our objective function in (3.8a) consists of two parts. Through the first term we min-
imize the Wasserstein distance between the probability distribution with n scenarios and the
probability distribution with n¯ selected scenarios, and with the remaining terms, we mini-
mize the distances between the distributions’ first and second moments. Constraint (3.8b)
allows a scenario k′ to be assigned to scenario k only if scenario k is selected as a repre-
sentative scenario. Constraint (3.8c) defines the number of representative scenarios that we
choose. Constraint (3.8d) ensures that every realization is assigned to one representative
scenario. Constraint (3.8e) establishes the number of realizations assigned to each repre-
sentative scenario. Constraints (3.8f)-(3.8h) compute deviations of the first two moments
of the n¯-sample distribution from the sample price index mean, variance, and covariance of
the n-sample distribution. Constraint (3.8i) ensures that we select representative scenarios
from the upper tail of the distribution with 1,000 scenarios. Constraints (3.8f)-(3.8h) are
linearized in our implementation. Non-negativity and binary variable requirements must
hold.
To improve computational tractability of Model (3.8), we employ strategies outlined
by Klotz and Newman [36], such as solving the dual when subject to primal degeneracy,
explicitly employing Devex pricing to determine the outgoing variable in iteration of the
Simplex method at the root node, and rounding our calculated distances to three decimal
places to avoid storing excessive digits. These actions yield reasonable run times (given in
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Table 3.2), which allow us to solve Model (3.8). In Section 2.6 we investigate the stability
of solutions from our multi-stage stochastic program using these distributions. We solve
Model (3.8) in AMPL, using CPLEX version 12.6.2.0 to a 0.05% relative tolerance on a Dell
Power Edge R430 server with a 1TB hard drive and 32GB of RAM. Table 3.3 shows the
mean, variance, and covariance of the optimized probability distribution of the distribution
with 1000 price realizations and the same measures for the distribution resulting from the
reduction to six representative scenarios at each purchase decision point. We specify that
Model (3.8) must select at least one scenario from the (1−α) = 10% tails.
Table 3.2: Run times required to choose six representative scenarios and assign probabilities
to each using Model (3.8).
Year Run time (sec)
2017 1397
2018 472
2019 912
2020 1080
Table 3.3: Mean, variance, and covariance for price samples in scenario one and those re-
sulting from solving Model (3.8).
Samples in distribution Statistic 2017 2018 2019 2020
1000 µ 0.009 0.004 -0.003 -0.001
1
6 µ 0.017 0.004 0.001 0.006
1
1000 µ 0.025 0.010 -0.007 -0.007
2
6 µ 0.003 0.010 -0.009 0.004
2
1000 σ2 0.220 0.223 0.219 0.226
1
6 σ2 0.189 0.203 0.194 0.206
1
1000 σ2 0.297 0.305 0.295 0.304
2
6 σ2 0.275 0.283 0.265 0.279
2
1000 σ 0.055 0.058 0.055 0.059
12
6 σ 0.052 0.057 0.051 0.057
12
Figure 3.6 and Figure 3.7 show the realizations that Model (3.8) selects as repre-
sentative scenarios relative to a histogram showing the frequency of all realizations in each
year. Figure 3.6 shows the realizations selected for price index 1, and Figure 3.7 shows those
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Figure 3.8: Representative scenarios for price index residuals 1 and 2
(3.4) to include the effects of seasonality and generate the error terms, ǫrk = (ǫrk,ǫrk),k =
t 1t 2t
1,...,n¯,t ∈ {2,...,T}. We next apply our linear trend (3.2) to generate prices, Pk =
t
(Pk,Pk),k = 1,...,n¯. We transform the resulting index prices using Equation (3.1) to
1t 2t
generate the coal prices for each supply node, demand node, coal type, purchase and delivery
years,andscenarioinourmulti-stageprogram. Thisresultsinatractablenumberoflocation,
quality, and date-of-delivery specific coal prices that are representative of a larger number
of price realizations to be used as input to our coal procurement problem.
3.5 Multi-stage stochastic programming model
Stochastic programming is a powerful branch of operations research that allows for
decisions to be made under uncertainty. Multi-stage stochastic optimization allows for these
decisions to be made over multiple time periods. Stochastic optimization methodology (in-
cluding multi-stage) are discussed by Shapiro [45] and Birge [46]. Birge [47] outlines different
stochastic programming algorithms and methodologies, and highlights their various applica-
tions including finance, manufacturing, telecommunications, and transportation.
The goal of our multi-stage stochastic program is to minimize the expected cost and
conditional value-at-risk to fill coal demand over multiple time periods in which purchase
decisionscanbemade. InSections3.3and3.4,pricesevolveweekly. Inthissection,purchases
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decisions are made annually using the weekly prices available at the start of each period.
For simplicity of notation, we refer to t by year in the section; however, it is understood
that t is first week of the annual decision stage. We use the representative scenarios selected
with Model (3.8) in each time period to construct a scenario tree to model outcomes of price
uncertainty in each year of our multi-stage stochastic program horizon. The root node of our
tree contains today’s known prices, and each subsequent node contains prices under different
scenarios. Each node level represents future decision making points, which in our model
occur once a year, and each branch represents a different scenario path that is defined by
price realizations over the time periods in our model horizon. This scenario tree branches
six times in each time period, yielding a total of 1,296 paths by 2020, as can be seen in
Figure 3.9.
Figure 3.9: Scenario tree with six scenarios each year.
Undereachoftheserealizations,everycoaltypeconsideredinModels(3.16)and(3.17)
has a price that results from a transformation of the underlying index prices from Equation
(3.1), which makes coal type-specific adjustments for transportation, quality differences, and
premiums for future delivery. We note that Model (3.8) refers to scenario k, while Models
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(3.16) and (3.17) refer to scenario ω . Scenario k is the weekly realization that is used in
t
determining how prices evolve over each year, while scenario ω is the weekly price scenario,
t
k, that is realized at the time that purchases are made in each stage of our multi-stage
stochastic problem, which in our case is annually.
We use the probability distributions generated by solving Model (3.8), as we describe
in Section 3.4, coupled with models (3.1), (3.2), (3.4), and (3.6) as input to our facility
locationmodel. TheoutputofModel(3.8)yieldstheprobabilityofeachoftherepresentative
scenarios occurring in a time period, which we can calculate as:
zk
pk = , (3.9)
n
k∈K
X
where zk is the number of scenarios assigned to representative scenario k in time t, and n is the
t
total number of scenarios each time period. If k is the scenario realized when purchase deci-
sions are made in our multi-stage stochastic program then K = Ω , we define the probability
t
of each scenario ω in Models (3.16) and (3.17) as:
t
pωt = pk, (3.10)
and we define the coal costs to be:
ck = cωt (3.11)
sdθt′t sdθt′t
In Section 3.6 we develop two formulations to minimize the conditional value-at-risk.
However, we also have requirements on the physical delivery and consumption of coal that
must be met. We now present these coal procurement constraints, we we use to characterize
the region in which our set of feasible purchase decisions, X, is defined. In Section 3.6, we
require our decisions to belong to X.
We must fill demand in each time period, and can do so with coal delivered or
from inventory. At each demand node, we must meet certain fuel quality specifications,
both because not every generator can burn every type of coal and because we must meet
various emissions regulations. At certain demand nodes, we can satisfy these requirements
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by blending various types of coal in order to obtain a satisfactory final product that will
be burned. We separate coals into groups based on their qualities; those that have similar
characteristics are viewed as the same type at a plant. The use of two similar coals to serve
demand at the same node is therefore not considered to be blended. Instead, maximum
blending constraints are only enforced for dissimilar coals.
We allow for stockpiling at each demand node as each plant we consider has a coal pile
for inventory storage. The utility has policies in place regarding the minimum and maximum
inventory levels they must carry. Additionally, we account for coal that is already contracted
to be delivered before the start of the model horizon.
3.5.1 Multi-stage stochastic program formulation
Sets
• s ∈ S: set of all supply nodes
• d ∈ D: set of all demand nodes
• θ ∈ Θ: set of all coal types
• β ∈ B: set of all coal type groups
• π ∈ Π: set of all coal characteristics; Π ={heat content, sulfur content, ash content,
moisture content}
• s ∈ S : subset of supply nodes that can send coal to demand node d
d
• d ∈ D : subset of demand nodes that can receive coal from supply node s
s
• θ ∈ Θ : subset of coal types available at supply node s
s
• θ ∈ Θ : subset of coal types available in coal type group λ
λ
• t ∈ T : set of all time periods: t ∈ {1,2,...,|T |}
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• ω ∈ Ω : set of all scenarios at time t
t t
• ω ∈ a(ω ): ancestor scenarios in time t of scenario ω in time t
t−1 t t
Parameters
• δ : demand at node d in time t (tons/period)
dt
• σ : supply of coal type θ at node s in time t (tons/period)
sθt
• cωt : time-discounted cost per ton of coal type θ purchased in time t′ for delivery in
sdθt′t
timet(t′ ≤ t),travelingfromsupplynodestodemandnodedunderscenarioω ($/ton)
t
• κ : quantity of characteristic π in coal type θ (heat content in MMBtu/ton, sulfur,
πθ
ash and moisture contents in % of total weight)
• l : minimum quantity of characteristic π for demand node d in the year beginning
πdt
with week t (heat content in MMBtu/ton, sulfur, ash and moisture contents in % of
total weight)
• u : maximum quantity of characteristic π for demand node d in the year beginning
πdt
with week t (heat content in MMBtu/ton, sulfur, ash and moisture contents in % of
total weight)
• µ : maximum number of coal type groups that can serve demand at node d in the
dt
year beginning with week t
• M: a sufficiently large number
• I : minimum inventory level of coal type θ at demand node d in the year beginning
dθt
with week t (tons)
• I : maximum inventory level of coal type θ at demand node d in t (tons)
dθt
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• Iˆ : initial inventory level of coal type θ at demand node d (tons)
dθ0
• pω: probability of scenario ω
t
• φ : tons of coal type θ already purchased from supply node s for demand node d
sdθt
for delivery in the year beginning with week t (tons)
Variables
• xω : tonsofcoal typeθ purchasedintheyearbeginningwith weekt′ fordeliveryin
sdθt′t
the year beginning with week t at demand node d: from supply node s under scenario
ω (t′ ≤ t) (tons)
• νω : tons of coal type θ burned at demand node d in the year beginning with week
dθt
t under scenario ω (tons)
1 if coal type group λ serves demand at node d under scenario ω in time t
• zω :
dλt
0 otherwise
• Iω : inventory of coal type θ at demand node d at the end of the year beginning
dθt
with week t under scenario ω
We define X to be in the feasible region defined by the following constraints.
νωt ≥ δ ∀d ∈ D,t ∈ T ,ω ∈ Ω (3.12a)
dθt dt t t
θ∈Θ
X
xωt + xωt∈a(ωt) ≤ σ ∀s ∈ S,θ ∈ Θ ,t ∈ T ,ω ∈ Ω (3.12b)
sdθtt sdθtt sθt s t t
d X∈Ds
(cid:0)
t
X
t′ ′∈ ≤T t:
(cid:1)
l νωt κ νωt ≤ u¯ νωt ∀π ∈ Π,d ∈ D,t ∈ T ,ω ∈ Ω
πt dθt πθ dθt πdt dθt t t (3.12c)
θ∈Θ θ∈Θ θ∈Θ
X X X
νωt ≤ Mγωt ∀d ∈ D,λ ∈ Λ,t ∈ T ,ω ∈ Ω (3.12d)
dθt dλt t t
θ X∈Θλ
γωt ≤ µ ∀d ∈ D,t ∈ T ,ω ∈ Ω (3.12e)
dλt dt t t
λ∈Λ
X
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Iωt = Iωt −νωt + xωt +φ ∀d ∈ D,θ ∈ Θ,t ∈ T ,ω ∈ Ω (3.12f)
dθt dθ,t−1 dθt sdθt′t sdθt t t
s X∈Sd (cid:0)t
X
t′ ′∈ ≤T t:
(cid:1)
I = Iˆ ∀d ∈ D,θ ∈ Θ (3.12g)
dθ0 dθ0
I ≤ Iωt ≤ I ∀d ∈ D,t ∈ T ,ω ∈ Ω (3.12h)
dt dθt dt t t
θ∈Θ
X
xωt ,Iωt ,νωt ≥ 0,γωt binary
sdθt′t dθt dθt dλt
∀s ∈ S,d ∈ D,θ ∈ Θ ,t ∈ T ,t′ ∈ T : t ≤ t,t′′ ∈ T : t′′ < t′,ω ∈ Ω
s t t
In the following discussion, each constraint must hold for every time period and under
all scenarios. Constraint (3.12a) ensures that all demand is met and constraint (3.12b)
enforces coal supply limits for each coal type. Constraint (3.12c) ensures that minimum
and maximum coal specification requirements are met at each demand node and for each
specification. Constraint (3.12d) only allows coal from a specific coal group to fill demand if
the binary variable zωt assumes value one with a Big-M constraint. Constraint (3.12e) limits
dλt
the number of coal groups that can be blended at each demand node. Constraint (3.12f)
balances inventory at each demand node. Constraint (3.12g) defines initial inventory for
each coal type at each plant at the start of the model time horizon, and constraint (3.12h)
enforces inventory levels at each plant. Non-negativity and binary variable requirements
must hold.
3.6 CVaR Formulations for Multi-Stage Models
We seek to minimize expected cost as well as the conditional value-at-risk, or CVaR ,
α
or the expected loss given losses exceed the value-at-risk (VaR ). VaR is defined to be the
α α
threshold which losses will not exceed with probability α, and is a common risk measure
in investment decisions. However, VaR is neither convex nor coherent, while CVaR is
α α
both. We favor a convex risk measure when solving optimization problems because the
mathematical properties of convex problems allow us to quickly find and draw conclusions
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about our solution. A coherent risk measure is preferred because it satisfies the following
criteria, as defined by Artzner et al. [48] (ρ is a risk measure and X and Y are random
outcome):
1. Translation invariance:
ρ(X +γ ·r) = ρ(X)−γ
which ensures that the addition of an amount γ to a risk-free investment in a portfolio
will decrease the risk measure by γ.
2. Subadditivity:
ρ(X +X ) ≤ ρ(X )+ρ(X )
1 2 1 2
which provides for the diversification of a portfolio; the risk of a composite holding is
at most equal to the sum of the risk measures of the individual components.
3. Positive homogeneity:
ρ(λ,X) = λρ(X)
which ensures that increasing the portfolio size by a factor will increase the risk by the
same factor.
4. Monotonicity:
∀X ≤ Y,ρ(Y) ≤ ρ(X)
which states that if Y outperforms X in all scenarios, Y will have a lower risk measure
These properties favor portfolios that demonstrate investment practices such as diver-
sification by determining them to be less risky, which make coherent risk measures preferable
to non-coherent ones. In our model we calculate VaR and minimize CVaR , as introduced
α α
in [49]. Philpott et al. show in [50] that by doing so, results in minimum values for both
CVaR and VaR .
α α
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When minimizing risk in a multi-stage setting, we need ensure our measure is time
consistent. Specifically, we need to make certain that we properly categorize risk in future
time periods; the characteristics that qualify a coherent risk measure must hold for all stages
and decisions cannot depend on information from future time periods. Shapiro discusses the
timeconsistencyofrisk measuresin [51]. Wenowexploretwotimeconsistentformulationsof
a multi-period model that minimizes both the expected cost and conditional value-at-risk of
procuring coal to serve future demand under price uncertainty. In each, we use the following
parameters:
Parameters
• Zωt: arbitrary random variable in time t under scenario ω
t t
• α: specified probability level
• λ: weight on risk term
Additionally, we suppress all indices but the time and scenario for costs, cωt, and
t
decisions xωt to simplify notation in this discussion. The risk measure in each time period
t
is defined to be:
ρ Z |ω = (1−λ)E Z |ω +λCVaR Z |ω (3.13)
t t t−1 t t−1 α t t−1
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
The expected value and the CVaR terms are weighted by λ and (1−λ), respectively,
and the CVaR is indexed by the probability level, α. In each time period, both the expected
value and risk term are conditioned on the outcome of the previous stage. We can write this
as a time consistent nested model, with the risk term defined recursively as:
min c x +ρ [c x +ρ [c x +···+ρ [c x ]]] (3.14)
1 1 2 2 2 3 3 3 T T T
x1,...,xT
While this is time consistent, implementing in a multi-stage setting is non-trivial
and can be cumbersome. As an alternative, Dupaˇcova´ and Kozmik [2] develop a second
multi-period CVaR model, which is also referred to as expected conditional value-at-risk, or
E-CVaRbyHomem-de-MelloandPagnoncelli[52]. ComparedtoanestedCVaRformulation,
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E-CVaRisalsotimeconsistentbutcanbeeasiertoimplementasitaveragestheriskinfuture
stages, rather than defining it recursively. The objective function of our E-CVaR problem
can be written in a general sense as:
min c x +E ρ [c x |ω ] +···+E ρ [c x |ω ] (3.15)
x1,...,xT
1 1 ω1 2 2 2 1 ωT−1 T T T T−1
h i h i
In both (3.14) and (3.15), the risk term in time t is dependent on the outcome in
time t−1, though (3.14) defines this dependency in a recursive nature while (3.15) employs
a conditional expectation.
In addition to the parameters needed to formulate our multi-stage risk models, we
define the following variables for the implementation of both a nested multi-stage CVaR
model and E-CVaR model to minimize expected cost and conditional value-at-risk for our
coal procurement problem:
Variables
• zωt: arbitrary random variable in time t under scenario ω
t t
• yωt: auxiliary variable used to calculate nested CVaR in time t under scenario ω
t t
• ζωt: α-level quantile of a cost term in time t under scenario ω ($)
t t
3.6.1 Nested CVaR minimization model for coal procurement
We develop a notationally compact way to implement a nested CVaR model by
capturing the recursion implicit in a nested formulation with auxiliary variables yωt. We do
t
this for our coal procurement problem in the following recursive manner:
minimize
c1 x1 +y1 (3.16a)
sdθ1t sdθ1t 1
Xs∈S d X∈Dsθ X∈ΘsXt∈T (cid:16) (cid:17)
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3.6.2 E-CVaR minimization model for coal procurement
An E-CVaR formulation can be simpler to implement than that of the nested model
because rather than defining the risk for subsequent time periods recursively, it instead
averages later-stage uncertainty with a conditional expectation. We construct this for our
coal-procurement problem in the following manner:
minimize
pω1cω1 xω1 (3.17a)
1 sdθ1t′ sdθ1t′
Xs∈Sd X∈Dsθ X∈Θst X′∈T ω Xt∈Ωt
T
λ
+ (1−λ) pωtcωt xωt +λ pωt−1ζωt−1 + pωtzωt
t sdθ,t,t′ sdθ,t,t′ t−1 t−1 1−α t t
Xt=2( Xs∈Sd X∈Dst
X
t′ ′∈ <T t:ω Xt∈Ωt ωt−X1∈Ωt−1 ω Xt∈Ωt )
(3.17b)
subject to:
cωt xωt −ζ −zωt ≤ 0 ∀t ∈ T : t ≥ 2,ω ∈ Ω (3.17c)
sdθt′t sdθt′t t−1 t t t
Xs∈S d X∈Dsθ X∈Θst X t′ ′∈ ≤T t:h i
xωt ∈ X ∀s ∈ S,d ∈ D ,θ ∈ Θ ,t ∈ T : t′ ≤ |T |,t ∈ T : t ≤ t′,ω ∈ Ω (3.17d)
sdθ,t,t′ s s t t
xωt ,zωt,ζωt ≥ 0
sdθt′t t t
∀s ∈ S,d ∈ D,θ ∈ Θ ,t ∈ T ,t′ ∈ T : t′ ≤ t,t′′ ∈ T : t′′ < t′,ω ∈ Ω
s t t
Inourobjective, weminimizetheexpectedcoaltoprocurecoalaswellastheexpected
conditional value-at-risk. Because the CVaR term in the objective results in a non-linearity
due to the positive part operator, we replace the nonlinear term with auxiliary variable in-
cludeandconstraint(3.17c)tolinearizetheobjective. Allofthecoalprocurementconstraints
from (3.12) hold, as do non-negativity and binary variable requirements.
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3.7 Data and results
Model input data are provided by a public utility in the United States that operates
service territory in multiple states and whose generation portfolio includes coal, natural gas,
nuclear, andrenewableresources. Wearegivenannualfueldemandatcoal-firedpowerplants
forthenextfiveyears, aswellassuppliers(mines)thatcanreacheachplantviarailtransport
and the quality specifications of coals originating at each mine. The quality specifications of
interest to the utility are heat content (given in MMBtu per ton), ash content, sulfur content,
and moisture content (all given as a percent by weight). Each plant can burn coal whose
qualities fall within a range for the specifications included in the model. The utility provides
information about the number of different coal types that can be blended to meet demand,
as well as minimum and maximum inventory that can be held at each plant. Finally, we are
provided details about the amounts of coal that have already been purchased for delivery
overthehorizon. Thepricesandscenarioprobabilitiesareoutputsfromourregressionmodel
and scenario selection model. Because we use historical data to develop our price scenarios,
we will rerun this model each year when we need to make the next set of purchasing decisions
as additional price data is available and can better inform our models.
We solve both the multi-stage expected CVaR Model (3.17) and the nested CVaR
Model(3.16), usingpricesfromourscenariogenerationandselectionprocedure. Wegenerate
1,000 scenarios for the first week of each future time period (2017, 2018, 2019, and 2020).
We then use Model (3.8) to form probability distributions comprised of six selected scenarios
with first and second moments that most closely match those of the distributions with 1,000
scenarios that we are trying to approximate. We use the output of Model (3.8), together
withourpricemodels, (3.1), (3.2), (3.4), and(3.6)toconstructcoalpricesandcorresponding
probabilities for each chosen scenario.
When solving our model, we need to choose λ (the weight on the risk term in the
objective function) and α (the probability level used to define the value-at-risk and the
conditional value-at-risk). Our selection of both parameters can greatly affect our solution:
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λ close to one weights the objective function to minimize the risk term exclusively. If α is
defined to be close to one, the value-at-risk is going to be higher because the it is unlikely a
large loss will occur. Because CVaR is the expected value of losses given that the value-at-
risk is exceeded, a higher value-at-risk implies that only higher loss values are considered in
the calculation of CVaR. This necessitates that an α near one will produce a larger CVaR.
Additionally, in Model (3.16), the effects of choosing proper λ and α are compounded by
the nested nature of the formulation. Specifically, we are concerned with the term λ ; if
1−α
this value is larger than one, over multiple stages the risk term in the objective function is
multiplied by a very large number, which can affect our solution. Likewise, if this value is
much smaller than one, the risk term reduces to almost zero, and the objective function only
minimizes the expected cost, It is therefore in our best interests to keep λ ≈ 1 to ensure
1−α
we do not either over or understate the weight of the risk term.
We first solve Models (3.16) and (3.17) with λ set to zero; in this case we are ignoring
the risk term and only minimizing the expected cost to procure coal. We expect both models
to produce the same value because they differ in how they calculate risk, but not expected
cost. Confirming this, we next choose different values of α and λ, solve both models, then
calculate the expected costs and CVaR terms. The expected cost terms from both Models
(3.16) and (3.17) are shown Figure 3.10 and Figure 3.11, respectively, while the risk terms
are shown in Figure 3.12 and Figure 3.13.
In both models, we observe that as the objective function weight is shifted from the
expected cost term to the risk term, the expected cost increases and the risk term decreases.
The overall objective function is better served by minimizing the more heavily-weighted
term; i.e., when the risk term carries the most weight, we are willing to face higher expected
costs to reduce CVaR. Additionally, we note that as α increases, the value of the risk term
increases; as the probability level used to define CVaR increases, the expected value of the
losses exceeding the value-at-risk is larger because more extreme tail losses are included in
the calculation of the conditional expectation.
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Model (3.16) produces expected cost terms that do not show much movement until
almost all objective function weight is placed on the risk term. Likewise, the risk term
does not show much variation as λ increases. This may be due to the nested nature of the
(3.16) skewing the weight the risk term carries in the objective function, as addressed in our
discussion of selecting λ and α.
To better understand the effects that introducing the risk term has on the model
behavior, we next consider values of λ that are very close to zero. Figure 3.14, Figure 3.15,
Figure 3.16, and Figure 3.17 show the expected cost and risk terms from both Models (3.16)
and (3.17) for λ less than .1.
Figure 3.14: Expected cost term from Model (3.16) for varying values of α and λ near 0.
Figure 3.14 and Figure 3.15 show the expected cost term for Models (3.16) and (3.17),
respectively. Model (3.16) grows in expected cost as λ increases to .05, then flattens, while
Model (3.17) shows a much more gradual rise in expected cost. We see this same behavior
in the risk term in Figure 3.16 and Figure 3.17, with Model (3.16) producing a CVaR which
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Figure 3.17: Risk term from Model (3.17) for varying values of α and λ near 0.
shows only very slight reductions as λ increases and Model (3.17) generating a CVaR that
decreases gradually but more significantly than in Model (3.16). Furthermore, the risk term
from Model (3.16) remains larger than that from Model (3.17), while the expected cost
from Model (3.16) remains lower than that from Model (3.17), which may be a result of the
disproportional effects of the λ factor in the nested model. This indicates that an expected
1−α
CVaR model may be better suited to solving a multi-stage stochastic program seeking to
minimize risk; the E-CVaR model incurs an increase in expected cost in order to drive down
the risk term, which we do not see in the nested model. If the risk term calculation is overly
sensitive to λ in the nested model, increasing the expected cost may not serve to reduce
1−α
the risk term in order to decrease the overall objective.
In both Models (3.16) and (3.17), we note that we do not see significant variation in
either expected cost or risk term as we vary α and λ. In fact, we see less than a 1% change
in cost over the parameter values we choose. This suggests that solving a stochastic program
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1. After obtaining the solutions to the multi-stage stochastic programs, Models (3.16)
and (3.17), we extract the optimal purchase strategies.
2. We next generate 100 paths, which consist of weekly price index realizations in each
time period in which we make purchases.
3. In each time period, we next match the new, generated price index scenarios,
pkˆ
to
i
the original price index realizations, pk, such that we minimize the Euclidean distance
i
between index prices:
dkkˆ = (pkˆ −pk)2 +(pkˆ −pk)2
1 1 2 2
q
4. We then match the new realizations to the scenarios in each time period of Models
(3.16)and(3.17); becausethesetofweeklypricescenariosrealizedatthetimepurchase
decisions are made define ω , and we can use
pkˆ
to define
ckˆ
, which we can map to
t i sdθt′t
cωt (because there exists ω = k when purchases are made), so long as the structure
sdθt′t t
of our scenario tree holds in the matched scenarios.
The new scenarios are thus assigned to those used to solve Models (3.16) and (3.17).
We assume that if we implement the purchase decisions suggested by solutions, and future
prices differ from those we use as input, we would adjust our decisions to match those under
the scenario that is most similar to our observations. We calculate the coal type specific
costs with Equation (3.1), and use these costs to determine the total cost to procure coal
under each new price scenario. Finally, we average the total cost across all 100 additional
price realizations to arrive at the expected cost to procure coal under out-of-sample testing,
which is given in Table 3.6.
We use our out-of-sample methodology to calculate the expected cost of coal procure-
ment under new realizations for solutions obtained by solving Models (3.16) and (3.17) for
α = .9 and λ = .5, i.e., CVaR defined with probability 90% and equal weight placed on
the objective function terms. Both Models (3.16) and (3.17) produce a higher expected coal
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Table 3.6: Cost to procure coal under out-of-sample scenarios using the purchase plan rec-
ommended by Models (3.16) and (3.17) ($MM)
Expected cost from model Out-of-sample expected cost
Model (3.16) 2,404.83 2,469.60
Model (3.17) 2,406.25 2,471.39
procurement cost under new scenarios, which is expected given that these additional realiza-
tions were not used to solve our multi-stage stochastic program. We note that both models
show similar behavior under new prices, with expected costs within 1% of one another.
3.8 Conclusion
We develop a methodology for coal procurement for a public utility that must make
large purchase decisions in the face of price uncertainty. We first build a regression model
that incorporates a linear trend as a function of both time and gas prices, a periodic trend
to describe the effects of seasonality, and a VAR(1) model to capture autocorrelation and
the interaction between the two price indices. We next generate and select scenarios to best
represent future price outcomes while maintaining tractability of our multi-stage stochastic
program. We select representative scenarios such that the first and second moments of the
approximating distribution they comprise very closely match the first and second moments
of the distribution from which we sample.
We next minimize the expected cost to procure coal and the conditional value-at-risk.
We develop a simplified, nested formulation for minimizing CVaR in a multi-stage stochastic
environment that is time consistent and formulate our multi-stage coal procurement problem
using this nested CVaR model, as well as using an expected CVaR model. We compare the
the results of these two implementations, and show that the nested model seems to be more
sensitive to the choice of parameters λ and α. This suggests that E-CVaR may be a better
method to use in order to hedge against future price uncertainty. Finally, we develop an
out-of-sample testing methodology and evaluate the performance of our model under new
price realizations.
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We note that we see very little change in the expected coal procurement cost and
CVaR term, suggesting that a stochastic program, even one incorporating a risk measure,
maynotbenecessaryforcoalprocurementplanningduetothehighconfidencewithwhichwe
can construct price forecasts. It may be beneficial to work to incorporate a larger number
of scenarios to better capture marginal cases; to do this, we will focus future work on
speeding up our facility location model using indexed sets to help restrict the number of
decision variables that assign realizations to representative scenarios. Specifically, we suggest
a scheme that only considers assignment variables that fall below a threshold distance. This
will allow us to select more than six representative scenarios each year to better approximate
all generated scenarios and potentially increase the value of solving a stochastic program.
Additionally, we may work to increase the price uncertainty by considering the variability
associated with gas prices or transportation costs. Currently, we assume these are known
with certainty but in reality, these two elements demonstrate significant variation. Finally,
it may be a valuable exercise to extend this work to other commodities that can be procured
through forward purchases, such as natural gas.
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CHAPTER 4
DEVELOPING A COAL PURCHASE STRATEGY UNDER PRICE UNCERTAINTY
FOR A MAJOR UTILITY
In Chapter 3, we develop a methodology for determining a coal procurement strategy
under price uncertainty. We now apply our methodology to a public utility in the United
States to assess its five-year coal purchase plan, and expect the Utility to implement our
recommendations when it purchases coal in April, 2017. This paper will be submitted to
Interfaces and is co-authored by Alexandra Newman and David Morton.
4.1 Abstract
A major electrical utility produces power for residential and commercial customers in
the United States and is seeking to reevaluate its coal procurement strategy to reduce its
fuel costs. The Utility currently practices a forward purchase strategy, which it has been
questioning given historically low coal prices. We propose a new policy for the Utility by
developing a multi-stage stochastic program to minimize the expected cost and conditional
value-at-risk of purchasing coal to serve demand over five years while meeting physical con-
straints such as supply and demand, plant inventory coal quality, and blending capabilities.
Our findings include adapting current demand hedging practices to purchase coal closer to
the time of delivery instead of requiring forward purchases; we show that doing so yields a
significant decrease in expected cost. We demonstrate that these results hold under out-of-
sample testing and conclude with a discussion of the effects of increased price variability on
our solution. We project that implementation of our purchase plan will save the Utility $151
million in coal procurement costs over five years.
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4.2 Introduction
We develop a coal procurement strategy considering a five-year time horizon for a
major utility (to which we refer as the Utility) in the United States that is faced with
future price uncertainty. The Utility is a publicly traded company that serves customers in
multiple states, is comprised of different operating subsidiaries, functions as both a regulated
electric and natural gas utility, provides transmission service, and includes a marketing
group that trades both electricity and commodities. As a participant in multiple energy
markets operated by independent system operators (ISOs), the Utility is subject to oversight
by the Federal Energy Regulatory Commission (FERC), the North American Reliability
Corporation (NERC), and numerous state Public Utility Commisions (PUCs). The Utility
employs over 6,000 people and prides itself on its environmental and community involvement
initiatives.
The Utility owns and operates generation facilities to serve electricity and gas demand
in multiple states. At its instantaneous peak, the Utility has the capacity to produce more
than 17,000 MW of power for over 3.5 million electricity customers, and its generation
portfolio includes multiple resource types, including coal, gas, nuclear, and renewables, such
as wind and solar (Figure 4.1 and Table 4.1).
Figure 4.1: The Utility’s generation by fuel type. Coal is the largest resource, contributing
to 46.1% of the Utility’s capacity.
While the Utility employs a diverse resource mix, coal-fired generation constitutes
the largest share of its portfolio, accounting for over 7,400 MW, or 43.5%, of its capacity.
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Table 4.1: The Utility’s generation resources by fuel type, separated into thermal, renewable,
and other resources. *Wind generation is based on installed capacity and is an intermittent
resource that is available when conditions exist to support generation.
Type Plants Units Net Dependable Capacity (MW)
Coal 12 23 7,409
Natural Gas 25 70 6,877
Nuclear 2 3 1,594
Hydro 26 79 377
Wind 3 238 327*
Solar 4 4 0.1
Other 4 19 435
Total 76 438 17,019
Its coal plants range in size from a single generator that is capable of producing 500 MW
to multiple generators with a combined capacity of over 2,000 MW. The Utility purchases
coal on an annual basis from multiple sources (mines) and transports the coal to its plants
by train. As such, a mine is only a viable option to supply a plant if the two are connected
by rail.
As a primary producer of an essential good, the Utility is faced with two main goals:
it is expected to provide reliable, uninterrupted power, and it is required to do so as cheaply
as possible to its consumers. The Utility has a responsibility to keep costs at reasonable
levels for its customers because it essentially operates as a monopoly. As such, it consistently
seeksstrategiestoreduceitspowerproductioncosts, whicharederivedfrommultiplefactors,
including building and maintaining power plants and the transmission system, power grid
operation, and fuel. Fuel, including purchase and transportation, has the largest impact
on customer bills; between 2004 and 2014, according to figures provided by the US Energy
InformationAgency(EIA),fuelprocurementaccountsforbetween75-80%ofcoalproduction
and 35-79% of total power costs [38]. Our objective is to minimize the amount the Utility
spends buying coal by optimizing its purchase strategy; reducing the financial burden of coal
procurement will have a significant impact on overall power generation.
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The Utility contracts coal from mines that are accessible from its plants by rail;
generators must be approved to receive different coal types through test burns. Typically,
generators are engineered for certain quality specification targets (though newer generators
have more flexibility in this respect) and utilities tend to adhere to recommended operation.
The Utility burns sub-bituminous coal at most of its plants (though some have the capac-
ity to burn higher-quality bituminous coal), which ranges from 8,500 - 13,000 BTu/lb with
15-45% moisture content, and low sulfur and ash content (less than 2% and less than 10%,
respectively), depending on the coal’s origin [53]. Once a coal type is approved for consump-
tion at a plant, the Utility can engage in variable-length purchase contracts with mines and
can negotiate with railroad companies to transport coal from mine to plant; however, our
work is primarily concerned with fuel procurement and assumes that transportation costs
are known and enduring.
Though it has the flexibility to buy coal at any time, the Utility typically makes large
procurement decisions once a year, choosing to purchase coal for delivery in the current
year or to contract volumes for delivery in future time periods, though it pays a premium
to do so. At present, the Utility follows a long-term procurement strategy that seeks to
fill specific percentages of demand at each plant in advance (Table 4.2). Today’s price of
coal for delivery in future years is known with certainty. Making purchases now limits the
Utility’s exposure to commodity market volatility; purchasing coal for delivery in future time
periods is a hedge against price uncertainty. In this sense, the premium that the Utility pays
for future deliveries is a measure of its risk aversion. This policy, a standard in strategic
planning, shields customers from unforeseen fluctuations resulting from increased fuel prices.
While its current purchase plan protects the Utility and its customers against price
increases, it necessarily results in forgone savings opportunities afforded by potential future
price drops. This strategy is a product of previous experience and has been approved by
upper-level management because it has historically provided sought-after risk abatement for
theUtility. However,inrecentyears,coalpricesintheUnitedStateshavedroppeddrastically
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Table 4.2: The Utility’s current purchase strategy. Annual demand is filled using the given
hedging scheme, with advance purchases defined by the percentage of demand they comprise.
For example, by 2016, the Utility needs to have purchased enough coal to serve between 70%
and 90% of its 2017 demand.
Purchase 2016 2017 2018 2019 2020
year (Current year) (1-year hedge) (2-year hedge) (3-year hedge) (4-year hedge)
2016 100% 70-90% 40-65% 15-35% 0%
([38]) which has affected the financial impact of the Utility’s purchases on its overall cost of
operation. With this decline in coal prices, the Utility is reassessing the appropriateness of
its future purchase strategy.
Coal price movements have significant financial impact on the Utility and its cus-
tomers. Because coal is the largest resource in electricity production, fluctuations greatly
affect customer bills. A discussion of the United States coal market trends and factors in-
fluencing prices can be found in [14] and [54]. Since these works have been published, the
U.S. market has seen a drastic evolution; Figure 4.2 shows the historical movements over
the past four years of the prices of two index coals against which the Utility benchmarks its
resources.
This recent decline in coal prices can be attributed to multiple factors, including a
drastic reduction in gas prices, decreases in coal production and transportation costs, and
environmental regulations that have reduced coal demand. Price decreases are expected to
continue because many of these factors represent fundamental shifts in United States coal
production and power generation. As such, the Utility is seeking to reevaluate its forward
procurementstrategybecauseitislessconcernedwithrisksresultingfromexposuretofuture
price spikes, and instead is interested in the potential savings that adjusting its purchase
strategy may afford its customers due to the decreasing nature of coal prices.
Whendeterminingthefinancialburdenoffuturedecisions,itisusefultoconsiderarisk
measure to help capture the effects of future price uncertainty. We incorporate conditional
value-at-risk (CVaR) into our model to give weight to this variability. CVaR is defined to
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be the expected portfolio losses given they exceed some predefined value; in this sense, it
is a measure of the most extreme outcomes. Additionally, it is convex and coherent [48],
which is especially valuable in financial planning because it naturally encourages investment
practices such as diversification and investment in options that are overall less risky. We
implement a multi-stage nested model that minimizes both the expected cost to procure
coal and CVaR [10]. Minimizing CVaR is appealing to a risk-averse provider of an essential
good, such as the Utility, because, in addition to minimizing costs, it reduces exposure to
unfavorable price movements.
We develop a multi-stage stochastic optimization model to determine a five-year coal
strategy for the Utility by minimizing the expected cost and conditional value-at-risk of
purchasing coal. We use a regression model to predict future coal prices and a facility
location model to select representative price scenarios to include in our optimization model.
We minimize the expected cost and conditional value-at-risk to procure coal by making
purchase decisions over a five-year horizon. We are faced with the same physical constraints
described in [10] such as supply and demand, inventory balance, coal quality specifications,
and coal blending. Figure 4.3 shows our methodology and how information moves between
models, which we refer to as (PPR), (PFL) and (PSP).
This chapter is organized as follows: we first outline the scenario-generation and
selection techniques we use to develop our model input. We then present our multi-stage
stochastic model formulation, and describe the data used to define our parameters. We
present our results, including a discussion of the benefits of our proposed coal-purchase
strategy compared to the Utility’s current plan and a recommendation for implementation.
We assess the performance of our solution under simulated future price scenarios using a
technique for out-of-sample testing and discuss the effects of increased price variation on our
results. The mathematical structure of our models can be found in Sections 4.9, 4.10, and
4.11.
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Figure 4.3: Process diagram for models (PPR), (PFL), and (PSP) and corresponding infor-
mation flow.
4.3 Price regression and scenario generation
As we are minimizing cost subject to future price uncertainty, we use a regression
model to generate different price scenarios, the mathematical details of which can be found
in Section 4.9. The coal prices we include in our stochastic program (PSP) are determined
by their quality specifications and are a result of a linear transformation of underlying index
prices, the historical movements of which are shown in Figure 4.2. Our regression model
(PPR) predicts the value of these indices, which we then use to derive the price of each
of the coal types our model uses. The transformation from index to coal price includes
transportation from mine to plant, a quality adjustment to account for differences in coal
specifications between the index and the coal type price, and a premium to purchase coal for
delivery in a future year. To this end, high-quality coal that is purchased for future delivery
and which must be transported a long distance is the most expensive coal the utility can
procure.
Our regression model (PPR), the mathematical structure of which is discussed in
[10], uses historical price data ranging from January 2012 to March 2016 (Figure 4.4). With
this model, we systematically remove various trends which explain portions of the price
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variability, yielding random residuals that are independent and are identically and normally
distributed. At this point, we can sample new error terms and add back the trends to
generate coal price forecasts that follow historical behavior. To do this, we remove the
downward trend and dependency on natural gas prices, as well as the effects of seasonality
from the residuals of our linear fit (Figure 4.5). We then fit a vector autoregressive lag-one
model (VAR(1)) to the remaining residuals to remove autocorrelation and dependencies of
the price indices on one another (Figure 4.6). Removing the effects of the VAR(1) model
produces price residuals that follow a normal distribution from which we can generate future
price realizations. Model (PPR) is based on time series analysis and regression modeling
found in [40] and [41].
Figure 4.4: Historical coal index prices used to build our regression model.
Once the effect of correlated prices is removed, we are left with price residuals that are
independent and identically and normally distributed. From this, we can sample new price
residuals and apply the trends we have modeled with (PPR) to produce coal price forecasts.
The fit and appropriateness of this model are discussed in detail in [10].
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4.4 Scenario selection
After we use our regression model (PPR) to generate scenarios, we choose the price
realizations to include in our stochastic optimization model (PSP) that best represent all
sampled values. The mathematical formulation for our scenario selection problem (PFL)
can be found in Section 4.10.
Using model (PPR), we generate 1,000 realizations for each time period in which
a purchase decision must be made [10]. Because of the exponential nature of multi-stage
stochastic optimization problems, we reduce the number of realizations to improve tractabil-
ity while maintaining characteristics of the 1,000-scenario distribution. A facility location
model (PFL) is used to choose a small number of representative scenarios at each point at
which a new purchase decision must be made. The remaining realizations are assigned to a
chosen scenario such that the Euclidean distance is minimized, which is an approximation for
scenario proximity. The probability of each selected scenario is determined by the number of
realizations it is assigned. The resulting distribution, which consists of the selected scenarios
and their probabilities, approximates the original 1,000-scenario distribution in its first and
second moments. We require that scenarios are selected from the tails of the 1,000-scenario
distribution because CVaR focuses on extreme values. This methodology is discussed in
detail in [44] and [43]. We apply our regression models and linear price transformation from
(PPR) to the selected price scenarios to generate price inputs for our multi-stage stochastic
program (PSP).
4.5 Literature Review
To determine the optimal coal purchase strategy under future price uncertainty, we
solve a multi-stage stochastic program. Discussions of stochastic programming and its appli-
cations can be found in [45], [46], and [47]. Electrical utility planning offers opportunities for
the application of multi-stage optimization as the majority of decisions must be made using
forecast data as input. Wallace and Fleten [55] outline energy optimization models that in-
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clude uncertainty. Other examples in the literature include Ahmed et al.[56] and Shuna and
Birge [57], both of whom develop models for capacity expansion under uncertainty. Fleten
and Kristofferson [58] examine hydroelectric production planning in a day-ahead market.
Swaroop et al. [59] formulate and solve a stochastic unit commitment and dispatch model
subject to wind and load uncertainty. Most closely related to our work is that of Ni et al.
[60], who discuss the problem of uncertainty in future commodity prices; while they focus on
the problem of portfolio re-balancing to develop a generalized optimal hedging strategy in
the face of volatile commodity prices, we examine specific physical applications and include
practical constraints such as blending to meet quality specifications and stockpiling.
4.6 Model formulation
We now present a discussion of our multi-stage stochastic programming formulation
(PSP), which determines a purchase strategy such that we minimize the cost to procure coal
and the conditional value-at-risk. The mathematical structure of (PSP) can be found in
Section 4.11. (PSP) uses price scenarios generated by our regression model and selected by
our facility location model at input.
We seek to minimize the cost and conditional value-at-risk to procure coal for a
five-year time horizon. Coal purchases are made once a year using available price data.
Sets in our model consist of demand nodes (plants), supply nodes (mines), time-indexed
stages, and scenarios. We consider the following time periods: (1) the period in which
coal is purchased, (2) the period in which it is delivered and (3) the period in which it is
burned. We define different coal types and groups in order to enforce quality constraints and
blending capabilities at each plant. Our types are differentiated by source, by price, and by
characteristic, and the groups consist of coal types that are similar in quality (as specified
by the Utility). Coal blending is a process that combines drastically different types such
that the final product is homogeneous. We define viable paths between supply nodes and
the demand nodes that can be reached by rail. As we are solving a multi-stage stochastic
program, we also need to specify ancestor and descendant scenarios in order to define how
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scenarios are related between time periods.
We require that demand is met every year in which coal is burned, at every plant,
and under every scenario. In our model, we do not include plants that are co-owned and
operated or scheduled for imminent retirement. Demand may be met by coal purchased
prior to the development of this model, purchased in the current year at known prices, or
purchased in future years, and the supply of each coal type cannot be violated. We require
that purchases in a given year cover at least a minimum percentage of demand in a future
year, according to the Utility’s purchase strategy. We eliminate this constraint to determine
the effects the relaxation of such a policy has on the Utility’s coal costs.
The coals that the Utility can purchase are defined by source as well as quality, given
by energy content, sulfur content, ash content, and moisture content. Each of the Utility’s
plants has coal specifications resulting from generator capability or emissions requirements.
We allow for blending of coals at each plant to meet these requirements. The coal quality
at each plant is thus a weighted average of the qualities of the coals that are blended to
serve demand. We require minimum and maximum quality measurements to be met at each
demand node, in each period in which coal is burned, and under every scenario. We also
restrict blending capabilities at each plant as most plants cannot blend more than two coals
to serve demand. We enforce blending restrictions with a Big-M constraint; so long as a coal
group is used, we allow any coal type from that group to serve demand at the plant. We
limit the number of coal groups that can be burned with a knapsack constraint that enforces
an upper bound at each demand node, in each period in which coal is burned, and under
every scenario.
We account for coal arriving and being consumed at each plant through inventory
balance constraints. At each demand node, coal can be held between time periods in a
stockpile from which the plant draws to burn. At the beginning of our model run, we are
given initial inventory levels at each plant. At the end of each subsequent time period, the
amount of coal stored is the sum of the inventory from the previous time period and the
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deliveries that occur less the coal burned in that time period. Stockpiles must stay between
a minimum and maximum level at each plant. We index our inventory by coal type as coals
are blended and burned once they leave the pile. Figure 4.7 shows the process by which coal
moves from supply nodes to the plant where it is burned.
Figure 4.7: Coal is purchased for delivery either immediately or in future years. When it is
scheduled for delivery, it is transported from the source, where it is mined, to a power plant,
where it is blended with other coal types and stored in a stockpile. When it is needed, coal
is removed from the stockpile and burned.
4.7 Model results
We first include a constraint enforcing the Utility’s current purchase strategy by re-
quiring coal to be procured in advance of the delivery year, using the pre-defined percentages
in ??. We refer to this version of (PSP) as (PSP). The Utility’s current purchase plan is
1
similar to what we find by solving (PSP) in that it is required to follow the procurement
1
strategy we capture with our demand-hedge constraint. The Utility is not currently solving
an optimization model to determine its purchases; we can use the solution to represent the
cost of the Utility’s current plan, though our solution may actually have a lower overall cost.
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We next remove the demand hedge constraint and determine the optimal purchase
strategy for the Utility without any forward purchase obligations; we refer to this model as
(PSP) and show that it is in the Utility’s best interest to adjust its demand hedging strategy.
2
We next generate additional scenarios to determine how well our optimal solutions behave
under out-of-sample scenarios; again, we compare the performance of our solutions from
models (PSP) and (PSP). Finally, we explore how the Utility’s purchase plan would change
1 2
in the face of increased price volatility.
We run our multi-stage stochastic program (PSP) coded in AMPL, using CPLEX
Version 12.6.2.0 on a Dell Power Edge R430 server with a 1TB hard drive, two internal Xeon
processors, and 32GB of RAM.
Our model produces a procurement plan that minimizes both the expected cost of
coal and CVarR. Our results show that the Utility’s current purchase strategy must be
suboptimal; the expected cost to procure coal over five years is about 6% lower in (PSP)
2
than it is in (PSP). Table 4.3 shows the objective function value in each case.
1
Table 4.3: Expected cost to procure coal ($MM)
(PSP) (PSP) Savings ($) Savings (%)
1 2
Expected coal cost 2,557 2,406 151 6.3%
Removing the requirement to procure a certain percentage of coal for future delivery
produces an expected savings for the Utility of $151 million over a five year period. Coal
prices are not expected to undergo any significant increases over the next five years, so the
Utility can wait to purchase coal until the year in which it is to be delivered. Doing so
allows the Utility to realize expected savings generated by price decreases as well as avoid
premiums for purchasing coal for delivery in future years. Figure 4.8 shows the difference in
purchase strategy with and without the demand hedge constraint included.
(PSP) approximates the Utility’s current strategy, in which purchases are made mul-
1
tiple years in advance. The results from (PSP) indicate that the Utility should wait to buy
2
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coal until the year in which it must be delivered in order to minimize both expected costs
and CVaR. However, it is unlikely that the Utility would delay all coal purchases until the
year in which coal is needed to serve load. As a provider of a necessary good, the Utility
prefers to incur a higher cost to protect itself against overexposure to price volatility; though
forecasts show that coal prices are unlikely to increase, the Utility’s policy is to make de-
cisions to protect its customers against relatively unlikely events. As such, we assume that
the Utility will not completely forgo a coal procurement strategy, but may instead choose to
reduce the requirement for forward purchase volumes. Figure 4.9 and Figure 4.10 show how
the expected cost to procure coal and CVaR terms decrease as we relax the percentage of
demand that must be met in advance.
Figure 4.9: Expected coal procurement costs as the demand hedge requirement is relaxed.
There is an approximately linear dependence of both the expected cost to procure
coal and CVaR terms over a five-year horizon on the percent of demand the Utility requires
to be filled in advance of delivery. Though the Utility is unlikely to commit to foregoing all
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Figure 4.10: CVaR term as the demand hedge requirement is relaxed.
forward purchases, it can still realize significant savings by reducing the amount of coal it
currently purchases for future years. We next assess how both solutions perform under new
price realizations.
4.7.1 Model validation
We perform out-of-sample testing todetermine how our solution performs given prices
that differ from those we use to solve our multi-stage stochastic program. This methodology
assumes that the Utility chooses to follow the purchase strategy determined by our multi-
stage stochastic program for the next five years, and that upon seeing future coal price
realizations, it purchases coal as recommended under the price scenario to which reality is
most similar; i.e., the scenarios whose price components are closest in Euclidean distance
[10]. To simulate this, we generate fifty new, independent price scenarios over the horizon
to evaluate the performance of the model solutions from (PSP) and (PSP). We match each
1 2
new realization to one of the scenarios used to solve (PSP) and implement the decision
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recommended in that scenario. We then average the expected cost of coal procurement
across all fifty new price scenarios. Our out-of-sample testing results are given in Table 4.4.
Table 4.4: Out-of-sample coal procurement cost ($MM)
(PSP) (PSP) Savings ($) Savings (%)
1 2
2,609 2,471 137 5.6%
Under new price scenarios, our recommendation to curtail forward purchases results
in a lower expected coal procurement cost. When implemented under price realizations that
are not used as model input, we see a 5.6% savings when we remove the demand hedge
constraint compared to the coal procurement cost found when adhering to the Utility’s
current purchase plan. The expected cost using out-of-sample testing is similar to those
we find solving (PSP), which shows that our model yields stable results; the Utility can
expect that implementing our solution will not result in large fluctuations in procurement
costs due to different price realizations. We acknowledge that as we move forward and have
additional historical data with which we can refine our price model, we will resolve our multi-
stage stochastic optimization model to inform and potentially restructure coal purchases in
future years. However, our out-of-sample testing methodology provides insight as to how
our proposed solution will perform as future prices are realized.
We next discuss the performance of (PSP) and (PSP) under greater price volatility.
1 2
The behavior under heightened price volatility is valuable because the Utility tends to be
risk averse and therefore wants to understand the effects of increased uncertainty.
4.7.2 Increased price volatility
Usingourpriceforecast, ourcoefficientofvariationisabout6%. Tohelpinformfuture
purchase decisions, we increase the coefficient of variation to determine how the forward
purchase strategy changes. We see that both objective function and the conditional value-
at-risk are larger as the coefficient of variation grows. Figure 4.11 shows the expected cost to
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consists of a linear trend which defines dependency on time and gas prices (4.2), a periodic
trend which explains the seasonality of price movements (4.3), and a vector autoregressive
model, which determines the effects of prices on one another from one time period to the
next (4.4). We then use the linear transformation, (4.1), to develop coal prices that are
adjusted for location, quality, and the timing of delivery.
Linear transformation from index price to coal type price:
ck
sdθt′t
= I iθ(pk
it′
+q iθ)+r
sd
+a t′t, (4.1)
k∈K
X
where
• pk : the price of index coal i in time t′ under realization k ($ per ton)
it′
• I : a binary indicator that is 1 if price index i is used to derive the cost of coal θ
iθ
• q : a quality adjustment made to price index i to yield the cost of coal θ. The
iθ
magnitude and direction of the quality adjustment is made based on the difference in
coal quality specifications ($ per ton)
• r : is the transportation cost of sending coal from supply node s to demand node
sd
d ($ per ton)
• a t′t: is added to reflect the premium paid in time t′ for delivery in a future time
period, t ($ per ton)
Linear trend:
P = α+β t+β g +ǫ , (4.2)
t 1 2 t t
where
• t: the week of the observation
• P : the observed vector of the price indices at week t; P = (P ,P )′
t t 1t 2t
• β : a 2×1 vector of time coefficients for the linear trend; β = (β ,β )′
1 1 11 12
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• β : a 2×1 vector of gas coefficients for the linear trend; β = (β ,β )′
2 2 21 22
• g : the price of natural gas at time t
t
• α = (α ,α )′: the intercept for the linear trend
1 2
• ǫ t = (ǫ 1t,ǫ 2t)′: an error term at time t, whose additional structure is explained by
Model (4.3)
Periodic trend:
Pr = αr +βrsin(m˜ )+βrcos(m˜ )+ǫr, (4.3)
t 1 t 2 t t
where
• Pr: a vector of observed detrended price indices at week t; Pr = (Pr,Pr)′
t t 1t 2t
• αr = (αr,αr): the intercept of the periodic trend
1 2
• βr: a 2×1 vector of the sine coefficients of the periodic trend; βr = (βr ,βr )′
1 1 11 12
• βr: a 2×1 vector of the cosine coefficients of the periodic trend; βr = (βr ,βr )′
2 2 21 22
• m˜ : the month into which time t falls, transformed into radians; m˜ = mt · π
t t 360 180
• ǫr = (ǫr ,ǫr ): an error term at time t whose structure is explained by Model (4.4)
t 1t 2t
Vector autoregressive lag-one model
Prr = a +A Prr +ǫrr, (4.4)
t 0 1 t−1 t
where
• Prr: a vector of detrended and deseasonalized price indices at week t
t
• a : a 2×1 vector of constants
0
• A : a 2×2 matrix of coefficients
1
• ǫrr = (ǫrr,ǫrr)T: an error term at time t. The ǫrr are independently and normally
t 1t 2t t
distributed, N(0,Σ), where Σ is the 2×2 correlation matrix
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1 if index price residual i in realization k is in the 1−α tail
• τ ik :
of the distribution with n = 1000 scenarios
0 otherwise
Variables
1 if realization k is chosen as a representative scenario
• yk:
0 otherwise
1 if realization k′ is assigned to representative scenario k
•
xk′k:
0 otherwise
• zk: nu mber of realizations assigned to representative scenario k
• eµ: deviation of the mean of VAR(1) residuals for price index i from the distribution
i
with
• nˆ: scenarios and the mean of the VAR(1) residuals for price index i from the distribu-
tion with n scenarios
• eσ2 : deviation of the variance of VAR(1) residuals for price index i from the distri-
i
bution with
• nˆ: scenarios and the variance of the VAR(1) residuals for price index i from the dis-
tribution with n scenarios
• eσ: deviation of the covariance of the VAR(1) residuals for price indices i and j from
ij
the distribution with nˆ: scenarios and the covariance of the VAR(1) residuals for price
indices i and j from the distribution with n scenarios
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• π ∈ Π: set of all coal characteristics; Π ={heat content, sulfur content, ash content,
moisture content}
• s ∈ S : subset of supply nodes that can send coal to demand node d
d
• d ∈ D : subset of demand nodes that can receive coal from supply node s
s
• θ ∈ Θ : subset of coal types available at supply node s
s
• θ ∈ Θ : subset of coal types available in coal type group λ
λ
• t ∈ T : set of all time periods: t ∈ {1,2,...,|T |}
• ω ∈ Ω : set of all scenarios in time t
t t
• ω ∈ ∆(ω ): descendant scenarios in time t of scenario ω in time t−1
t t−1 t−1
• ω ∈ a(ω ) : ancestor scenarios in time t of scenario ω in time t
t−1 t t
Parameters
• δ : demand at node d in time t (tons/period)
dt
• σ : supply of coal type θ at node s in time t (tons/period)
sθt
• cωt : cost per ton of coal type θ purchased in time t′ for delivery in time t (t′ ≤ t),
sdθt′t
traveling from supply node s to demand node d under scenario ω ($/ton)
• κ : quantity of characteristic π in coal type θ (heat content in mmBTU/lb, sulfur,
πθ
ash and moisture contents in % of total weight)
• l : minimum quantity of characteristic π for demand node d in the year beginning
πdt
with week t (units as given in κ definition)
• u¯ : maximum quantity of characteristic π for demand node d in the year beginning
πdt
with week t (units as given in κ definition
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• µ : maximum number of coal type groups that can serve demand at node d in the
dt
year beginning with week t
• M: a sufficiently large number
• I : minimum inventory level of coal type θ at demand node d in the year beginning
dθt
with week t (tons)
• I¯ : maximum inventory level of coal type θ at demand node d in t (tons)
dθt
• Iˆ : initial inventory level of coal type θ at demand node d (tons)
dθ0
• pωt: probability of scenario ω in time t
• ψ dt′t: fraction of total demand in time t at node d that must be filled by the year
beginning with week t′
• ρ : tons of coal type θ already purchased from supply node s for demand node d
sdθt
for delivery in the year beginning with week t (tons)
• Zωt arbitrary random variable in time t under scenario ω
t t
• α : specified probability level
• λ : weight on risk term
Variables
• xωt : tons of coal type θ purchased in the year beginning with week t′ for delivery
sdθt′t
in the year beginning with week t at demand node d from supply node s under scenario
ω (t′ ≤ t) (tons)
t
• νωt : tons of coal type θ burned at demand node d in the year beginning with week
dθt
t under scenario ω (tons)
t
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1 if coal type group λ serves demand at node d under scenario ω in time t
t
• υωt :
dλt
0 otherwise
• Iωt : inventory of coal type θ at demand node d at the end of the year beginning
dθt
with week t under scenario ω
t
• zωt: arbitrary random variable in time t under scenario ω
t t
• ζωt: α-level quantile of a cost term in time t under scenario ω ($)
t t
minimize
pω1cω1 xω1 (4.6a)
1 sdθ1t′ sdθ1t′
Xs∈Sd X∈Dsθ X∈Θst X′∈T ω Xt∈Ωt
T
λ
+ (1−λ) pωtcωt xωt +λ pωt−1ζωt−1 + pωtzωt
t sdθ,t,t′ sdθ,t,t′ t−1 t−1 1−α t t
Xt=2( Xs∈Sd X∈Dst
X
t′ ′∈ <T t:ω Xt∈Ωt ωt−X1∈Ωt−1 ω Xt∈Ωt )
(4.6b)
subject to
cωt xωt −ζ −zωt ≤ 0 ∀t ∈ T : t ≥ 2,ω ∈ Ω (4.6c)
sdθt′t sdθt′t t−1 t t t
Xs∈S d X∈Dsθ X∈Θst X t′ ′∈ ≤T t:h i
νωt ≥ δ ∀d ∈ D,t ∈ T ,ω ∈ Ω (4.6d)
dθt dt t t
θ∈Θ
X
xωt + xωt∈a(ωt) ≤ σ ∀s ∈ S,θ ∈ Θ ,t ∈ T ,ω ∈ Ω (4.6e)
sdθtt sdθtt sθt s t t
d X∈Ds
(cid:0)
t
X
t′ ′∈ ≤T t:
(cid:1)
l νωt κ νωt ≤ u¯ νωt ∀π ∈ Π,d ∈ D,t ∈ T ,ω ∈ Ω
πt dθt πθ dθt πdt dθt t t (4.6f)
θ∈Θ θ∈Θ θ∈Θ
X X X
νωt ≤ Mγωt ∀d ∈ D,λ ∈ Λ,t ∈ T ,ω ∈ Ω (4.6g)
dθt dλt t t
θ X∈Θλ
γωt ≤ µ ∀d ∈ D,t ∈ T ,ω ∈ Ω (4.6h)
dλt dt t t
λ∈Λ
X
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Iωt = Iωt −νωt + xωt +φ ∀d ∈ D,θ ∈ Θ,t ∈ T ,ω ∈ Ω (4.6i)
dθt dθ,t−1 dθt sdθt′t sdθt t t
s X∈Sd (cid:0)t
X
t′ ′∈ ≤T t:
(cid:1)
I = Iˆ ∀d ∈ D,θ ∈ Θ (4.6j)
dθ0 dθ0
I ≤ Iωt ≤ I ∀d ∈ D,t ∈ T ,ω ∈ Ω (4.6k)
dt dθt dt t t
θ∈Θ
X
(xω sd′ t θ′ t′ ′∈ ′tA(ωt) +ρ sdθt) ≥ ψ dt′tδ
dt
∀d ∈ D,t ∈ T ,t′ ∈ T : t′ < t,ω
t
∈ Ω
t
(4.6l)
s X∈Sdθ X∈Θst tX′ ′′ ′∈ ≤T t′:
xωt ,Iωt ,νωt ,zωt,ζωt ≥ 0,γωt binary
sdθt′t dθt dθt t t dλt
∀s ∈ S,d ∈ D,θ ∈ Θ ,t ∈ T ,t′ ∈ T : t ≤ t,t′′ ∈ T : t′′ < t′,ω ∈ Ω
s t t
We are minimizing the expected cost and the conditional value-at-risk to procure
coal. Constraint (4.6c) linearizes the positive part operator that is used to define CVaR.
Constraint (4.6d) ensures that demand is met at each plant and constraint (4.6e) enforces
supply limits at each mine. Constraint (4.6f) maintains coal quality requirements over coal
blends that are used to serve demand. Constraints (4.6g) and (4.6h) together ensure that
at most a maximum number of different coals can be blended at a plant. Constraints (4.6i)
- (4.6k) are inventory balance constraints, define initial inventory, and bound the size of
the stockpile at each plant. Constraint (4.6l) is our hedging constraint that defines the
percentage of demand that must be filled a specified number of years in advance according
to the Utility’s purchase strategy; we solve the model both with this included and removed.
Finally, we have non-negativity and binary requirements on our variables.
We run this model coded in AMPL, using CPLEX Version 12.6.2.0 on a Dell Power
Edge R430 server with a 1TB hard drive, two internal Xeon processors, and 32GB of RAM.
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CHAPTER 5
CONCLUSIONS AND FUTURE WORK
In this chapter, we summarize the major contributions of our work and recommend
future work.
Thefirstcontributionofourresearchisamixed-integeroptimizationmodelthatsolves
for optimal thermal coal shipping patterns given deterministic price data, global supply,
demand, and quality requirements, and shipping constraints. The results of this model
provideinsighttoresponsestovariousscenarios, whichincludevariationsinpriceandimport
and export volume requirements, and can be used to inform policy and understand market
position for both coal consumers and producers.
Our second contribution is a methodology that can be used to determine purchase
strategy for future delivery of physical commodities. This methodology consists of a price
regression model, a scenario reduction technique, and a multi-stage stochastic program that
together provide a forward purchase strategy under uncertainty. The price regression model
includes (i) a linear trend to model dependencies on time and other factors (which, in our
case, includes natural gas prices); (ii) a periodic trend to model the effects of seasonality;
and (iii) a vector auto-regressive lag-one model to capture autocorrelation. This regression
model is used to generate future price scenarios. We use a facility location model to reduce
the number of scenarios to provide input to our stochastic program while keeping the model
tractable and capturing the properties of the distribution consisting of generated prices. The
resulting scenarios are included in a multi-stage stochastic program that seeks to minimize
the expected cost and conditional value-at-risk of forward purchases.
Our third contribution is a generalized and notationally compact multi-stage stochas-
tic program for minimizing CVaR. We develop a nested model that is time consistent, cap-
turing the conditional nature of the risk measure in each time period on that of previous
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time periods. This formulation can be written succinctly for multiple time periods and is
therefore more straightforward to implement than models that require explicit notation for
the risk term in every stage.
A fourth contribution is our comparison of an expected conditional value-at-risk (E-
CVaR)andournestedmodelformulation. WeshowthatE-CVaRseemstobeabetteroption
for minimizing risk over a multi-stage problem as it is less sensitive to input parameters λ
and α then the nested model.
A fourth contribution is a methodology for out-of-sample testing that can be used in
a multi-stage environment. This includes generating new price realizations which we match
to the scenarios that were used to solve the model, and implementing the purchase decision
recommended. We can then determine the cost of following our model solution under new
price realizations, and can use this as a method for assessing model performance.
Finally, we implement an E-CVaR model for an electric utility that is reassessing
its procurement strategy given changing coal market conditions. Specifically, we reevaluate
the benefit of a demand hedging program when future coal prices are not expected to see
large increases. We apply our end-to-end methodology as follows: (i) we use our price
regression model to forecast future coal prices given historical observations; (ii) we select
representative scenarios from the generated realizations using our facility location model;
(iii) we run an E-CVaR model to minimize both the cost to purchase coal and CVaR and
determine that it is economically beneficial to reduce the Utility’s requirement for forward
coal purchases. We observe the effects on our solution if we increase price uncertainty and
conclude that significantly greater volatility, relative to our forecast, would be necessary to
encourage substantial forward purchases.
There are several natural extensions for our work. First, we recommend implementing
our methodology for a commodity that faces greater future price volatility than coal, such
as oil or gold. While our solution is beneficial and informative to a risk averse company
such as a public utility, coal prices may not face enough price volatility to justify solving a
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stochasticprogram. However, thereissignificantincentivetoquantifyingandminimizingthe
risk associated with building contracts for commodities that do face large price uncertainty.
This could be very valuable for mining companies that need to make investment decisions
that are highly impacted by price variation.
Indevelopingourcoalprocurementmodel, weassumethattheonlystochasticinputis
the price of coal. In reality, a utility making fuel purchase decisions faces multiple uncertain
elements, including transportation costs, demand, and prices of competing resources. We
may find larger benefit to solving a stochastic program if we incorporate the uncertainties
associated with some of these factors.
Other extensions include varying the number of representative scenarios selected with
the facility location model to understand the behavior of the stochastic program and the
additional computational efforts required. In Chapter 3, we suggest a scheme which would
allowustoproducealargernumberofrepresentativescenariostobetterapproximatepossible
realized prices.
Finally, we acknolwedge that while the global thermal optimization model we discuss
in Chapter 2 is sufficient for the needs of RungePincockMinarco, we necessarily ignore real
aspects of the international coal market by assuming a deterministic problem. There is
likely value to be found in extending this to a stochastic program by incorporating variable
elements such as coal prices, shipping costs, and foreign exchange rates.
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ABSTRACT
Solventextractionisanefficientandeffectivemethodfortheindustrial-scalerecoveryand
purification of metals from mixed raw materials. Understanding the molecular-scale forces
driving extraction remains an area of active research despite over 70 years of experience
with industrial-scale solvent extraction processes. This thesis presents a series of studies
on the extraction chemistry of two extractants relevant to applied separations. The ex-
tractant tributyl phosphate (TBP) is used in the Plutonium Uranium Reduction Extraction
(PUREX) process for recovering uranium and plutonium from used nuclear fuel. The extrac-
tantN,N,N’,N’-tetraoctyldiglycolamide(TODGA)isbeingconsideredforuseintheActinide
Lanthanide Separation (ALSEP) process for separation and purification of lanthanides and
minor actinides from PUREX process wastes.
Distribution data collected for the extraction of trace amounts of fission and corrosion
products by TBP in the presence of bulk uranium were mostly consistent with extraction
through a traditional solvation mechanism. The exeception was the decreased extraction of
low valence transition metals with increasing uranium concentration, which suggests another
extraction mechanism such as reversed micelle formation. Distribution data collected for
lanthanides in a TODGA solvent extraction system were also consistent with a traditional
solvation mechanism. The selectivity trend for TODGA across the lanthanide series was
found to follow the amount of water co-extracted with each lanthanide, suggesting that the
extraction of lanthanides by TODGA is impacted by species in the outer coordination sphere
of the extracted complexes.
Diffusion NMR spectroscopy was used to determine the sizes and interactions of colloidal
TBP aggregates in 20% TBP samples containing nitric acid, zirconium, and uranium. The
aggregate sizes calculated from diffusion experiments were similar to those found previously
using small angle neutron scattering (SANS). However, diffusion experiments suggested the
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ACKNOWLEDGMENTS
First, I would like to thank my advisor, Dr. Jenifer C. Shafer, for her endless patience,
kindness, and support. I am very grateful to have been one of the first students advised by
such an exceptional and accomplished scientist. This thesis would not have been possible
without her guidance.
I would also like to thank the engineers and scientists who helped me as I got to do some
of the most fun research I could never have imagined when I started. I am grateful to Dr.
Yuan Yang for her patient NMR spectroscopy instruction. I still have so much more to learn!
Thank you to Dr. Ross J. Ellis for his mentorship at Oak Ridge National Laboratory and
delightful discussions about solvent extraction. Thank you to my thesis committee for their
input along the way, and for helping me avoid dead ends. And thank you to many others,
who helped me find chemicals in strange labs or trained me on new equipment or pointed
me in the direction of so many interesting ideas. This project has been successful because
of the many ways they contributed.
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CHAPTER 1
INTRODUCTION
Solvent extraction is an important technique for separating components in a solution
based on differences in their distribution between two immiscible solvents. Prior to the
advances in nuclear technology that occurred in the 1940s, solvent extraction was primarily
used for laboratory scale separations of organic compounds[1]. Its widespread use in large
scale inorganic systems can be traced back to the design and construction of a uranium
purification plant at the Mallinckrodt Chemical Works in St. Louis during World War II.
This plant scaled up an analytical method for the determination of uranium that relies
on the extraction of uranium by ether[2, 3]. At an industrial scale, solvent extraction has
the benefit of being able to be run as a continuous, multistage process. The speed and
effectiveness of such separations have made it an attractive choice for recovering pure metals
from ores and other raw materials in the metallurgical industry. Recovery of these metals
would otherwise require a series of selective batch precipitations and crystallizations, which
have a low throughput and limited ability to produce metals of very high purity[4, 5].
Despite over 70 years of experience in the design and construction of solvent extraction
plants, a comprehensive theoretical framework for predicting the thermodynamic properties
of these systems at high solute and extractant concentrations remains an elusive goal. An
accurate predictive model of metal partitioning under these conditions would allow for the
design and optimization of process flowsheets with a minimum of liquid-liquid equilibrium
distribution data, reducing the costs associated with the development of new extraction
plants[6]. It could also be used to efficiently determine trace metal compositions of puri-
fied metals produced in different solvent extraction facilities using computational process
modeling. This latter application is relevant to the field of nuclear forensics, where such
trace metal compositions could be used to identify the origin of interdicted nuclear material
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produced by solvent extraction[7]. The development of a rigorous thermodynamic model for
predicting the behavior of solvent extraction systems requires a fundamental understanding
of the composition, structures, and interactions of chemical species in both the aqueous and
organicphasesofasolventextractionsystem. Thermodynamicmodelscombiningtheoretical
insight and experimental data have already been developed for predicting the behavior of
aqueous phase species in concentrated systems[8–11]. However, no similarly well-developed
correlations exist for organic phase species.
This research focuses on understanding the chemical behavior and structures of or-
ganic phase species in tributyl phosphate (TBP) and N,N,N’,N’-tetraoctyl diglycolamide
(TODGA) extraction systems under conditions similar to those that might be found in in-
dustrial scale processes. The chemical structures of these neutral solvating extractants are
shown in Figure 1.1. TBP is the extractant used in the Plutonium Uranium Reduction
Extraction (PUREX) process for selectively recovering uranium and plutonium from irra-
diated nuclear fuel[12]. The PUREX process was developed in the early 1950s, and is the
predominant industrial-scale reprocessing method used in the nuclear industry. The more
recently developed TODGA extractant is part of the class of neutral diglycolamide deriva-
tives whose extraction behavior was first investigated in the early 2000s[13]. TODGA has
a high affinity for trivalent f-block elements and has been proposed for use in the Actinide
LanthanideSeparation(ALSEP)processforrecoveringminoractinidesandlanthanidesfrom
PUREX raffinates[14]. TODGA may also have applications in the separation of light and
heavy lanthanides[15].
A multipronged approach is used to understand the extraction chemistry of TBP and
TODGA in systems with high aqueous phase concentrations of inorganic solutes and high
organic phase concentrations of extractant. In the past, fundamental studies of molecular-
scale solvent extraction behavior have focused on systems in which the concentrations of
metal and extractant were on the order of one to ten mM. The results of these studies have
been assumed to apply under concentrated conditions, although the evidence to support
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Figure 1.1: Tributyl phosphate (TBP), left, and N,N,N’,N’-tetraoctyl diglycolamide
(TODGA), right, are solvating extractants with applications in industrially-relevant sep-
arations processes.
such an extrapolation is limited. Here, the focus is on solvent extraction systems with metal
and extractant concentrations that are on the order of ten to one hundred mM—one to two
ordersofmagnitudegreaterthaninmanypaststudies. Theexaminationofsolventextraction
systemsmorecloselyapproximatingtheconcentratedconditionsfoundinindustrialprocesses
eliminates the need to extrapolate from data collected under dilute conditions, and allows
for a direct investigation of extraction chemistry in applied, industrial-scale separations.
The methods used to understand the extraction of metals by TBP and TODGA include
both routine and state-of-the-art techniques. The bulk extraction chemistry of TBP and
TODGA is understood using distribution studies and comprehensive chemical characteri-
zation of organic and aqueous phase samples at equilibrium under varying conditions of
acid and metal loading. These methods have been used to understand solvent extraction
of organic and inorganic compounds for over 100 years[16]. The microscopic structures of
extracted species are investigated using diffusion nuclear magnetic resonance (NMR) spec-
troscopy, rheology, and small angle neutron scattering (SANS). These methods have been
usedforatleast40yearstocharacterizecolloidalsystems. Viscositymeasurementsoforganic
phase solvent extraction samples have been used over a similar period of time. However, the
application of diffusion NMR spectroscopy and small angle scattering techniques to solvent
extraction systems is a relatively recent development, occurring only within the last 20 years.
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In brief, the sequence of work covered in this document proceeds from the use of es-
tablished to more sophisticated techniques to advance the overall goal of understanding the
molecular-scaleforcesandstructuresfoundintheorganicphaseofTBPandTODGAsolvent
extraction systems. The remainder of this chapter will cover essential background related
to the extraction of inorganic species by TBP and TODGA. This includes the industrial
applications of TBP and TODGA, the traditional understanding of their extraction mecha-
nisms, and a survey of recent work. It ends with a listing of specific research objectives and
a summary of the chapters making up the body of this thesis.
1.1 Industrial Solvent Extraction Processes: TBP and TODGA
Nuclear reprocessing is used to separate and recycle the components of used nuclear
fuel that have potential for further use in energy generation. It also reduces the volume
of radioactive waste that must be stored in long-term geological repositories. The PUREX
and ALSEP processes were both developed to aid in the management of nuclear waste. In
the PUREX process, an organic phase consisting of 30 v/v % TBP dissolved in an aliphatic
hydrocarbon diluent preferentially extracts tetravalent plutonium and hexavalent uranium,
leaving trace metal impurities behind in an immiscible aqueous phase[17]. This aqueous
feed initially consists of irradiated nuclear fuel dissolved in molar nitric acid. A series of
continuous, countercurrent solvent extraction steps follow, using process equipment such
as mixer-settler banks or centrifugal contactors. After extraction, uranium and plutonium
are stripped back into the aqueous phase one at a time by controlling the oxidation state
of plutonium and the aqueous phase acidity. These separation steps result in two product
streams of uranium and plutonium, which then undergo additional purification. Finally,
these streams are converted to uranium and plutonium solids, often oxides. The PUREX
process is shown schematically in Figure 1.2.
The waste stream that results from the PUREX process contains minor actinides, lan-
thanides, and transition metals. The volume of radioactive waste produced after PUREX
reprocessing can be further reduced by separating and fissioning the minor actinides in fast
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Figure 1.2: PUREX process schematic. Plutonium and uranium are often converted to their
oxide forms after final purification.
reactors. However, trivalent lanthanides in PUREX wastes have chemistries similar to triva-
lent minor actinides but cannot be put in a fast reactor due to their significant nuclear
interactions with neutrons. The established approach to the separation of these elements on
an industrial scale requires two different solvent extraction processes - one to separate the
trivalent lanthanides and actinides from the PUREX waste stream, and one to separate the
minor actinides from the lanthanides. The proposed ALSEP process attempts to separate
trivalent lanthanides and actinides from PUREX raffinates and each other in a single solvent
extraction process.
IntheALSEPprocess,trivalentlanthanidesandactinidesarefirstseparatedfromPUREX
raffinates by extraction by TODGA or a similar diglycolamide derivative in the presence of
an acidic organophosphorus extractant and an aliphatic hydrocarbon diluent[14]. TODGA
has a high affinity for trivalent f-block elements, and is a particularly strong extractant
for the lanthanides[13]. Initially, TODGA extracts both trivalent actinides and lanthanides
from the PUREX raffinates into the organic phase, which is prevented from splitting into
two organic phases by the acidic extractant. The minor actinides are then stripped from
the organic phase by an aminopolycarboxylic acid and citrate buffered aqueous phase. This
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aminopolycarboxylic acid complexes the minor actinides, which are not as strongly retained
by the solvent as the lanthanides. The lanthanides are then stripped from the organic phase
by a more strongly complexing aminopolycarboxylic acid capable of competing with the
extracting power of TODGA, and the solvent is recycled.
1.2 Traditional Description of Extraction by Solvation
Traditionally,theextractionofinorganicspeciesbyneutralsolvatingextractantslikeTBP
andTODGAhasbeenconsideredtoproceedthroughtheformationofdiscretestoichiometric
solvates in the organic phase. This extraction mechanism can be understood as a series of
steps in which water-solvated anions and cations associate in the aqueous phase to form
neutral species, which are then solvated by unionized extractant molecules and transferred
to the organic phase[18]. In the case of TBP and TODGA, the phosphate and amide groups
belonging to each molecule are sufficiently basic to fully dehydrate the extracted metal, so
thatbothextractantsinteractdirectlywiththecationintheextractedcomplex. Thenumber
ofTBPorTODGAmoleculesrequiredtoextractamononuclearcomplexisdependentonthe
oxidation state of the metal according to the extraction equilibria shown in Equations 1.1–
1.6.
M3+ +3NO − +3TBP ↽⇀ M(NO ) TBP (1.1)
(aq) 3 (aq) (org) −−−− 3 3 3(org)
M4+ +4NO − +2TBP ↽⇀ M(NO ) TBP (1.2)
(aq) 3 (aq) (org) −−−− 3 4 2(org)
MO 2+ +2NO − +2TBP ↽⇀ MO (NO ) TBP (1.3)
2 (aq) 3 (aq) (org) −−−− 2 3 2 2(org)
M3+ +3NO − +3TODGA ↽⇀ M(NO ) TODGA (1.4)
(aq) 3 (aq) (org) −−−− 3 3 3(org)
M4+ +4NO − +3TODGA ↽⇀ M(NO ) TODGA (1.5)
(aq) 3 (aq) (org) −−−− 3 4 3(org)
MO 2+ +2NO − +3TODGA ↽⇀ MO (NO ) TODGA (1.6)
2 (aq) 3 (aq) (org) −−−− 2 3 2 3(org)
The equilibria in Equations 1.1–1.6 were inferred from the chemical compositions of or-
ganic phases after being fully saturated with metal, and batch distribution studies under
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highly dilute conditions.[19, 20] These latter studies were performed under the assumption
that extracted species exist in solution as a limited number of distinct chemical species,
analogous to traditional coordination compounds[19]. The compositions of these species
were determined by measuring the distribution ratios of individual metals while varying the
extractant concentration in the organic phase. The distribution ratio of a metal is its equi-
librium organic phase concentration divided by its aqueous phase concentration. When the
logarithm of the distribution ratio is plotted versus the logarithm of the extractant concen-
tration, the slope of the resulting line corresponds to the number of extractant molecules in
the extracted complex. This analysis requires assumptions of ideal solution behavior in the
organic phase, constant activity coefficients of aqueous phase species, and a constant free
extractant concentration in the organic phase[18].
The experimental conditions in these distribution studies were chosen to give validity to
all of the necessary assumptions. Systems with low overall metal and extractant concen-
trations in the organic phase approximate ideal behavior, while the use of a large excess of
extractant supports the assumption of a constant free extractant concentration. Constant
aqueous phase activity coefficients can be assumed if a constant ionic strength is maintained
in the aqueous phase. The stoichiometries of extracted species found using distribution stud-
ies can be verified by comparison with bulk composition measurements in metal-saturated
systems, and have typically been assumed to apply to the concentrated systems used in
industrial scale processes. Deviations from ideal behavior in concentrated systems are then
attributed to non-ideality, rather than the formation of different extracted species[21]. If
the degree of this non-ideality can be quantified, all of the thermodynamic properties of a
solvent extraction system can be predicted, including equilibrium compositions and phase
behavior.
Because of TBP’s long history of use in the solvent extraction field, a substantial body
of literature exists reporting equilibrium compositions of TBP solvent extraction systems
under a wide variety of conditions. This includes distribution data for systems with high
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concentrations of the major actinides as well as distribution data for trace amounts of nearly
all of the elements in the periodic table. The distribution data trends for individual metals
are consistent with the equilibria in Equations 1.1–1.6. However, there has not previously
been distribution data available in the open literature for trace amounts of many fission
product and corrosion impurities in the presence of bulk uranium, conditions similar to
those found in the PUREX process. Such data could be used as further validation of these
equilibria.
TheamountofliteraturedataavailableforsolventextractionsystemsinwhichTODGAis
theonlyextractantismuchlessthanthatforTBP.Whiledistributiondatafortraceamounts
of many elements are available, there is a limited amount of distribution data available
for concentrated systems. The distribution data that exist show that TODGA’s affinity
for the lanthanides increases across the series to gadolinium, beyond which the measured
distribution ratios are approximately constant. Because of the great chemical similarity
among the trivalent lanthanides and their steadily decreasing ionic radii, this behavior seems
anomalous. It would be expected that there would be a consistent trend in distribution ratio
across the series. The collection of TODGA distribution data for individual lanthanides
from across the series at macroscopic concentrations could provide insight into the extraction
mechanisms at work, as well as suggest a reason for the high selectivity of TODGA for the
light lanthanides.
1.3 TBP Non-ideal Solution Behavior and Third Phase Formation
OrganicphasesinsolventextractionsystemscontainingTBPinanaliphatichydrocarbon
diluent are observed to split into two liquid phases in equilibrium with the aqueous phase
under conditions of high acid and metal loading. The upper, light organic phase consists
primarily of diluent, while the middle, heavy organic phase consists primarily of extractant
and extracted solutes. This phenomenon is referred to as third phase formation. In the
traditional approach to solvent extraction, third phase formation was thought to occur be-
cause of the limited solubility of the extracted metal species in the nonpolar diluent. The
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improved solubility of extracted metal species in nonpolar aromatic and polar solvents, such
as alcohols, supports this hypothesis[22]. However, difficulties associated with determining
the activities of organic phase species have made the development of a model capable of de-
scribing this phase transition within the framework of non-electrolyte solution theory alone
a challenging task that has received limited attention in the literature. Most explanations
for the molecular-scale forces responsible for third phase formation in the relevant literature
have been limited to speculation supported by observations of macroscale phenomena.
A computational study by Colon et al. used thermodynamic modeling to predict metal
partitioning under applied conditions in a TBP solvent extraction system also containing
water, nitric acid, uranyl nitrate, and n-dodecane[23]. This approach predicted equilib-
rium phase compositions by minimizing the Gibbs free energy of the system, computed by
modeling the activities of aqueous phase species using the Pitzer model and assuming that
the organic phase behaves as an ideal mixture. When apparent thermodynamic parameters
for organic phase species derived from equilibrium data are used in this model, it is able
to accurately reproduce experimentally determined equilibrium aqueous and organic phase
compositions. This modelling approach is effective as long as the system is not close to the
third phase limit. However, the assumption of ideal behavior by organic phase species makes
it incapable of describing the separation of the organic phase into two liquid phases.
Recently, progress has been made in explaining third phase formation in solvent extrac-
tion systems through the application of concepts and methods developed to describe the
microscopic behavior of colloidal suspensions, a type of complex fluid. In 1991, Osseo-Asare
suggested that TBP, which is a surface-active amphiphile like many inorganic extractants,
might form water, acid, and metal-containing reversed micelles with TBP aggregation num-
bers greater than those determined in batch distribution studies under dilute conditions. In
1999, Erlinger et al. proposed that third phase formation observed in a diamide extractant
organic phase could be attributed to substantial aggregation resulting from attractive inter-
actions between acid and metal-containing reversed micelles in competition with dispersive
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thermal energy. In order to analyze small angle X-ray scattering (SAXS) data for organic
phase diamide samples, the samples were assumed to consist of reversed micelles modeled
as monodisperse hard spheres interacting through surface adhesion. This interparticle in-
teraction model, developed by Baxter in 1968, was used to determine the sizes of diamide
aggregates and the strength of the attractive interactions between them by varying the val-
ues of the aggregate diameter and stickiness parameter,
τ−1,
until a good fit to the data by
the model was achieved. The stickiness parameter is directly proportional to the strength
of the attractive interactions between adhesive hard spheres, such that higher values of
τ−1
correspond to stronger attractive interactions.
Since2003, theBaxtermodelhasbeenusedtointerpretSAXSandSANSdatafororganic
phase TBP samples containing water, nitric acid, and several metals[24–30]. The results for
TBP aggregate sizes and interactions in these small angle scattering investigations have
been consistent across samples containing different metals and inorganic acids, which were
assumed to have been extracted in TBP reversed micelles modeled as monodisperse adhesive
hard spheres. These SAXS and SANS experiments suggest that TBP aggregates consist of
two to five TBP molecules interacting through a strong attractive surface potential with
τ−1
values between approximately 6 and 12. Assuming a thin square well potential with a
well width of 10% of the hard sphere diameter, this corresponds to well depths ranging from
approximately 1.6 to 2.3 k T, with deeper well depths found in samples at compositions
B
approaching third phase formation. This trend in the strength of the attractive interactions
between aggregates supports the idea that third phase formation results from increasing
attractive interactions with increasing metal and acid extraction.
While small angle scattering studies have been successful in determining an approxi-
mate empirical threshold value for
τ−1
beyond which third phase formation takes place in
TBP systems, they do not relate this phase transition to specific thermodynamic quantities
characteristic of the components in these systems. The Baxter model is useful insofar as it
suggests a mechanism for third phase formation, however, there is no way to predict this
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phase boundary from the composition of a system. Given the identities of an extractant,
diluent, and metal, the Baxter model does not suggest a way to predict the greatest con-
centration of metal that can be extracted into the organic phase before a third phase is
formed by relating the
τ−1
for a system to its composition. For the results of these small
angle scattering experiments to have practical significance, they must be connected back to
some fundamental property of the systems under investigation. The development of ther-
modynamic models for solvent extraction systems based on a molecular understanding of
extraction requires still further experimental investigation. The most useful solvent extrac-
tion solution model would be able to quantify the activities of organic phase species in such
a way that both extraction equilibria and the third phase transition could be predicted from
a minimum of experimental data.
1.4 Colloidal and Molecular Approaches to Understanding Extracted Species
Relating the results of small angle scattering experiments to the structures formed by
TBP in solution requires first, a general understanding of the species formed in solution
and their dominant length-scales, and, second, a model that can be used to attribute bulk
measurements of equilibrium phenomena, like small angle scattering patterns, to specific
chemical species. The first requirement impacts the nature of the theoretical treatment
needed to fulfill the second requirement because the simplest model capable of capturing
the behavior of a solute depends on the size of the solute relative that of the solvent. Prior
small angle scattering experiments with TBP solvent extraction samples have relied on the
assumptionthatTBPextractedspeciesarecolloidal. TBPspeciesweretreatedasmesoscopic
particles suspended in a continuum fluid, similar to reversed micelles or macromolecules.
This assumption allows small angle scattering data to be interpreted with relative ease using
simplified particle scattering models. However, the justification for using this approach relies
almost entirely on speculation that such structures may exist, rather than material evidence
that they do[31, 32].
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One way to establish that organic phase TBP samples are colloidal systems is to compare
the results of different experimental methods for a set of identical samples, interpreting the
data from a colloidal perspective. This is the approach used in this thesis. The theoretical
treatment of colloidal systems is relatively straightforward because they fall in the hydro-
dynamic regime. A simplified model of particles moving in a hydrodynamic field can be
used to relate experimentally-accessible quantities in systems in the hydrodynamic regime
to molecular-scale forces and structures. The hydrodynamic field in this model represents
the solvent. If the experimental results for different experimental methods using a colloidal
approach are in agreement, this suggests that the colloidal treatment of TBP extracted
species is valid. However, if the results are contradictory, this suggests that considering
organic phase TBP solvent extraction samples as molecular solutions may be a more appro-
priate model to use in understanding small angle scattering data and extraction by TBP in
general.
The theoretical treatment of molecular solutions is much more challenging than the treat-
ment of colloids. This is because of the need to explicitly address the interactions among
the different species in a molecular solution, which strongly impact their equilibrium proper-
ties. The shapes of the solvent and solute molecules, the directionality and strength of their
interactions, and the mass densities of each type of molecule impact chemical behavior and
structures in these systems. Models relating small angle scattering and other experimen-
tal measurements to molecular-scale phenomena must take these attributes into account.
Because of this complexity, all-atom molecular dynamics (MD) simulations are the most
effective means of modeling behavior in molecular solutions. However, MD models must be
rigorously developed and validated before they can be used to relate molecular-scale phe-
nomena to experimental observations. Regardless, if TBP does not form colloidal species,
such computational methods will be important in understanding solvent extraction at a
fundamental level.
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1.5 Research Objectives and Thesis Summary
The primary objectives of the work described in this thesis and their underlying hypothe-
ses are as follows:
Objective 1: CollectdistributiondatainTBPandTODGAsolventextractionsystems
under concentrated conditions that have not been previously characterized in the
literature.
Hypothesis 1.1: The extraction of trace metals by TBP and lanthanides by TODGA
adhere to a traditional solvation mechanism.
Hypothesis 1.2: TheselectivityofTODGAforthelightlanthanidesresultsfromouter
coordination sphere effects.
Objective 2: Compare the results of diffusion NMR spectroscopy and SANS studies
of organic phase TBP samples interpreted from a colloidal perspective.
Hypothesis 2.1: Like small angle scattering techniques, diffusion NMR spectroscopy
can be used to characterize the nitric acid, uranium (VI) nitrate, and zirconium
(IV) nitrate species extracted by TBP.
Objective 3: Assess the use of colloidal models to describe organic phase TBP samples
by comparing the results of diffusion NMR spectroscopy, rheology, and SANS
studies of samples under concentrated (PUREX-like) conditions.
Hypothesis 3.1: Nitric acid, uranium (VI) nitrate, and zirconium (IV) nitrate are
extracted by TBP as colloidal species under PUREX-like conditions.
Chapters 2 and 3 describe batch distribution studies performed to determine the dis-
tribution ratios of metals in industrially-relevant solvent extraction systems under applied
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CHAPTER 2
DISTRIBUTION OF FISSION PRODUCTS INTO TRIBUTYL PHOSPHATE UNDER
APPLIED NUCLEAR FUEL RECYCLING CONDITIONS
Modified from a paper published in Industrial & Engineering Chemistry Research1
Anna G. Baldwin2, Nicholas J. Bridges3, Jenifer C. Braley4
2.1 Abstract
Tributyl phosphate (TBP) is an important industrial extractant used in the Plutonium
Uranium Redox Extraction (PUREX) process for recovering uranium and plutonium from
used nuclear fuel. Distribution data have been assessed for a variety of fission and corro-
sion product trace metals at varying uranium concentrations under representative PUREX
extraction (3 M HNO ) and stripping (0.1 M HNO ) conditions. As might have been antic-
3 3
ipated, the extraction of most trace metals was found to decrease or remain constant with
increasing uranium concentration. In contrast, the extraction of some low valence transi-
tion metals was found to increase with increasing uranium concentration. The increase in
extraction of low valence transition metals may be related to TBP forming reverse micelles
instead of recovering uranium as a classical UO (NO ) (TBP) coordination complex. The
2 3 2 2
low valence transition metals may be being recovered into the cores of the reverse micelles.
Also unanticipated was the lack of impact the TBP degradation product, dibutyl phosphate
(DBP), had on the recovery of metals in batch distribution studies. This is possibly related
to the batch contacts completed in these experiments not adequately recreating the multi-
1Adapted with permission from Industrial & Engineering Chemistry Research 2016 55 (51), 13114-13119.
Copyright 2016 American Chemical Society.
2Primary author and researcher
3Co-author, Savannah River National Laboratory
4Corresponding author and advisor
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stage aspects of industrial-scale uranium extraction done using mixer settlers or centrifugal
contactors.
2.2 Introduction
The extractant tributyl phosphate (TBP) has been used for the industrial-scale recov-
ery of actinides in nearly all stages of the nuclear fuel cycle over the past 60 years[33]. In
that time, a substantial amount of TBP extraction data has been published under numerous
experimental conditions[34]. Extraction studies have examined the partitioning of various
metals and metal compositions[35–40], the impact of acid types and concentrations[41], and
diluents[42]. Data collected in support of industrial-scale processes used to separate uranium
from used nuclear fuel forrecyclinghavefocused on thecombined extraction behavior ofhigh
concentrations of uranium, low concentrations of plutonium and select problematic fission
products[17, 43–45]. Similarly, TBP extraction data collected in support of nuclear waste
remediation processes have focused on the recovery of a variety of fission and corrosion prod-
ucts in the presence of very low concentrations of uranium[46]. As a whole, these data have
been essential to the scale-up of solvent extraction processes involving TBP. However, the
solvent extraction behavior of many fission and corrosion products simultaneously recovered
in the presence of bulk uranium remains an area where data is lacking in the open literature.
The design and predictive modeling of industrial scale solvent extraction processes rely
on data collected under conditions closely approximating those found in the final process.
Industrial-scale solvent extraction processes using TBP for uranium recovery have been re-
fined to the extent that the recovery of poorly extracted, low-concentration metals in the
presence of bulk uranium may need to be assessed to further optimize these separations[47].
A predictive understanding of fission and corrosion products recovery by TBP in these pro-
cesses also has application in the field of nuclear forensics. Uranium is used in the manufac-
ture of nuclear weapons, and high-purity uranium products are often the result of processes
using TBP as an extractant. Trace metal fingerprints arising from the minimal partition-
ing of fission and corrosion products could be characteristic of the processes used in the
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production of any uranium-containing material[7]. Fully understanding these trace metal
partitioning patterns may help determine the history of recovered, illegally-trafficked ura-
nium. Ideally, these trace metal fingerprints would be determined through direct chemical
analysis of product samples. However, in the case where such samples are difficult to obtain
or nonexistent, predictive process modeling could be used to determine characteristic trace
metal compositions of uranium products from different industrial processing facilities.
Tributyl phosphate is used for the separation and purification of uranium as part of
the Plutonium Uranium Reduction Extraction (PUREX) process[12, 48–50]. In a typical
PUREX process, uranium and plutonium in used nuclear fuel are separated from fission
and corrosion product impurities in a series of continuous, countercurrent solvent extraction
stages. A high concentration of nitric acid is used both to dissolve the nuclear fuel and
as a salting-out agent to facilitate the extraction of uranium and plutonium nitrate by the
organic phase containing 30 v/v% TBP in a long-chain aliphatic hydrocarbon diluent. A
low concentration of nitric acid is used in the stripping and recovery of these purified metals
back into the aqueous phase following reduction of plutonium to the trivalent state. In
the course of several treatment and recycle steps, the TBP extractant is exposed to high
levels of radioactivity and a high concentration of acid. The exposure of TBP to these acid
and radiation conditions results in degradation products which are known to impact the
decontamination of the plutonium and uranium products[51–53]. The primary degradation
product is dibutyl phosphate (DBP), an organic-soluble cation exchange extractant. Limited
data also exists regarding the extraction chemistry of trace metals in the presence of DBP
under PUREX relevant conditions. This information is also crucial to accurate predictive
modeling of PUREX-type processes.
Inthiswork,theextractionbehaviorofselectedtracemetalsinthepresenceofbulkuranyl
nitrate under representative PUREX conditions were determined using multiple radiotracers
simultaneously in a series of batch extraction experiments. A PUREX feed simulant was
designedandproducedbasedonpreviouslypublishedcalculationsofnuclearfuelcomposition
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arising from a typical nuclear power reactor. Changes in the distribution ratios of the
metals in this simulant with varying organic phase uranium concentrations and aqueous
nitric acid concentrations were explored. Finally, data on the effect of the TBP degradation
product, DBP, on trace metal distribution ratios under extraction and stripping conditions
were collected.
2.3 Experimental
2.3.1 PUREX Feed Simulant Composition
The PUREX feed simulant is a combination of fission and corrosion product radiotracers,
and bulk uranyl nitrate. The calculated composition of pressurized water reactor (PWR)
Zircaloy-4 clad UO fuel at approximately 3% enrichment after 20 MWd/kgM burnup was
2
used as the basis for choosing representative fission products. This has been calculated
previously by Guenther et al. using the ORIGEN2 computer program[54]. Only elements
with a concentration greater than 10-5 g/g U were included in the simulant. Fission product
gases, such as xenon or krypton, were not included, nor were palladium and rhenium because
of their high cost and previously reported minor extraction by TBP[40]. The important
corrosion products chromium, manganese, iron, and nickel were identified from Izumida et
al. and are consistent with PUREX raffinate compositions given in Choppin et al.[55, 56].
These corrosion products were included in the simulant.
The final choice of fission and corrosion product radiotracers used in the PUREX feed
simulant took into account the availability of strong, unique gamma peaks after neutron
activation of each element, and their importance in the PUREX process. Table 2.1 shows
the final composition of the PUREX feed simulant used in these studies, the anticipated
trace metal and uranium concentrations in a typical PUREX feed, utilized radioisotopes,
half-lives, and unique gamma peak energies. The activities of the radiotracers used in the
PUREX simulant were kept approximately equal to maintain a low detector dead time while
also ensuring that each unique peak had sufficient counts. As a result, the concentrations
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of trace metals deviate in most cases from those found in a typical PUREX feed. These
differences would be expected to have minimal impact on the distribution ratios measured
due to their overall low concentrations. However, future work could benefit from using
real used nuclear fuel in the measurement of trace metal distribution ratios under PUREX
conditions.
2.3.2 Materials
All chemicals were used as received without further purification. All metal compounds
used were of analytical grade. Chromium, iron, rubidium, strontium, and cesium stock so-
lutions were prepared by dissolving their respective solid nitrates in 0.01 M HNO . The
3
stock solution containing molybdenum (VI) was prepared by dissolving solid ammonium
molybdate in 0.01 M HNO . The ruthenium (III) nitrosyl nitrate stock solution was pre-
3
pared by dissolving solid ruthenium nitrosyl nitrate in 0.01 M HNO . Lanthanum (III)
3
and samarium (III) nitrate stock solutions were prepared by dissolving the solid oxides in
concentrated nitric acid. A cerium (III) nitrate stock solution was prepared by dissolving
solid cerium (IV) oxide in nitric acid and reducing the metal to cerium (III) with hydrogen
peroxide. The stock solution of zirconium nitrate was made from hydrous zirconium oxide
as described in Chiarizia et al. using zirconium (IV) chloride as a starting material[27]. The
details of these stock solutions are summarized in Table 2.2. Stock solution of ACS grade
uranyl nitrate hexahydrate (International Bio-Analytical Industries, Inc., Boca Raton,FL)
at varying concentrations of HNO (3, 0.01, and 0.001 M) were used in the preparation of
3
aqueous solutions.
Organic phases were prepared using 99+% tributyl phosphate from Acros Organics, 97%
dibutylphosphatefromAcrosOrganics, and99+%n-dodecanefrom AlfaAesar, asspecified.
The TBP was washed with 0.1 M NaOH for solvent extraction experiments involving the
addition of dibutyl phosphate to the organic phase. The purpose of this wash was to remove
acidic impurities (including DBP) from the TBP before adding in the desired amount of
DBP. Aqueous phases were prepared using metal stock solutions, ACS grade nitric acid from
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Macron Fine Chemicals and degassed, ultrapure (18 MΩ-cm) water.
Table 2.2: Chemicals used in the preparation of the PUREX feed simulant and irradiation
times.
Element Starting Material Irradiated Form Stock Solution Irradiation Time
(M) (hr)
Cr Cr(NO ) 9H O Liquid 1 3
3 3 2
·
Fe Fe(NO ) 9H O Solid 2.4 8
3 3 2
·
Rb RbNO Liquid 2 3
3
Sr Sr(NO ) Solid 1.2 8
3 2
Zr ZrO nH O Solid 0.3 8
2 2
·
Mo (NH ) MoO Liquid 1 3
4 2 4
Ru RuNO(NO ) Liquid 1 3
3 3
Cs CsNO Liquid 0.2 3
3
La La O Liquid 0.01 3
2 3
Ce CeO Liquid 0.47 3
2
Sm Sm O Liquid 0.001 3
2 3
2.3.3 Radiotracer Production
All radiotracers were produced by neutron irradiation of solid or solution-phase samples
of individual metals at the United States Geological Survey TRIGA Reactor (GSTR) in
Denver, CO.Specificsabouttheneutronfluenceandirradiationpositionshavebeenprovided
previously[57]. Metals irradiated as solids were dissolved in 0.01 M HNO after irradiation to
3
yield stock solutions at the concentrations given in Table 2.2, except for zirconium which was
dissolvedin8MHNO . Formetalsirradiatedasliquids, stocksolutionsattheconcentrations
3
given in Table 2.2 were irradiated directly. Irradiation times are also given in Table 2.2.
2.3.4 Solvent Extraction Studies
Distribution ratios of the trace metals in the PUREX feed simulant under representa-
tive PUREX conditions were determined using batch solvent extraction studies at ambient
◦
temperature (22 1 C). All batch extractions were run using 30% TBP in n-dodecane pre-
±
viously equilibrated with an aqueous phase containing only nitric acid. Nitric acid, uranium,
and DBP concentrations were varied as shown in Table 2.3. The high nitric acid conditions
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simulate the extraction step of the PUREX process. The low nitric acid conditions simulate
the stripping step. Extraction in the PUREX process is usually run at approximately 70%
saturation of the organic phase[17]. This saturation level corresponds to an initial aqueous
uranium concentration of 0.42 M in these batch extraction experiments. The variation in the
uranium concentration is inclusive of the conditions under which PUREX takes place, but al-
lows for observation of the effect of different uranium concentrations. Similarly, the variation
in concentration of DBP used here is inclusive of PUREX conditions. The DBP concentra-
tion in a typical PUREX stream has been previously determined to be approximately 170
ppm[58].
Table 2.3: Experimental matrix showing conditions under which trace metal distribution
ratios were determined.
Unit Operation Initial HNO Initial U Additional DBP
3
(M) (M) (ppm)
Extraction 3 0 - 0.6 0
Stripping 0.1 0 - 1.0 0
Extraction 3 0.4 0 - 400
Stripping 0.1 0.4 0 - 400
Stripping 0.01 0.4 0 - 400
For each experimental condition, 5 mL volumes of the organic and aqueous phases were
used. Immediately before contact, 20 µL of a combined radiotracer solution containing the
fission and corrosion product elements given in Table 2.3 was added to the aqueous phase.
The organic and aqueous phases were contacted by vortex mixing for 15 minutes. The phases
were separated by centrifugation. Subsamples of each phase were collected and counted on a
well-type high-purity germanium detector (Canberra GCW2523, FWHM resolution is 1.25
keV at 122 keV). A typical 4 mL organic phase sample was counted for 20 to 24 hours. A
typical, 500 µL aqueous phase sample was diluted up to a total volume of 4 mL to give a
sample geometry consistent with the organic phase samples and counted for approximately
anhour. Equilibriumorganicuraniumconcentrationsweredeterminedbyusingacalibration
curve unique to the well-detector used.
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2.3.5 Determination of Distribution Ratios
Gamma spectra of each equilibrium solvent extraction phase were analyzed to give the
peak areas of each radioisotope’s unique gamma peak. The average dead time for aqueous
samples was 2.6%. The average dead time for organic samples was 1.0%. Only peaks with at
least 1,000 counts, which corresponds to a counting error of 3%, were used. The count rate
wasthencalculatedfromthecounttimeandpeakarea. Thecountratesassociatedwitheach
radioisotope were decay corrected to give their count rates at the time of phase separation
by centrifugation. These were then adjusted for the different sample volumes collected from
each phasetogivethespecific count ratesin each phase. Thesecount rateswerethen divided
to give the distribution ratio (D) of each trace metal, as shown in Equation 2.1:
C
org
D = (2.1)
C
aq
where C is the decay-corrected count rate in the organic phase, and C is the decay-
org aq
corrected count rate in the aqueous phase. The relative uncertainty in the distribution ratio
was calculated by adding the relative standard deviations of the counts measured for each
organic and aqueous phase sample.
2.4 Results and Discussion
2.4.1 Fission and Corrosion Product Chemistry in the PUREX Process
The chemical states of all ions in the PUREX feed simulation solution were chosen to
simulate those found in real PUREX processes. The PUREX feed simulant fission products
with the most complex solution chemistry are ruthenium and zirconium. The solution chem-
istry of these metals in nitric acid have been studied extensively because of their established
roles as problematic impurities in the PUREX process[59]. Ruthenium is known to exist
in dissolved nuclear fuel primarily as various nitro- and nitrato- ruthenium (III) nitrosyl
complexes whose speciation is time- and concentration-dependent. Each species is extracted
substantially differently by TBP[60]. Likewise, the speciation of zirconium (IV) in dissolved
nuclear fuel is dependent on the history of the solution. Zirconium is known to polymerize
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in nitric acid solution particularly under conditions of low acid or high metal concentration.
Zirconium monomers are relatively well-extracted by TBP, while polymers are effectively
inextractable[53]. Rather than explore the effects of all the different potential ruthenium
and zirconium species on the distribution of these metals, these experiments were designed
to improve the reproducibility of ruthenium and zirconium extraction by timing dilution of
the ruthenium and zirconium stocks identically between batch extraction experiments. The
radiotracer spikes were added immediately (<10 minutes) before contacting the organic and
aqueous phases.
All other metals have stable solution chemistry in nitric acid solution. Molybdenum (VI)
exists as the molybdate ion in dissolved reactor fuel. The lanthanides exist as mononuclear
species in the trivalent state. While some cerium (IV) may form on dissolution of nuclear
fuel, cerium is found as cerium (III) in PUREX process streams. The transition metals
— chromium (III), iron (III), rubidium (I), strontium (II), and cesium (I) — are found as
mononuclear species in their respective dominant oxidation states[59].
2.4.2 Distribution under Extraction Conditions
The extraction of metals by TBP is commonly understood to proceed through the solva-
tion of a neutral metal salt according to the equilibrium relationships given in Equations 2.2
to 2.4[61]. The negative dipole on the phosphoryl oxygen of TBP interacts with the pos-
itively charged metal cation, resulting in the formation of a metal solvate. The nonpolar
butyl tails of TBP keep the resulting metal solvate dissolved in the aliphatic hydrocarbon
diluent.
M3+ +3NO − +3TBP ↽⇀ M(NO ) TBP (2.2)
(aq) 3 (aq) (org) −−−− 3 3 3(org)
M4+ +4NO − +2TBP ↽⇀ M(NO ) TBP (2.3)
(aq) 3 (aq) (org) −−−− 3 4 2(org)
MO 2+ +2NO − +2TBP ↽⇀ MO (NO ) TBP (2.4)
2 (aq) 3 (aq) (org) −−−− 2 3 2 2(org)
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The distribution of trace fission and corrosion products under representative PUREX
extraction conditions (3 M HNO in the aqueous phase) with increasing uranium concentra-
3
tion in the organic phase is given in Figure 2.1. The extraction of ruthenium (III) nitrosyl,
zirconium (IV), the lanthanides (III), and the molybdate anion were found to decrease with
increasing uranium concentration. This would be expected given the high affinity of TBP for
uranium (VI)and thelimitingconcentrationsofTBPrelativetouranium and otherfission or
corrosion products. As the amount of uranium increases in the solvent extraction system, it
displaces other metals from TBP extracted complexes. In contrast, the extraction of mono-,
di-, and trivalent transition metals appears to increase with increasing uranium concentra-
tion, suggesting a different mechanism for their extraction. If the extraction of these metals
were dependent on the same interaction between the phosphoryl oxygen of TBP and the
metal cation, a decrease in their extraction with increasing uranium concentration would
be observed. The observed minimum in the chromium distribution ratio curve could be the
result of both TBP extraction mechanisms. The decrease in distribution ratio could result
from competition for TBP between uranium and chromium at low uranium concentrations,
followed by an increase in the distribution ratio as chromium starts to be extracted by the
same mechanism as the rest of the low valence transition metals.
Given the extremely low concentrations of the mono-, di-, and trivalent transition metals
in the organic phases of these solvent extraction systems, the possibility exists that metal
recovery is occurring due to physical entrainment of the metal in the organic phase. The low
valence transition metals are at micromolar concentrations in the organic phases of these
◦
solvent extraction systems. For comparison, at 25 C, the solubility of water in pure n-
dodecane is 65 mg/kg solvent, which corresponds to a concentration of around 3 mM[62].
The extracted metals may not be associated with the TBP extractant, but could instead be
independently associated with either the n-dodecane diluent or the free water in the organic
phase. However, this does not explain the apparent dependence of the extraction of these
metals on the uranium concentration.
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Alternatively, these low valence transition metals may be being co-extracted with ura-
nium in the core of the reversed micellar structures formed in solution by TBP. Tributyl
phosphate is a surface-active amphiphile with a polar head and nonpolar tails that are nec-
essary for metal extraction. Such molecules are known to self-assemble into complex struc-
tures in solution, including reversed micelles[31]. Tributyl phosphate has been noted to form
reversed micellar structures in the presence of high concentrations of polar solutes, where
the polar solutes are located in the core of the micelle[24, 25, 27]. The low valence transition
metals may have sufficiently small hydrated radii, where the bulk of the hydrated radii are
made up of metal ion and not long range ordered water, that they are being co-extracted as
part of uranium-containing TBP reversed micelles[63]. Such a mechanism would explain the
observed increase in extraction of these metals with extracted uranium.
2.4.3 Distribution under Stripping Conditions
The distribution of trace fission and corrosion products under representative PUREX
stripping conditions (0.1 M HNO in the aqueous phase) with increasing uranium concentra-
3
tion in the organic phase is given in Figure Figure 2.2. The behavior of these metals under
stripping conditions is consistent with expectations. Measured distribution ratios remained
low for all metals and either stayed substantially the same or decreased with increasing
uranium concentration in the organic phase. Ruthenium was the only metal with apprecia-
ble distribution ratios at a low nitric acid concentration. This behavior has been observed
previously and attributed to reduced competition for free TBP between ruthenium nitrosyl
nitro- and nitrato- complexes and nitric acid at low nitric acid concentrations. These data
indicate that if most of these trace metals were in the organic phase following the extraction
step of the PUREX process, they would be quantitatively stripped into the aqueous phase
along with the uranium product. The lone exception is ruthenium, which would be partially
retained in the solvent, possibly contributing to its identity as a problematic trace metal
impurity in the PUREX process.
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processes and modeling characteristic trace metal signatures for nuclear forensics applica-
tions. The PUREX process for recovering uranium and plutonium from used nuclear fuel
is one of the most important industrial-scale processes using TBP as an extractant. In
this work, batch distribution studies have been completed under representative PUREX
extraction and stripping conditions to find the distribution ratios of select fission and cor-
rosion product trace metals in the presence of uranium. Metal extraction decreases for
most of these metals with increasing uranium concentration, as would be expected given the
commonly-understood mechanism of TBP extraction. However, under extraction conditions,
the distribution ratios of some low valence transition metals were found to increase with in-
creasing uranium concentration. This anomalous behavior must be taken into account when
modeling trace metal extraction in the PUREX process.
Batch distribution studies of TBP systems with added DBP were unable to demonstrate
the impact of DBP on the extraction of trace metals. This was unexpected because previous
studies on the impact of DBP on metal extraction by TBP in the absence of uranium
indicated that TBP and DBP extract some metals better than TBP alone[52, 64]. The
observedlackofimpactofsmallamountsofDBPontracemetalpartitioningislikelybecause
a single batch solvent extraction experiment cannot capture the effects of multistage solvent
extraction cycles on extraction behavior. This would be an appropriate area for further
consideration, as well as other experiments assessing the possibility of entrainment versus
reverse micellar recovery of low valence transition metals.
2.6 Acknowledgements
This invited contribution is part of the I&EC Research special issue for the 2017 Class
of Influential Researchers.
This material is based upon work supported by the U.S. Department of Homeland Se-
curity under Grant Award Number, 2012-DN-130-NF0001. The views and conclusions con-
tainedinthisdocumentarethoseoftheauthorsandshouldnotbeinterpretedasrepresenting
theofficialpolicies,eitherexpressedorimplied,oftheU.S.DepartmentofHomelandSecurity.
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CHAPTER 3
THE IMPACT OF OUTER-SPHERE COORDINATION ON THE LANTHANIDE
SELECTIVITY OF DIGLYCOLAMIDE EXTRACTANTS
3.1 Abstract
The extraction of adjacent lanthanides by the solvating extractant tetraoctyl diglyco-
lamide (TODGA) has been explored in distribution studies using a 0.25 M TODGA in n-
dodecane organic phase and aqueous phases with macroscopic amounts of lanthanides from
across the series in nitrate media. The limiting organic concentrations (LOCs) of La, Pr,
Sm, Gd, Dy, and Tm form an “S”-shaped curve when plotted against atomic number. The
distribution ratios for La, Pr, Sm, and Gd increase with decreasing ionic radius, and decrease
with increasing metal concentration, consistent with established extraction behavior. The
amount of water co-extracted with each lanthanide was found to increase with decreasing
ionic radius, from La to Gd, after which it levels off. This trend is similar to the trend
observed for the extraction of the lanthanides by TODGA across the series, and suggests a
relationship between co-extracted water and lanthanide extraction.
3.2 Introduction
The trivalent f-block elements are difficult to separate from one another due to their
similar charge densities and near-identical chemistries. Amide extractants were originally
considered for use in the separation of f-block elements after being described by Siddall in
1960[65]. Neutral diglycolamide extractants were first explored as alternatives to bidentate
malonamide ligands in the early 2000s[13]. Both classes of extractant are fully incinera-
ble, making them attractive for applications in the management of high level radioactive
waste. However, the additional ether oxygen in tridentate diglycolamide extractants greatly
increases their affinity for trivalent lanthanides and actinides over bidentate malonamide
ligands. While the extraction capabilities of numerous different diglycolamides have been
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explored in the literature, most work has focused on N,N,N’,N’-tetraoctyl diglycolamide
(TODGA), shown in Figure 3.1, due to its low solubility in water, high solubility in organic
diluents, and high affinity for trivalent f-block elements[66]. Work with these extractants
has focused on the utility of diglycolamides in the separation of trivalent actinides from lan-
thanides. However, the high affinity of diglycolamide extractants for trivalent lanthanides
suggests that they may also have applications in the separation of adjacent lanthanides.
Figure 3.1: Molecular structure of the solvating extractant N,N,N’,N’-tetraoctyl diglyco-
lamide (TODGA).
Traditional approaches to the solution-phase separation of adjacent lanthanides have
focused on the effect of the contraction of trivalent lanthanides’ ionic radii on their inner-
sphere coordination environment[67]. The trivalent lanthanides are hard acids with very
similar chemical and physical properties[63], making this small decrease in size across the se-
ries, shown in Figure 3.2, the primary difference that can be leveraged to accomplish mutual
separations in lanthanide mixtures. The primary effect of this contraction is the increased
strength of ion-dipole interactions between the lanthanides and extractant molecules with
increasing atomic number. In a solvent extraction system, this corresponds to slightly better
extraction of the heavier lanthanides. However, this simplistic understanding of the extrac-
tion chemistry of the lanthanides fails to include the possible effects of interactions occurring
beyond the first coordination sphere. Determining the nature of these interactions, which are
most effectively probed at high metal concentrations, may be essential to designing efficient
lanthanide separation processes.
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Figure3.2: Theionicradiiofthelanthnidesdecreasesteadilyacrosstheseries[63]. Extraction
of the lanthanides by TODGA increases from the light to mid-lanthanides, after which it
levels off[66].
The affinity of TODGA for the trivalent lanthanides does not follow a linearly increasing
trend across the lanthanide series as would be expected if the strength of the Coulombic
interactions between the lanthanide and extractant were the only property impacting ex-
traction. Instead, distribution data show a steady increase in extraction across the light
lanthanides and a break in this trend at Gd beyond which metal distribution is effectively
constant (Figure 3.2)[66]. In a previously reported study, this trend was attributed to a sim-
ilar but much less pronounced break in the energy difference between an aqueous-solvated
lanthanide ion and a diglycolamide-solvated lanthanide ion, measured relative the energy
of lanthanum solvation[68]. This difference is referred to as the aqueous-phase selectivity,
and takes into account only the energetics of inner sphere coordination. However, the weak
agreement between the experimental distribution data and aqueous-phase selectivities sug-
gests that there are additional factors affecting the observed selectivity trend that have not
yet been identified.
Priorworklookingatthestructureofdiglycolamide-lanthanidecomplexeshasdetermined
that the extractant amide and ether groups are directly coordinated to the central metal
cation in a tridentate fashion, while charge neutralizing anions are located in the outer-
sphere of the complex. This structure has been determined in the solid state for lanthanide
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and actinide nitrates[69, 70], and verified for solution-phase species in solvent extraction
samples[68, 71]. The calculation of aqueous-phase selectivities does not take these outer-
sphere anions into account. However, the behavior of outer-sphere species may provide an
additional basis for the observed selectivity.
Thepurposeofthisworkistoexpandunderstandingofthetrendinlanthanideextraction
across the series by the neutral diglycolamide extractant, TODGA, at high metal concentra-
tions. The extraction behavior of TODGA was investigated by measuring the equilibrium
organic and aqueous phase compositions of solvent extraction systems containing macro-
scopic amounts of La, Pr, Sm, Gd, Dy, and Tm. The maximum concentration of each
lanthanide that could be extracted into the organic phase before it splits into two phases
was also determined for these metals. This concentration is referred to as the limiting or-
ganic concentration (LOC) of the metal. Solvent extraction studies performed at high metal
concentrations could give further insight into the nature of TODGA extracted complexes
and trends in lanthanide selectivity.
3.3 Experimental
Stock solutions of approximately 0.5 M La(NO ) , Pr(NO ) , Sm(NO ) , Gd(NO ) ,
3 3 3 3 3 3 3 3
Dy(NO ) , and Tm(NO ) were prepared by dissolving 99.9% pure solid lanthanide nitrate
3 3 3 3
hydrates from Sigma Aldrich in 0.001 M HNO , to prevent hydrolysis. A stock solution of
3
3 M NaNO was prepared by dissolving ACS grade NaNO in water. All solutions were
3 3
standardized by inductively coupled plasma optical emission spectroscopy (ICP-OES) prior
touse. TODGAwas used asreceived from Marshallton Research Laboratories, and dissolved
in 99+% n-dodecane from Alfa Aesar to make a 0.25 M stock solution.
3.3.1 Solvent Extraction Studies
The 0.25 M TODGA solution was pre-equilibrated with water prior to all distribution
experiments. Aqueous solutions at the desired lanthanide concentration were prepared from
stock solutions, such that the equilibrium aqueous phase concentration of NO – under each
3
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experimental condition remained constant at 0.5 M NO –. For each extraction sample, equal
3
volumes of an aqueous and organic phase were combined in a plastic centrifuge tube and
◦
contacted on a vertical rotating wheel in a thermostated incubator maintained at 21 C.
Samples were allowed to rotate until equilibrium was reached, at least one hour. Samples
were centrifuged for five minutes at 3000 rpm to separate the phases, and aliquots taken
from each phase for analysis.
Metal concentrations in both phases were determined using ICP-OES. Aqueous phase
samples were prepared for analysis by adding a small sample of the separated aqueous phase
aliquot directly to a 4% HNO solution. Organic phase samples were prepared by strip-
3
ping the metal from the organic phase aliquot three times with equal volumes of 0.01 M
HNO . These solutions were combined and the HNO concentration adjusted to 4% HNO
3 3 3
for analysis by ICP-OES.
Organic phase water concentrations were determined using a Metrohm 831 Karl-Fischer
coulometer.
All metal and water analyses were performed in duplicate. The estimated uncertainty in
the measured metal concentrations is 2%. The estimated uncertainty in the measured water
concentrations is 2%.
3.3.2 Maximum Organic Phase Lanthanide Concentration
To determine the maximum amount of each lanthanide that could be dissolved in the
organic phase before the appearance of a third phase, solvent extraction studies were per-
formed as described in the previous section. A range of initial aqueous phase lanthanide
concentrations was used such that a third phase was observed for at least one of the con-
centrations. For La, the third phase appeared as a thin liquid phase located above the
aqueous phase and below the larger organic phase. For the other lanthanides, the third
phase appeared as a pale white gel or waxy solid in the upper organic phase. The initial
aqueous phase concentrations were refined until the organic phase limit was determined to
within an initial aqueous phase concentration of 2 mM. Once the conditions under which the
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LOC was reached were determined, ICP-OES was used to find the organic phase lanthanide
concentrations.
3.4 Results and Discussion
The investigated solvent extraction systems consisted of an organic phase with 0.25 M
TODGA dissolved in an n-dodecane diluent contacted with a 0.001 M HNO aqueous phase
3
containing macroscopic amounts of NaNO and lanthanides from across the series. The
3
organic phase TODGA concentration was chosen to be high enough that it could reasonably
be considered for use in an industrial-scale operation, where a high saturation of the organic
phasebymetalisdesired. Theaqueousphasecompositionwaschosentominimizetheimpact
of competitive extraction on the system. A salting out agent in the aqueous phase is required
toobtainapproximately50%extractionofthelightlanthanidesbya0.25MTODGAorganic
phase. Nitric acid, a salting out agent used in applied separations, competes with metals for
extractant, adding an additional experimental parameter that must be considered. In these
experiments, a background electrolyte consisting of approximately 0.5 M NaNO was used
3
to improve the extraction of the lanthanides without Na+ being extracted. A small amount
of nitric acid was added to the aqueous phase to prevent hydrolysis of the metals, for a final
acid concentration of 0.001 M HNO .
3
3.4.1 Limiting Organic Concentrations
According to solvent extraction[20] and EXAFS studies[68], trivalent lanthanides are
likely extracted as a 1:3 metal to TODGA species according to the following equilibrium
relationship:
M3+ +3NO − +3TODGA ↽⇀ M(NO ) TODGA (3.1)
(aq) 3 (aq) (org) −−−− 3 3 3(org)
This equation suggests a theoretical value for the maximum concentration of metal that can
be extracted into a 0.25 M TODGA organic phase. A fully saturated organic phase would
havea0.083Mmetalconcentration, correspondingtoallTODGAmoleculesbeingapartofa
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trimeric extracted metal species. However, like other neutral solvating extractants, TODGA
isknowntoformathirdphasebeforethissaturationconcentrationcanbereached[72]. Third
phase formation in a solvent extraction system occurs when the organic phase splits into an
upper, diluent-rich organic phase, and a heavy, extractant- and metal-rich organic phase in
equilibrium with the aqueous phase. Third phase formation is correlated with high metal
concentrations in the organic phase.
The limiting organic concentration (LOC) is defined as the greatest concentration of
metal that can be extracted into an organic phase before a third phase is formed. The LOCs
of six metals from across the light, middle, and heavy lanthanides (La, Pr, Sm, Gd, Dy, and
Tm) were found by incrementally changing the initial aqueous phase concentrations of each
metal. The result is an “S”-shaped curve, shown in Figure 3.3, in which the LOC starts
at a high concentration and decreases slowly across the light lanthanides, rapidly across the
early-middle lanthanides, then levels off beginning at the late-middle lanthanides. The point
of inflection of this curve roughly corresponds to Nd, and the complete leveling-off of the
LOCs takes place at Gd. The leveling off of the LOC takes place at the same point in the
seriesatwhichmetalextractionisalsoobservedtoleveloff. However, the“S”-shapedoesnot
correspondtootherimportantreportedphysicalorchemicalattributesofthelanthanideions
in solution[63], suggesting that this trend is not the result of a direct interaction between the
metal ion and extractant. Rather, it may be the result of outer-sphere coordination effects.
La comes the closest to the theoretical limit of saturation of the organic phase before
forming a third phase at an organic concentration of 0.077 M, followed by Pr at 0.070 M.
The remaining lanthanides reach the LOC at concentrations much less than the theoretical
saturation point of the organic phase. The closeness of the LOC to the saturation concentra-
tion of the organic phase validates the formation of the 1:3 metal to TODGA species for La
and Pr, and suggests that the 1:4 metal to TODGA species is not favored in these systems.
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Figure 3.3: Lanthanide LOCs in a TODGA solvent extraction system form an “S”-shaped
curve when plotted against atomic number.
3.4.2 Distribution Data
The distribution ratio, D, is a measure of metal extraction, and is calculated as shown
in Equation 3.2, where the organic phase concentration of a metal, [M] is divided by the
org
aqueous phase concentration, [M] .
aq
[M]
org
D = (3.2)
[M]
aq
In the absence of confounding factors like competitive extraction, distribution ratios are
expected to decrease with increasing metal concentration as saturation of the organic phase
is approached. This trend is observed in the distribution ratio values shown in Figure 3.4.
The distribution ratios of La, Pr, Sm, and Gd decrease exponentially with increasing metal
concentration in the system until the LOC is reached. The heavy lanthanides Dy and Tm
are sufficiently well-extracted that their aqueous phase concentrations were undetectable by
ICP-OES, preventing the calculation of distribution ratios for these metals. Because of the
high recovery of the heavy lanthanides, the leveling off of the distribution ratio from Gd
onward that has been observed in dilute extraction experiments could not be verified for
concentrated systems. However, the distribution data collected for La, Pr, Sm, and Gd
under concentrated conditions follow the same trend as that observed in dilute systems. For
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a given equilibrium organic phase concentration of metal, the distribution ratio increases
with increasing atomic number.
Figure 3.4: The distribution ratios of the lanthanides are inversely proportional to concen-
tration and ionic radii.
3.4.3 Trends in Water Co-extraction
The amount of water co-extracted with each lanthanide is shown in Figure 3.5. The
point at zero lanthanide concentration in the organic phase corresponds to the equilibrium
composition of a 0.25 M TODGA phase contacted with a 0.5 M NaNO and 0.001 M HNO
3 3
aqeuous phase. As expected, the amount of Na extracted into the organic phase was unde-
tectable by ICP-OES. The water concentration in the equilibrium organic phase increases
with increasing metal concentration at different rates depending on the identity of the co-
extracted metal. The equilibrium organic phase water concentration increases at the slowest
rate with increasing La concentration, followed by Pr, Sm, and Gd. At Gd the amount of
co-extracted water stops increasing, such that the increase in water concentration with metal
concentration is the same for Gd, Dy, and Tm.
For a constant organic phase lanthanide concentration, the observed trend in water co-
extraction follows the distribution ratio trend across the lanthanides. The number of water
molecules co-extracted with each lanthanide ion in the organic phase increases steadily to
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Gd, beyond which it remains approximately constant. Figure 3.6 shows the increase in the
organic phase water concentration for a constant lanthanide concentration of 0.014 M. These
values were determined from linear interpolation of the data shown in Figure 3.5. This trend
compares favorably with the trend in distribution ratios across the lanthanide series, also
shown in Figure 3.6. Such close agreement between these data suggests a relationship be-
tween the water co-extracted with a metal and how well that metal is extracted by TODGA.
However, the exact nature of this relationship is unclear.
Figure 3.6: The expected concentration of water in an equilibrium organic phase with a
0.014 M lanthanide concentration increases for the light lanthanides and levels off for the
heavy lanthanides. This trend in water co-extraction appears to closely follow the trend in
lanthanide extraction by TODGA.
Water is not a part of the inner coordination sphere of TODGA-extracted trivalent lan-
thanides. A time resolved laser-induced fluorescence spectroscopy (TRLFS) study on the
complexationofEubyTODGAdemonstratedthatwaterisnotcoordinatedtothemetal[73].
This finding was validated by solution-phase EXAFS studies of TODGA-extracted Pr, Nd,
Eu, Yb, and Lu, which showed the inner coordination sphere of these ions saturated by the
ligand[68]. These findings, as well as the low concentration of water in the organic phase
without metals, means that this water is likely being co-extracted in the outer coordination
sphere of the lanthanides, suggesting a link between the outer-sphere coordination chemistry
of the lanthanides and their extraction by TODGA. Such outer-sphere effects have been con-
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sidered as an explanation for the extraction chemistry of a malonamide extractant, although
the explicit molecular origins of these effects were not identified[74].
InthecaseofTODGA,theobservedtrendinextractionacrossthelanthanideseriescould
be explained by a decrease in the organic phase solubility of the extracted complexes beyond
Gdduetotheirhigherwatercontent. Tobesolubleintheorganicphase, TODGAcomplexes
must be net neutral species. However, the inner coordination sphere of an extracted trivalent
lanthanide is fully saturated by TODGA molecules, meaning that charge neutralizing nitrate
anions must be electrostatically associated in the outer coordination sphere. These nitrate
ions could be interacting strongly enough with water molecules to pull them into the organic
phase with the extracted lanthanides. As the radii of the lanthanides contract across the
series, the hydrocarbon chains of the coordinated TODGA molecules could become more
crowded, pushing the nitrate anions farther away from the metal center and beyond the
steric protection of TODGA’s nonpolar tails. This could increase the polar surface of the
nitrate available to interact with water, and, as a result, the amount of water associated with
the extracted complex. Any increase in the extraction of lanthanides by TODGA beyond
Gd, which would be expected if extraction was only impacted by inner sphere coordination
effects, could potentially be offset by the decrease in organic phase solubility of species with
more associated water, leading to the observed trend in extraction.
3.5 Conclusions
The trivalent lanthanides are difficult to separate because of similarities in their physical
and chemical properties. Most mutual separation processes take advantage of the small de-
crease in ionic radius that occurs across the lanthanide series. In solvent extraction systems,
this decrease would be expected to result in steadily increasing extraction across the series.
However, this trend is not observed with TODGA, for which lanthanide extraction has been
observed to increase across the light to middle lanthanides, then remain constant across the
heavy lanthanides. In this work, distribution studies in TODGA extraction systems with
high lanthanide concentrations were used to explore the basis for this observed extraction
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CHAPTER 4
TRIBUTYL PHOSPHATE AGGREGATION IN THE PRESENCE OF METALS: AN
ASSESSMENT USING DIFFUSION NMR SPECTROSCOPY
Modified from a paper published in The Journal of Physical Chemistry B1
Anna G. Baldwin2, Yuan Yang3, Nicholas J. Bridges4, Jenifer C. Braley5
4.1 Abstract
Diffusion nuclear magnetic resonance (NMR) spectroscopy was used to find the inter-
aggregate interactions and sizes of tributyl phosphate (TBP) aggregates containing varying
concentrations of uranium or zirconium and HNO in an n-dodecane diluent. The average
3
diffusion coefficients of TBP species were measured using a pulsed-field gradient stimulated
echo experiment with a longitudinal eddy-current delay (STE-LED). Inter-aggregate inter-
actions were determined by measuring the diffusion coefficient of TBP in a sample after a
series of dilutions with n-dodecane. The interaction-independent infinite dilution diffusion
coefficient was also calculated from these measurements. The sizes of TBP aggregates were
calculated from the infinite dilution diffusion coefficient using the Wilke-Chang equation. In-
teractions between TBP aggregates were observed to correspond to a hard sphere potential
with a repulsive component. Aggregate sizes found by NMR were comparable to literature
values found using small angle neutron scattering. The diffusion of TBP in heavy organic
third phases indicates that the third phase may be a bicontinuous structure like that found
in traditional surfactant systems.
1AdaptedwithpermissionfromTheJournalofPhysicalChemistryB 2016120 (47),12184-12192. Copyright
2016 American Chemical Society.
2Primary author and researcher
3Co-author, NMR research scientist
4Co-author, Savannah River National Laboratory
5Corresponding author and advisor
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4.2 Introduction
Solvent extraction is an important chemical separation process with applications in many
different industries. Solvent extraction separates components by leveraging differences in
their solubilities between two immiscible liquid phases, usually a less-dense organic and
a more-dense aqueous phase. For solvent extraction processes targeting metals, a metal
extractant in the organic phase must be hydrophobic enough to be insoluble in the aqueous
phase yet polar enough to interact with the metal of interest. As a result, extractants used
in the separation and purification of metals are usually amphiphiles consisting of a polar
headgroup attached to nonpolar hydrocarbon tails[75]. The polar headgroup interacts with
the extracted ionic species while the nonpolar hydrocarbon tails prevent the extracted metal
species from being soluble in the aqueous phase.
One of the most industrially-relevant metal extractants is tributyl phosphate (TBP), a
neutral solvating extractant used to separate plutonium and uranium from used nuclear fuel
in the Plutonium Uranium Redox EXtraction (PUREX) process[12, 48–50]. The structure
of TBP is shown in Figure 4.1. TBP has a density close to that of water and is usually dis-
solved in kerosene, an inert hydrocarbon diluent, to facilitate disengagement of the organic
phase[76]. Under conditions of high acid and metal loading, TBP and other neutral ex-
tractants are prone to the formation of liquid third phases, a problematic phenomenon not
usually observed in extraction systems containing acidic extractants[77, 78]. Third phase
formation interferes with solute recovery, and refers to the appearance of a second, heavy or-
ganic phase consisting primarily of extracted species in equilibrium with a light, diluent-rich
organic phase and the aqueous phase.
Historically, the extraction mechanisms of metals in solvent extraction systems have been
investigatedthroughbatchdistributionstudiesatlowsoluteandextractantconcentrations[16,
18, 61]. By analogy with traditional coordination compounds, extracted species were as-
sumed to exist in solution as a limited number of discrete stoichiometric solvates[19]. The
compositions of these solvates were determined by relating the metal concentration in each
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Figure 4.1: Molecular structure of tributyl phosphate (TBP).
liquid phase to experimental variables based on equilibrium extraction relationships and the
law of mass action. These studies were run under dilute conditions to eliminate the effects of
the activities of extracted species on the thermodynamic constants governing the extraction
equilibria. The small, discrete species found under dilute conditions have been assumed to
exist under the highly concentrated conditions in industrial-scale processes. Deviations from
ideal behavior in these concentrated systems were then attributed to non-ideality rather
than the formation of different extracted species[21]. However, this understanding of the
organic phase does not explain third phase formation, leading to the need for an improved
understanding of the molecular-scale forces at work in solvent extraction systems.
Interest in understanding the fundamental chemistry of solvent extraction, particularly
under applied separation conditions, has encouraged the application of the characteriza-
tion tools of colloid chemistry to determine the structure of the organic phase. In 1991,
Osseo-Asare suggested that solvating metal extractants, which are usually surface-active
amphiphiles, might form reversed micellar aggregates like those found in more traditional
surfactant systems at high concentrations of acid and metals[31]. A simplified representation
of such a proposed structure is shown in Figure 4.2. This important parallel between col-
loid chemistry and solvent extraction provided the impetus for small angle X-ray scattering
(SAXS) and small angle neutron scattering (SANS) studies of the morphology of organic
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phase species in solvating extractant systems[24–27, 30, 32, 79, 80]. These studies indicate
the formation of extractant aggregates containing more extractant molecules than had been
found as part of the coordination compounds whose stoichiometries were established by
dilute batch distribution studies.
Figure4.2: Asimplifiedrepresentationoftheproposedstructureofmetal-containingreversed
micellar TBP species.
Neutron and X-ray scattering are techniques used to probe the structure of nanoscale
particles in solution. In these techniques, a beam of either X-ray or neutron radiation is
passed through a sample and the resulting interference pattern is collected. These scattering
data are then either directly fit by an appropriate theoretical model or indirectly analyzed by
computing the Fourier transform of the scattering data and reversing correlation averaging
to yield particle size, shape, and interaction information[81, 82]. Recent work has used
scattering patterns derived from molecular dynamics simulations of sample systems in the
interpretation of experimental small angle scattering data[83–85]. The primary difference
between SAXS and SANS is in the nature of the interactions of X-ray and neutron radiation
with the sample. X-rays are scattered by areas of electron density, while neutrons interact
with atomic nuclei. However, mathematically identical principles of radiation scattering are
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