Robust Passivity-Based Control of Boost Converters in DC Microgrids
Michele Cucuzzella Riccardo Lazzari Yu Kawano Krishna C. Kosaraju Jacquelien M. A. Scherpen

All-Against-One Linear-Quadratic
 Differential Games
    
	The authors are with the Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL. Emails: shahriar@ece.ucf.edu, simaan@ucf.edu, qu@ucf.edu.
	This work was supported in part by the US National Science Foundation under grants ECCS-1308928.
Shahriar Talebi, Student Member, IEEE, Marwan A. Simaan, Life Fellow, IEEE,  Zhihua Qu, Fellow, IEEE,


    
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This work deals with the design of a robust and decentralized passivity-based control scheme for regulating the voltage of a DC microgrid through boost converters. 
A Krasovskii-type storage function is proposed and a (local) passivity property for DC microgrids comprising unknown ZIP' (constant impedance Z', constant current I' and constant power P') loads is established.
More precisely, the input port-variable of the corresponding passive map is equal to the first-time derivative of the control input. Then, the integrated input port-variable is used to shape the closed loop storage function such that it has a minimum at the desired equilibrium point. 
Convergence to the desired equilibrium is theoretically analyzed and the proposed control scheme is validated through experiments on a real DC microgrid.








§ INTRODUCTION
 
Distributed Generation (DG) and the necessity of storing energy require fundamental transformations of the conventional power generation, transmission and distribution systems <cit.>.
DG represents a conceptual solution to i) enhance the integration of Renewable Energy Sources (RES) in order to reduce the dependency on fossil fuels and CO_2 emissions, ii) increase the energy efficiency by reducing the transmission power losses, iii)  improve the service quality by enabling the operation of portions of the network  disconnected from the main grid and iv) minimize the costs for electrifying remote areas or re-powering the existing power networks due to the ever increasing electric demand. 
A set of multiple DG Units (DGUs), loads and energy storage devices interconnected through power lines is identified in the literature as a microgrid <cit.>. 

In the last decades, due to the prevalence of Alternating Current (AC) networks, the literature on microgrids mainly considered AC systems (see for instance <cit.> and the references therein).
However, the recent widespread use of RES as DGUs is motivating the design and operation of Direct Current (DC) microgrids <cit.>. 
Several devices (e.g. electric vehicles, electronic appliances, batteries and photovoltaic panels) can indeed be directly connected to a DC network avoiding lossy DC-AC conversion stages and the issues related to the frequency and reactive power control <cit.>. Besides the development of industrial, commercial and residential DC distribution networks, some examples of existing or promising DC microgrid applications are ships, mobile military bases, trains, aircrafts and charging stations for electric vehicles.
For all these reasons, control of DC microgrids and, consequently, DC-DC power converters is gaining growing interest.

In DC microgrids, control schemes are usually designed to achieve voltage stabilization and current (or power) sharing (see for instance <cit.> and the references therein). 
However, the dynamics of the power converters are often neglected or described by linear models (e.g., buck converters).
Differently, in this letter we design a robust and decentralized passivity-based control scheme for regulating the voltage of a DC microgrid through boost converters, the dynamics of which are nonlinear. 
Regulating the voltage towards the nominal value is required to ensure a proper operation of the connected loads and guarantee the network stability.



 §.§ Literature Review and Main Contributions

We provide now a brief comparison with some existing theoretical results dealing with the design of voltage controllers for boost converters. Simple tuning rules of passivity-preserving controllers are provided in <cit.>, while stability in presence of bounded input is analyzed in <cit.>.
However, only constant impedance loads are considered, the network dynamics are neglected and in <cit.> the load resistance is assumed to be known. 
In <cit.> and <cit.>, Plug-and-Play voltage controllers are proposed. 
More precisely, the controller designed in <cit.> is robust with respect to load uncertainties. However, the line dynamics are neglected and only local stability is established for the boost converter. In <cit.> the microgrid stability is proved considering bounded input. However, only constant current loads are considered and the controller requires local information (including the load) and the value of the resistance of the lines interconnecting with the neighboring nodes. Under the assumption that the equilibrium point is known, a novel nonlinear control law that takes into account the constraints of the control action is proposed in <cit.>. 

We can now list the main contributions of this work:


  §.§.§ Nonlinear model
 The considered microgrid model takes into account the nonlinear dynamics of boost converters and a possible meshed network topology, incorporating dynamic resistive-inductive lines and a general nonlinear load model (called ZIP') including constant impedance Z', constant current I' and constant power P'.


  §.§.§ Passivity framework
 A Krasovskii-type storage function <cit.> is proposed and a (local) passivity property for the considered DC microgrid is established.
More precisely, the input port-variable of the corresponding passive map is equal to the first-time derivative of the control input. Then, the integrated input port-variable is used to shape (input shaping methodology <cit.>) the closed loop storage function such that it has a minimum at the desired equilibrium point. Convergence to the desired equilibrium is established together with extremely simple tuning rules.


  §.§.§ Robustness
 The proposed control scheme is decentralized and robust with respect to unknown loads and other parameter uncertainty (e.g., line and filter impedances).


  §.§.§ Validation
 The proposed control strategy is verified through experiments on a real DC microgrid test facility at Ricerca sul Sistema Energetico (RSE), Milan, Italy, showing excellent closed-loop performance (see <cit.> and <cit.> for more information about the experimental setup where we have performed our tests).





 §.§ Outline

The present letter is organized as follows. The microgrid model is described in Section <ref>, while the control objective is formulated in Section <ref>. In Section <ref>, the proposed control scheme is designed and the stability of the controlled microgrid analyzed. In Section <ref>, the proposed control scheme is validated through experiments on a real DC microgrid and, finally, conclusions are gathered in Section VI.


 §.§ Notation

Let 0⃗ be the vector of all zeros of suitable dimension and let 1_n ∈^n be the vector containing all ones. The i-th element of vector x is denoted by x_i. 
 A steady state solution to system ẋ = ζ(x), is denoted by x,  i.e., 0 = ζ(x). 
 
 A constant signal is denoted by x^∗. Given a vector x∈^n, [x]∈^n× n indicates the diagonal matrix whose diagonal entries are the components of x. Let ∘' denote the Hadamard product, i.e., given vectors x, y ∈^n, ( x ∘  y) ∈^n is a vector with elements ( x ∘  y)_i := x_i y_i for all i=1,…,n. 
 Let A ∈^n × n be a matrix. In case A is a positive definite (positive semi-definite) matrix, we write A ≻ 0  (A ≽ 0). The n × n identity matrix is denoted by I_n.
 Given a set Ω, |Ω| represents the cardinality of Ω.






§ DC MICROGRID MODEL



The DC microgrid is represented by a connected and undirected graph 𝒢 = (𝒱,ℰ), where   𝒱 = {1,...,n} is the set of nodes and ℰ⊆𝒱×𝒱 represents the set of the resistive-inductive lines interconnecting the nodes.
Each node, which we call Distributed Generation Unit[Note that, we consider generation units only for the sake of simplicity and without loss of generality. In the experiments (see Section <ref>), the controlled nodes are indeed storage units, i.e., batteries.] (DGU), includes a DC-DC boost converter supplying an unknown load.
A schematic electrical diagram of the considered DC network including two DGUs interconnected by a power line  is illustrated in Fig. <ref> (see also Table <ref> for the description of the used symbols).



By applying the Kirchhoff's laws, the average[Under the condition that the Pulse Width Modulation (PWM) frequency is sufficiently high, the state of the system can be replaced by the average state representing the average inductor currents and capacitor voltages. Consequently, the switching control input is replaced by the so-called duty cycle of the converter.] governing dynamic equations[For the sake of simplicity, the dependence of all the variables on time t is omitted when it is clear from the context.] of the node i ∈𝒱 are the following:

    L_siİ_si   = - (1-u_i) V_i + V^∗_si
    
    C_iV̇_i    =  (1-u_i) I_si - I_li(V_i) -  ∑_k ∈ℰ_i^I_k,

where I_si: _≥0→, I_k: _≥0→, V_i:_≥0→_>0, I_li(V_i):_>0→_≥0, u_i : _≥0→ [0,1) and V^∗_si, L_si, C_i ∈_>0. Moreover, ℰ_i is the set of power lines connected to the DGU i and I_k is the current flowing on the line k ∈ℰ_i.
Let k be the power line interconnecting DGUs i, j ∈𝒱. Then, the dynamic of I_k in (<ref>) is given by

    L_kİ_k = (V_i - V_j) - R_k I_k,

with L_k,R_k ∈_>0.
Moreover, the term I_li(V_i) in (<ref>) represents the current demanded[The results presented in this work hold also in case of the so-called net generating loads, i.e., I_li(V_i)<0.] by the load i ∈𝒱 and (generally) depends on the node voltage V_i. 
In this work, we consider a general load model including the parallel combination of the following load components:


  * constant impedance: I_li = G^∗_liV_i, with G^∗_i∈_>0,

  * constant current: I_li = I^∗_li, with I^∗_li∈_≥0, and

  * constant power: I_li = V_i^-1P^∗_li, with P^∗_li∈_≥0.

To refer to the load types above, the letters Z', I' and P', respectively, are often used in the literature <cit.>.
Therefore, in presence of the so-called ZIP loads, I_li(V_i) in (<ref>) is given by

    I_li(V_i) = G^∗_liV_i + I^∗_li + V_i^-1P^∗_li.

The symbols used in  (<ref>)–(<ref>) are described in Table <ref>.


We represent the microgrid topology by using its corresponding incidence matrix D ∈^n × |ℰ|. The ends of edge k ∈ℰ are arbitrarily labeled with a + and a -. More precisely, one has that

    D_ik=
    
    +1      if i is the positive end of k
    
    -1      if i is the negative end of k
    
    0      otherwise.

The overall microgrid system (<ref>), (<ref>) in presence of ZIP loads (<ref>) can now be written compactly for all nodes i ∈𝒱 as follows:

    L_sİ_s   = - (1_n - u) ∘V +  V^∗_s
    CV̇   =   (1_n - u) ∘I_s - G^∗_l V - I^∗_l - [V]^-1P^∗_l + D I
    
    		 L İ   = -D^⊤ V - RI,

where I_s: _≥0→^n, V:_≥0→^n_>0, I: _≥0→^|ℰ|, u:_≥0→ [0,1)^n, V^∗_s∈^n_>0 and I^∗_l, P^∗_l ∈^n_≥0. Moreover, the matrices L_s,  C, G^∗_l, L and R have appropriate dimensions and are constant, positive definite and diagonal.






§ PROBLEM FORMULATION: VOLTAGE REGULATION


In this section, we formulate the control objective aiming at regulating the voltage of a boost-based DC microgrid.
First, we notice that for given u^∗, V^∗_s, G^∗_l, I^∗_l and P^∗_l, a steady state solution (I_s, V, I) to system (<ref>) satisfies


    V   =  ( I_n -[u^∗] )^-1 V_s^∗
    (I_n - [u^∗] ) I_s    = G^∗_l V + I^∗_l + [V]^-1P^∗_l - D I
    I   = -R^-1D^⊤V,


where from (<ref>) it follows that the boost output voltage V_i is higher than the  voltage source V^∗_si, i ∈𝒱, while (<ref>) implies[The incidence matrix D, satisfies 1_n^⊤ D = 0.] that current balance is achieved at the steady state, i.e., the total current 1^⊤(I_n - [u^∗] ) I_s injected by the boost converters is equal to the total current 1^⊤(G^∗_l V + I^∗_l + [V]^-1P^∗_l) demanded by the ZIP loads.
Moreover, in order to guarantee a proper functioning of the connected loads, it is required that the current balance is achieved at the desired voltage value.
Consequently, before formulating the control objective, we introduce the following assumption on the existence of a desired reference voltage for each DGU:
(Desired voltage)

There exists a constant desired reference voltage V_di^∗≥ V_si^∗ for all i ∈𝒱.



Given V^∗_d = [V_d1^∗, …, V_dn^∗]^T, the control objective is then formulated as follows:
(Voltage regulation)


    lim_t →∞   V(t) =    V =   V^∗_d.


Moreover, in order to permit the controller design in the next section, the following assumption is introduced on the available informations:

(Available informations)

The state variables I_si, V_i and the voltage source V^∗_si are locally available only at the DGU i. 

Consequently, the control scheme we design in Section <ref> to achieve Objective 1 needs to be fully decentralized, increasing the practical applicability of the proposed approach.

(Microgrid uncertainty)

Note that, according to Assumption <ref>, the parameters I^∗_l, P^∗_l, G^∗_l, L_s, L, C, R of the ZIP loads, lines and boost converters are not known. As a consequence, we need to design a control scheme that is robust with respect to the system uncertainty.







§ THE PROPOSED SOLUTION


In this section, we introduce the key aspects of the proposed decentralized passivity-based control scheme aiming at achieving Objective 1. More precisely, we first augment system (<ref>) with additional dynamics. Secondly, we propose a Krasovskii-type storage function <cit.> and establish a (local) passivity property for the augmented system. The input port-variable of the corresponding passive map is equal to the first-time derivative of the control input. Then, we use the integrated input port-variable to shape the closed loop storage function such that it has a minimum at the desired equilibrium point.


Consider the following auxiliary[The state variables and the control input of the auxiliary system are I_s, V, I, İ_s, V̇, İ, u and υ_c, respectively.] system:



    L_sİ_s =    - (1_n - u) ∘V +  V^∗_s
    CV̇ =      (1_n - u) ∘I_s - G^∗_l V - I^∗_l - [V]^-1P^∗_l + D I 
    
    		 L İ =    -D^⊤ V - RI 
    L_sÏ_s =    - (1_n - u) ∘V̇ +  υ_c ∘ V 
    CV̈ =     (1_n - u) ∘İ_s - υ_c ∘ I_s 
       - ( G^∗_l - [V]^-2[P^∗_l] ) V̇ + D İ
    
    		 L Ï =    -D^⊤V̇ - R İ
    u̇ =     υ_c,


which includes also the dynamics of the first-time derivative of the state and input of system (<ref>). 

Let the vector z := (I_s^⊤, V^⊤, I^⊤, İ_s^⊤, V̇^⊤, İ^⊤, u^⊤)^⊤∈𝒵 := { z ∈^5n+2|ℰ|: V ∈^n_>0, u ∈ [0,1)^n} denote the state of the auxiliary system (<ref>).
In order to establish a passivity property for system (<ref>), we first introduce the following set:





    𝒵_ZIP := { z ∈𝒵 : G_l^∗ - [V]^-2[P_l^∗] ≻ 0  }.

Then, the following result can be proved.

(Passivity property)

System (<ref>) is passive with respect to the supply rate υ_c^⊤( İ_s ∘ V - V̇∘ I_s ) and the storage function

    S(İ_s, V̇, İ) = 1/2İ_s^⊤ L_s İ_s + 1/2V̇^⊤ C V̇ + 1/2İ^⊤ L İ,

for all the trajectories z ∈𝒵_ZIP.



The storage function S in (<ref>) satisfies

    Ṡ   = - V̇^⊤( G^∗_l - [V]^-2[P^∗_l] ) V̇ - İ^⊤ R İ
        + υ_c^⊤( İ_s ∘ V - V̇∘ I_s )
       ≤υ_c^⊤( İ_s ∘ V - V̇∘ I_s ),

along the solutions z ∈𝒵_ZIP to system (<ref>), which concludes the proof.

(Insights on the proposed storage function S)

The storage function S in (<ref>) depends on the states İ_s, V̇, İ. This implies that S depends also on I_s, V, I and u, i.e., the entire state of the auxiliary system (<ref>). This is evident from replacing İ_s, V̇, İ by the corresponding dynamics (<ref>)–(<ref>), or rewriting S as follows:

    S(z) =    1/4( İ_s^⊤ L_s İ_s + V̇^⊤ C V̇ + İ^⊤ L İ)
       +1/4( f_I_s^⊤ L^-1_s f_I_s + f_V^⊤ C^-1 f_V + f_I^⊤ L^-1 f_I),

where f_I_s: ^n_>0× [0,1)^n → ^n, f_V: ^n ×^n_>0×^|ℰ|× [0,1)^n → ^n, and f_I: ^n_>0×^|ℰ|→ ^|ℰ| represent the right-hand sides of (<ref>)–(<ref>), respectively.
Moreover, it will be shown in Theorem <ref> that using (<ref>) to design the controller permits, differently from <cit.> and <cit.>, the achievement of Objective 1 despite the system uncertainty (see Remark <ref>). However, the cost of designing a robust controller is the need of information about the first-time derivative of the signals I_s and V (see Remark <ref>).

Before designing the controller and introducing the main result of this work, to be able to achieve Objective 1, we show that a unique steady state solution to system (<ref>)
always exists.
(Existence of a unique steady state solution)

Let Assumption <ref> hold. Given υ_c = 0 and V^∗_di > √(P^∗_li/G^∗_li), for all i ∈𝒱, there exists a unique steady state solution z = (I_s,  V = V^∗_d, I, 0⃗, 0⃗, 0⃗, u) ∈𝒵_ZIP to system (<ref>), satisfying

    u =     1_n -[V^∗_d]^-1 V_s^∗
    (I_n - [u] ) I_s =     G^∗_l  V^∗_d + I^∗_l + [V^∗_d]^-1P^∗_l - D I
    I =     -R^-1D^⊤ V^∗_d
    0⃗ =     V̇
    0⃗ =     İ_s
    0⃗ =     İ
    0⃗ =     υ_c,



The proof follows from setting the left-hand-side of system (<ref>) to zero.

We can now show the main result of this paper concerning the design of a controller that (provably) stabilizes system (<ref>) achieving Objective 1.



(Stability)

Let Assumptions <ref>-<ref> hold.
Consider system (<ref>) controlled by

    T_c υ_c = - K_c ( u - u^∗_d ) - ( İ_s ∘ V - V̇∘ I_s  ),

where u^∗_di = 1-V^∗_si/V^∗_di is the desired value of the duty cycle of the boost converter i ∈𝒱, T_c =  (T_c1, …, T_cn), K_c =  (K_c1, …, K_cn) and T_ci>0, K_ci >0 are the gains of the controller i ∈𝒱.
Given V^∗_di > √(P^∗_li/G^∗_li), for all i ∈𝒱, the equilibrium z = (I_s, V^∗_d, I, 0⃗, 0⃗, 0⃗,  u^∗_d) ∈𝒵_ZIP is  asymptotically stable.


Consider the desired closed-loop storage function

    S_d (İ_s, V̇, İ, u) = S(İ_s, V̇, İ) + 1/2 (u - u^∗_d)^⊤ K_c (u - u^∗_d),

where S is given by (<ref>). Noticing that S_d is function of the entire state of the auxiliary system (<ref>) (see Remark <ref>),
it is immediate to see that S_d attains a minimum at the equilibrium (I_s,  V^∗_d, I, 0⃗, 0⃗, 0⃗, u^∗_d), where V = V^∗_d follows from (<ref>) at steady state together with u = u^∗_d. Furthermore, S_d satisfies

    Ṡ_d    = - V̇^⊤( G^∗_l - [V]^-2[P^∗_l] ) V̇ - İ^⊤ R İ
         + υ_c^⊤( K_c ( u - u^∗_d ) + İ_s ∘ V - V̇∘ I_s )
       = - V̇^⊤( G^∗_l - [V]^-2[P^∗_l] ) V̇ - İ^⊤ R İ - υ_c^⊤ T_c υ_c,

along the solutions to system (<ref>). 
From the last line of (<ref>) it follows that S_d satisfies Ṡ_d ≤ 0 for all z ∈𝒵_ZIP.
Then, given ε > 0, choose r ∈ (0, ε] such that there exists a ball ℬ_r(z) ⊂𝒵_ZIP centred in z = (I_s, V^∗_d, I, 0⃗, 0⃗, 0⃗, u^∗_d) ∈𝒵_ZIP, i.e., 

    ℬ_r(z) := { z ∈𝒵_ZIP: ||z -z|| ≤ r }⊂𝒵_ZIP.

 Moreover, let α denote the minimum value of S_d on the boundary of ℬ_r(z), i.e., α = min_||z - z|| = r S_d(z). Since S_d is positive definite, then α > 0. Take β∈ (0,α), then the set
 
    Ω_β := { z ∈ℬ_r(z): S_d(z) ≤β}

is compact, positively invariant and in the interior of ℬ_r(z) (see <cit.>).





Let now E denote the set of all points in Ω_β where Ṡ_d = 0, i.e.,

    E := { z ∈Ω_β: V̇ = 0⃗, İ = 0⃗, υ_c = 0⃗}.

Moreover, let M be the largest invariant set in E. Then, by LaSalle's invariance principle <cit.>, every solution starting in Ω_β approaches M as t approaches infinity. As a consequence, in the largest invariant set M, from (<ref>) and (<ref>) we obtain Ï_s = 0⃗ and Ï = 0⃗, respectively. Moreover, a straightforward  computation shows that in M the third-time derivative of the voltage V satisfies

    CV^(3) = -( T_c^-1[I_s]^2 + ( G^∗_l - [V]^-2[P^∗_l] ) )V̈,

along the solutions to system (<ref>), implying that also V̈ is equal to zero in M. Consequently, from (<ref>) we obtain İ_s=0⃗ in M, and from (<ref>) we can conclude that, in the largest invariant set M, u = u^∗_d, implying from (<ref>) that V asymptotically converges to V^∗_d. We finally conclude the proof observing from (<ref>) and (<ref>) that also I_s and I converge to a constant value satisfying (<ref>).





(Robustness property)

Note that controller (<ref>) requires the first-time derivative of the current I_s and voltage V, which can be estimated in finite time by implementing for instance the well known Levant's differentiator <cit.>. Moreover, we observe that the use of İ_s, V̇ gives robustness properties to the proposed  controller with respect to loads, lines and boost parameter uncertainty. Controller (<ref>) requires indeed only the knowledge of u^∗_d, which depends on V^∗_s.


(ZI' loads)
We observe that in case of only ZI' loads, i.e., P^∗_l=0⃗, the results developed in this section can be strengthened. The absence of constant power loads implies indeed that the passivity property of system (<ref>) and the result of Theorem <ref> hold in the whole set 𝒵. This immediately follows by noticing that P^∗_l=0⃗ implies 𝒵_ZIP≡𝒵.







§ EXPERIMENTAL RESULTS





In order to validate the proposed control scheme, experimental tests have been performed on the DC microgrid test facility at RSE. The electrical scheme, a photo and the layout of the setup are shown in Figs. <ref>–<ref>.
The RSE's DC microgrid is unipolar with a nominal voltage of 380 V and includes a ZIP load, with a maximum power of 30 kW at 400 V, a DC generator with a maximum power of 30 kW, which emulates a PV plant, and two storage devices, based on high temperature NaNiCl batteries, each of them with an energy of 18 kWh and a maximum power of 30 kW for 10 s. The batteries are connected to the DC network through 35 kW bidirectional boost converters. All the other parameters of the RSE's DC microgrid are reported in Table <ref>.


In order to regulate the voltages V_2 and V_4 at the nodes 2 and 4 towards the corresponding desired value V^∗_d= 380V, the control strategy proposed in Section <ref> (with T_c = 1e7 and K_c = 1e9) is implemented through dSpace controllers.
The currents I_l1(V_1) and I_l3(V_3) demanded by the load and generated by the PV emulator are treated as disturbances.
In the following, we arbitrarily assume the passive sign convention[I_l1(V_1) ≥ 0, I_l3(V_3)≤ 0, I_s1, I_s2 > 0 (I_s1, I_s2 < 0) if the batteries charge (discharge).]. 



In the first scenario the system is in a steady state condition with zero power absorbed by the load or provided by the generator. Each battery converter regulates its output voltage at the desired value V^∗_d= 380V. At the time instant t = 5 the load (see Fig. <ref>) or the PV emulator (see Fig. <ref>) absorbs/generates 20 until the time instant t = 45. 
 From Fig. <ref> and in Fig. <ref>, one can observe that, after a transient due to the load/generator variations, the system exhibits a stable performance. This clearly shows the robustness of the proposed controller with respect to the  system uncertainty.


In the second scenario the system is in a steady state condition with a constant power equal to 20 provided by the generator. Each battery converter regulates its output voltage at the desired value V^∗_d= 380V. At the time instant t = 5 the desired value V^∗_d2 is changed to 375V and at the time instant t = 45 also the desired value V^∗_d4 is changed to 375V (dashed line). From Fig. <ref>, one can observe that the system exhibits a stable performance tracking the new desired voltage value. A similar scenario is illustrated in Fig. <ref>. Tracking capabilities are generally essential to couple voltage controllers with higher-level control schemes that modifies the voltage reference of each node, in order to achieve power sharing among the nodes of the microgrid. 

Finally, we note that in the discussed scenarios, only the voltages V_2 and V_4 are controlled and the deviations from the desired value during the load and generator variations are less than 4%. In the uncontrolled nodes, the deviations of the voltages V_1 and V_3 from the desired value are less than 7%. These deviations are due to the line impedances between the controlled and uncontrolled nodes.















§ CONCLUSIONS


In this paper a decentralized passivity-based control scheme is designed to regulate the voltage of a DC microgrid through boost power converters. 
Using a Krasovskii-type storage function, a (local) passivity property for the considered DC microgrid is established.
More precisely, the integrated input port-variable is used to shape the closed loop storage function.
Convergence to the desired equilibrium is proven in presence of the so-called ZIP' (constant impedance Z', constant current I' and constant power P') loads, showing robustness with respect to system parameter uncertainties.
The proposed control scheme is validated through experimental tests on a real DC microgrid, showing excellent closed-loop performances.




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