$H_{\infty}$-Control of Grid-Connected Converters: Design, Objectives and Decentralized Stability Certificates
Linbin Huang Huanhai Xin Florian Dörfler










ℋ_∞-Control of Grid-Connected Converters: Design, Objectives and Decentralized Stability Certificates
    Linbin Huang, Huanhai Xin, and Florian Dörfler
L. Huang is with the College of Electrical Engineering at Zhejiang University, Hangzhou, China, and the Department of Information Technology and Electrical Engineering at ETH Zurich, Switzerland. (Email: huanglb@zju.edu.cn)
H. Xin is with the College of Electrical Engineering at Zhejiang University, Hangzhou, China. (Email: xinhh@zju.edu.cn)
F. Dörfler is with the Department of Information Technology and Electrical Engineering at ETH Zurich, Switzerland. (Email: dorfler@ethz.ch)
This research was supported in part by the 2020 Science and Technology Project of State Grid Power Company Limited (“Power Grid Strength Evaluation and Optimization to Accommodate High-penetration Renewables”) and in part by ETH Zurich Funds.
    December 30, 2023
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The modern power system features high penetration of power converters due to the development of renewables, HVDC, etc. Currently, the controller design and parameter tuning of power converters heavily rely on rich engineering experience and extrapolation from a single converter system, which may lead to inferior performance or even instabilities under variable grid conditions. In this paper, we propose an ℋ_∞-control design framework to provide a systematic way for the robust and optimal control design of power converters. We discuss how to choose weighting functions to achieve anticipated and robust performance with regards to multiple control objectives. Further, we show that by a proper choice of the weighting functions, the converter can be conveniently specified as grid-forming or grid-following in terms of small-signal dynamics. Moreover, this paper first proposes a decentralized stability criterion based on the small gain theorem, which enables us to guarantee the global small-signal stability of a multi-converter system through local control design of the power converters. We provide high-fidelity nonlinear simulations and hardware-in-the-loop (HIL) real-time simulations to illustrate the effectiveness of our method.



H-infinity control, admittance modeling, power converters, small gain theorem, stability, small-signal stability.




§ INTRODUCTION


The penetration rate of power-electronic converters in the modern power system is ever-increasing mainly because of the rapid development of renewables, energy storage systems and high-voltage DC transmission (HVDC) systems. The high controllability of power converters is making the power system more flexible, which allows auxiliary services such as frequency support, voltage support and oscillation damping to be provided for the power grid <cit.>.

Commonly, power converters facilitate multiple control loops to achieve different control objectives. For example, in a typical grid-following converter, a phase-locked loop (PLL) is used for grid synchronization; a current control loop is used for current control and fast current limitation; an active power control loop is used for active power tracking; and a voltage control loop is used to regulate the terminal voltage <cit.>. The corresponding control structure, i.e., how these loops are interconnected and tuned, is currently based on analysis of a single converter together with rich engineering experience and physical intuition. Compared with conventional synchronous generators, power converters generally present much more complex dynamics in a wide frequency range due to these multiple loops, thereby posing great challenges to the analysis, operation, and control of future low-inertia power grids <cit.>.


It has been revealed that the control loops inside a converter are strongly coupled with each other, deteriorate the performance and even lead to instabilities under variable grid conditions <cit.>. The coupling among different loops complicates the parameter tuning of converters as proper parameters can hardly be directly obtained from a tractable model. The most common method to deal with this issue is to model the whole converter system, analyze how certain ranges of the parameters affect the stability of the converter (through eigenvalue loci, Nyquist diagrams, etc.), and then pick some acceptable parameter sets to be tested in a real system <cit.>. However, this method may not lead to the optimal parameter set in terms of damping ratio or stability margin because eigenvalue loci or Nyquist diagrams can hardly deal with multiple parameters (e.g., the converter may have eight parameters to tune if it has four loops). It is even more challenging to design optimal and stabilizing controller with regards to multi-converter systems, as the converters can be strongly coupled and lead to complex interacting dynamics.


One convenient way to achieve optimal performance is to use ℋ_2/ℋ_∞-optimal control synthesis. In <cit.> and <cit.>, ℋ_∞ and gain scheduled ℋ_∞ controllers were used in microgrids to perform robust control. However, only a simplified model of the converter (without current control loop) is considered, which may result in inferior performance for converters that have a current control loop.

An optimal voltage control problem for islanded power converters was posed in <cit.> and solved using ℋ_∞ synthesis. In <cit.>, ℋ_∞ synthesis was used to design the converter's output current controller and improve the robustness against variable grid impedance. In terms of harmonic suppression, ℋ_∞ and repetitive control techniques can be used for designing output voltage controllers to reject harmonic disturbances from nonlinear loads or the public grid, as presented in <cit.>.


The above methods will result in a dynamic controller whose order is the same as that of the system, which further complicates the system dynamics. It is also possible to obtain fixed-structure fixed-order optimal controllers by ℋ_2/ℋ_∞ synthesis <cit.>. For example, a robust frequency control was obtained in <cit.> through ℋ_∞ loop shaping design of a static control gain matrix to improve the grid frequency dynamics. A fixed-structure current controller based on ℋ_∞ synthesis was developed in <cit.> to guarantee robust stability and performance for power converters.
In <cit.>, an ℋ_2-optimal tuning method was applied to optimize the PI gains in HVDC systems. In <cit.>, ℋ_2-optimal design is applied to allocate the virtual inertia of grid-forming and grid-following converters in low-inertia systems. It is noteworthy that grid-forming converters and grid-following converters, which present distinct dynamic behaviors, have been widely integrated in modern power grids. It was shown in <cit.> that grid-forming and grid-following converters can be distinguished by their small-signal responses to grid frequency disturbances in the frequency domain. Based on this finding, we will analyze how to specify a converter as grid-forming or grid-following with regards to small-signal dynamics  for the purpose of guiding the ℋ_∞-control design. Note that in this paper, akin to <cit.>, we consider PLL-based converter as one prototypical grid-following type and droop-controlled converters as one prototypical grid-forming type.


Both ℋ_2 and ℋ_∞ optimal controllers are capable of stabilizing the system. However, the existing ℋ_2/ℋ_∞-optimal design for power converters generally considers a single-converter system, which can only ensure the system stability when the converter is connected to an infinite bus, but provides no guarantee on the stability of a multi-converter system.
Moreover, the control objective of stable grid synchronization under variable grid conditions was not included in the ℋ_2/ℋ_∞-optimal design in the existing literature, which may result in inferior synchronization performance and poor robustness.

In this paper, we propose a fixed-structure ℋ_∞-control design framework to perform optimal, robust, and multivariable control for grid-connected power converters. Our ℋ_∞-control design considers multiple control objectives (grid synchronization, active power and voltage regulations) simultaneously. We elaborate on how to achieve the specified control objectives by choosing proper weighting functions. Moreover, we show that by choosing different weighting functions, the converter can be conveniently specified as grid-forming or grid-following in terms of small-signal dynamics. Furthermore, we propose a decentralized stability criterion based on the small gain theorem, which enables to ensure the global stability of a multi-converter system through local ℋ_∞-control design of the converters. We show that the resulting ℋ_∞-optimal controller has the anticipated dynamic performance, robustness against variable grid conditions, and it guarantees on the overall system stability. We illustrate our results by means of high-fidelity simulations and hardware-in-the-loop (HIL)  real-time simulations.

The rest of this paper is organized as follows: Section II presents the ℋ_∞-control design framework for the grid-connected power converters. Section III discusses the control objectives and the corresponding weighting functions. Section IV proposes the decentralized stability criterion for multi-device systems and shows how it can be incorporated in the ℋ_∞-control design framework. Detailed simulation and HIL real-time simulation results are provided in Section V. Section VI concludes the paper.




§ ℋ_∞-CONTROLLER FOR CONVERTERS



In this section we present the design setup of our fixed-structure ℋ_∞-controller for power converters, and compare its structure with conventional control schemes.





 §.§ Converter system descriptions


Fig. <ref> shows a three-phase power converter which is connected to a power grid via an LCL filter. In the control scheme of this converter, the three-phase voltage and current signals are represented by two-dimensional vectors in the synchronously rotating dq frame (through Park transformation). Usually, the rotating frequency of this dq frame is generated by a grid-synchronization unit, e.g., PLL, emulated swing equation or droop control. Here this frequency (denoted by ω + ω_0, and ω_0 is the nominal value) is generated by the ℋ_∞-controller, as depicted in Fig. <ref>.
A current control loop is used to make the converter-side current (I_ Cd and I_ Cq) track their reference values (I_ Cd^ ref and I_ Cq^ ref) and implement fast current limitation <cit.>.
In the global dq frame (with constant rotating frequency ω_0), the converter-side voltage vector is denoted by U'_ C, which is determined by the PWM signals; the converter-side current vector is I'_ C; the grid-side current vector is I'; the capacitor voltage vector of the LCL filter is V'; and the grid-side voltage vector is U', as labeled in Fig. <ref>.

Let Y(s) be the converter's admittance matrix (linearized model), which is a 2 × 2 transfer function matrix reflecting how a perturbation from the terminal voltage affects the converter's current output, i.e., I' = Y(s)U'. It has been demonstrated that this admittance matrix dominates the stability of a converter system since it describes the input/output characteristics of the converter as seen from the point of the AC grid, and for the detailed derivation process we refer to <cit.>, <cit.> or <cit.>.











 §.§ ℋ_∞-control design setup


The converter system in Fig. <ref> (not including the ℋ_∞-controller K) can be modeled as

    [[ z; y ]] = [[ P_11(s) P_12(s); P_21(s) P_22(s) ]][[ w; u ]] ,

where u is the control input of the system, y is the measured output for the ℋ_∞-controller, w and z are the input/output signals chosen to quantify the performance of the system <cit.>. Fig. <ref> shows how the elements of w enter the system as disturbances, and how z is chosen from the system variables.
To quantify the power and voltage tracking performance, we choose w_1, w_2, w_3 and w_4 to be the disturbance inputs imposed on the power and voltage reference signals, as shown in Fig.1. The disturbances w_5 and w_6 are used to incorporate the admittance matrix in the ℋ_∞-controller for decentralized stability certification, which will be elaborated upon in Section IV. The disturbance signal w_7 is used to specify the grid-synchronization performance of the system, as grid synchronization requires the angle output (i.e., θ) to reject disturbances that enter the closed loop.

In this paper, u is chosen as u=[I_ Cd^ ref  I_ Cq^ ref  ω]^⊤, and y is the vector of voltage/power signals and their integrals

    y = [ [ y_1; y_2; y_3; y_4; y_5; y_6; y_7 ]] = [ [      V_d^ ref - V_d + I_qX_v + w_1; 1/s(V_d^ ref - V_d + I_qX_v + w_1);      V_q^ ref - V_q - I_dX_v + w_2; 1/s(V_q^ ref - V_q - I_dX_v + w_2);                 P_ ref - P_E + w_3;            1/s(P_ ref - P_E + w_3);                 Q_ ref - Q_E + w_4;                                    ]] ,


where V_d^ ref and V_q^ ref are d-axis and q-axis voltage references, V_d and V_q are the d-axis and q-axis voltage components, P_ ref and Q_ ref are active and reactive power references, P_E and Q_E are converter's active and reactive power, and X_v is the virtual reactance to enhance the system flexibility. In addition to the virtual impedance loop <cit.>, other auxiliary control loops (e.g., for damping oscillations <cit.>, improving power quality <cit.>, etc.) can also be conveniently included in the ℋ_∞-control design framework by modifying the power/voltage references or the current control loop, which changes the model in (<ref>) and possibly also the specification of the signals (u,y,w,z).


 We remark that the integrals in y determine the system's steady state. To be specific, the integrals of voltage and active power signals regulate the voltage and active power to their references in steady state. The integral of reactive power is not included in y since the steady-state value of reactive power is determined by the power flow when the voltage is regulated to the reference value.
Also, other expected steady states, e.g., voltage droop, can be conveniently configured by changing the integrals in y.






The aforementioned voltage regulation is similar to some existing voltage control loops (e.g., the ac voltage control loop in droop-controlled converters <cit.>) in the sense that the dq-axis components of V' in the controller's rotating coordinate are regulated to their reference values. The frequency of the controller's coordinate (i.e., ω) is determined by the ℋ_∞-controller to achieve grid synchronization, which will be elaborated upon below.


The performance output vector z is chosen as

    z     = [z_1  z_2  z_3  z_4  z_5  z_6  z_7  z_8  z_9  z_10]^⊤
       = [V_d  V_q  y_1  y_3  P_E  Q_E  θ  y_5  [I'_d  I'_q]G(s)^⊤]^⊤ ,

which quantifies the tracking and disturbance-rejection performance of the voltage, power and angle signals. Note that G(s) is a 2 × 2 transfer matrix chosen to enable a decentralized stability criterion, which will be elaborated in Section <ref>.

Note that the disturbance signals w_1 ∼ w_7 and the performance output vector z are chosen for specifying the control objectives and quantifying the system performance in the ℋ_∞ control design. In other words, they are chosen in order to derive a proper transfer function matrix in (<ref>) which describes the response characteristics of the system. On this basis, we can use weighting functions to shape the characteristics of the system and achieve certain control objectives, which will be detailed in the next section.


Finally, we remark that the transfer function matrix in (<ref>) can be conveniently obtained by first deriving the state-space model of the system with the  inputs/outputs specified in Fig. <ref> and then transforming this state-space model to the input/output transfer function matrix. Another way is to  derive the transfer function matrix by incorporating the input/output configuration (defined in Fig. <ref>) and then modifying the existing frequency-domain models (e.g., those in <cit.> and <cit.>) of power converters (including the dynamics of LCL, current control loop, etc.).

The ℋ_∞-controller can be formulated as

    u=Ky
     ,

where K ∈ℝ^3 × 7 is a static parameter matrix.



Two Noteworthy Cases



The controller in (<ref>) can be regarded as a generalized form of PLL-based controller (grid-following type) and frequency droop controller (grid-forming type). For example, we obtain a typical PLL-based controller <cit.> for the converter by setting

    K =[ [      0      0      0      0  K_ PP  K_ PI      0; -K_ VP -K_ VI      0      0      0      0      0;      0      0 -K_ω P -K_ω I      0      0      0 ]],

where K_ PP and K_ PI are the PI gains of the active power control loop, K_ VP and K_ VI are the PI gains of the voltage control loop, K_ω P and K_ω I are the PI gains of the PLL (V_q^ ref can be set as zero to achieve voltage orientation).

Moreover, a frequency droop controller <cit.> (with a cascaded voltage/current control structure) can be obtained by

    K =
    [ [ K_ VP K_ VI     0     0     0     0     0;     0     0 K_ VP K_ VI     0     0     0;     0     0     0     0   K_f     0     0 ]] ,

where K_f is the frequency droop coefficient.



 §.§ ℋ_∞-control formulation


In the following, we discuss how K can be obtained optimally by solving an ℋ_∞-optimization problem so as to make the converter have optimal and multivariable performance.

By combining (<ref>) and (<ref>) we obtain the closed-loop transfer function matrix of the system as

    z     = { P_11(s)+P_12(s)K [I-P_22(s)K]^-1P_21(s) }w
       Δ = P(K)(s)w ,

which indicates that the design of K will affect the closed-loop performance of the system.

The standard ℋ_∞-optimal control problem is to find a stabilizing controller K by solving

    min_K W(s)P(K)(s)_∞ = min_K max_ωσ̅[W(jω)P(K)(jω)] ,

where ·_∞ denotes the ℋ_∞ norm, W(s) is the user-defined diagonal weighting transfer function matrix, and σ̅(·) denotes the largest singular value.

However, in (<ref>), the transfer functions from all the inputs to a particular output share the same weighting function, which is not a suitable setting for the multi-objective design in this paper. For example, sometimes we expect z_1, which is the d-axis voltage V_d, to track a disturbance in the reference w_1, and meanwhile rejects the other disturbances in w. However, these objectives cannot be achieved simultaneously when z_1 has the same weighting function for all the inputs in w. In addition, standard algorithms to solve (<ref>) result in a high-dimensional dynamic controller with the same order as that of the system <cit.>, which may complicate the system dynamics.

To deal with these problems, in this paper, we consider K as a matrix of static gains to ensure the simplicity and thus implementability of the resulting controller. The corresponding ℋ_∞-optimal control problem can be solved with the algorithm in <cit.>. Moreover, in order to achieve multi-objective design, we specify the shape of P(K)(s) by solving

    min_K 𝒲(s) ∘ P(K)(s)_∞
    
    =   min_K max_ωσ̅[𝒲(jω) ∘ P(K)(jω)] ,

where ∘ denotes the entrywise product of matrices, and 𝒲(s) is the weighting transfer function matrix (not necessarily diagonal) which has the same dimension as P(K)(s).
The entrywise weighting functions in (<ref>) enable us to specify the shape of every entry of P(K)(s) and thus provide more flexibility than (<ref>).
We will discuss the design of the weighting transfer function matrix 𝒲(s) in next section.



 §.§ Grid-forming and grid-following dynamics


The different settings for K will inevitably result in different dynamic behaviors, e.g., making the converter behave like the grid-following or grid-forming type in terms of small-signal frequency/angle tracking dynamics. Fig. <ref> plots the closed-loop Bode diagrams of P_77(K)(s) under droop control (<ref>) and under PLL-based control (<ref>). Note that P_77(K)(s) reflects how the controller rejects angle disturbances (from the grid) via the closed loop, which can be thought of as one single-input-single-output sensitivity function of the system <cit.>.

Observe that the PLL-based controller has a higher bandwidth than the droop controller, which indicates that the PLL-based controller has higher tracking speed but at the same time also higher sensitivity to grid disturbances. These observations are consistent with <cit.> analyzing the complementary sensitivity and showing that grid-following converters generally have higher control bandwidth to track the grid frequency.





§ CONTROL OBJECTIVES AND WEIGHTING FUNCTIONS





When solving the ℋ_∞-optimal control problem, the weighting transfer function matrix 𝒲(s) is used to specify the shape of P(K)(s) and thus the performance objectives of the system. To be specific, the entry of 𝒲(s) in the i  th row and j  th column, i.e., 𝒲_ij(s), shapes how the i th output is affected by the j th input, i.e., the transfer function P_ij(K)(s).






 §.§ Brief review on weighting functions for ℋ_∞-control



The weighting functions 𝒲(s) in ℋ_∞-control design in (<ref>) can be in fact considered as the expected upper bounds for shaping the transfer function matrix of the system P(K)(s). To illustrate this point, we explain how to choose a weighting function to specify reference tracking or disturbance rejection from a particular input to a particular output.

To begin with, we recall the concepts of sensitivity function and complementary sensitivity function <cit.>. The sensitivity function describes how the output rejects the disturbance from the input, while the complementary sensitivity function characterizes how the output tracks the input. Therefore, the Bode diagram of the sensitivity function is distinguished from that of the complementary sensitivity function. For example, Fig. <ref> plots the Bode diagrams of P_53(K)(s) and P_83(K)(s) when the droop controller in (<ref>) is applied. The transfer function P_53(K)(s) describes how the active power tracks its reference P_ ref, i.e., a complementary sensitivity function. The transfer function P_83(K)(s) describes how the active power error rejects the disturbance from P_ ref, i.e., a sensitivity function. It can be observed that the magnitude of the complementary sensitivity function approaches one (0 dB) in the low-frequency range and approaches zero in the high-frequency range. In contrast, the magnitude of the sensitivity function approaches one (0 dB) in the high-frequency range and approaches zero in the low-frequency range.



Based on the above characteristics of sensitivity function and complementary sensitivity function, we summarize the following two basic steps for choosing weighting functions to achieve reference tracking and disturbance rejection.


Step 1) Decide whether the output aims at tracking the input or rejecting the disturbance from the input. For instance, if the output aims at tracking the input, then the inverse of the weighting function should be the shape of a complementary sensitivity function, i.e., the magnitude approaches one in the low-frequency range and approaches zero in the high-frequency range.

Step 2) Decide the expected tracking/disturbance-rejection speed by assigning the bandwidth for the inverse of the weighting function.


For more details on choosing the weighting functions we refer to <cit.> and will further elaborate on the converter system below.



 §.§ Specification of the weighting functions for converters



In the following, we list the control objectives of grid-connected power converters and discuss how these objectives can be achieved by the proper design of weighting functions, which follows from the two steps listed above in Section <ref>.

i) Grid synchronization: As displayed in Fig. <ref> (a), we choose 𝒲_77(s) as

    𝒲_77(s) = s+s_1_77/s+s_2_77

with s_1_77=0.2 and s_2_77=0.0001 such that the internal phase θ (z_7) has disturbance-rejection capability against w_7. According to Fig. <ref>, the converter can conveniently emulate grid-forming or grid-following (small-signal) dynamics by choosing different 𝒲_77(s) to shape P_77(K)(s). Note that the choice of 𝒲_77(s) in Fig. <ref> (a) will lead to grid-forming (small-signal) dynamics since the high gains appear only in the low-frequency range (hence the bandwidth of P_77(K)(s) is limited).



ii) Power regulation: The active power should track its reference with fast dynamics, hence we choose 𝒲_83(s) as

    𝒲_83(s) = s+s_1_83/s+s_2_83 ,

where s_1_83=5 and s_2_83=0.0005 such that the tracking error is well eliminated in the low-frequency range, as displayed in Fig. <ref> (b).
In addition, we choose 𝒲_53(s) as

    𝒲_53(s) = s^2/ω_2_53^2 + 2ξ_2_53 s/ω_2_53 + 1/s^2/ω_1_53^2 + 2ξ_1_53 s/ω_1_53 + 1

so as to suppress high-frequency disturbances for the power regulation. The magnitude of 𝒲_53(s) is displayed in Fig. <ref> (b), with the parameters chosen as ω_1_53 = 7 × 10^5, ξ_1_53 = 0.35, ω_2_53 = 1000 and ξ_2_53 = 0.8.
Moreover, we choose 𝒲_55(s) to be equal to 𝒲_53(s) such that the active power control is capable of rejecting disturbances from the power grid voltage.

Considering that when the terminal voltage remains constant, the change of active power flow also affects the steady-state value of the reactive power, we choose 𝒲_63(s) as

    𝒲_63(s) = k_ QT×s^2/ω_2_63^2 + 2ξ_2_63 s/ω_2_63 + 1/s^2/ω_1_63^2 + 2ξ_1_63 s/ω_1_63 + 1

with k_ QT = 1/6, ω_1_63 = 7 × 10^5, ξ_1_63 = 0.35, ω_2_63 = 1000 and ξ_2_63 = 0.8. As shown in Fig. <ref> (b), 𝒲_63(s) has a lower gain than 𝒲_53(s) in the low-frequency range due to the fact that the change of active power reference has less impact on the reactive power than on the active power, and the high gain of 𝒲_63(s) in the high-frequency range is used to suppress high-frequency disturbances.

iii) Voltage regulation: The d-axis and q-axis voltage components should track their reference values with fast dynamics, hence we choose 𝒲_31(s) and 𝒲_42(s) to be

    𝒲_31(s) = 𝒲_42(s) = s+s_1_31/s+s_2_31

with s_1_31 = 20 and s_2_31 = 10^-5 in order to eliminate the tracking error in the low-frequency range, as shown in Fig. <ref> (c). Furthermore, 𝒲_11(s) and 𝒲_22(s) are chosen to be

    𝒲_11(s) = 𝒲_22(s) = s^2/ω_2_11^2 + 2ξ_2_11 s/ω_2_11 + 1/s^2/ω_1_11^2 + 2ξ_1_11 s/ω_1_11 + 1

with ω_1_11 = 7 × 10^5, ξ_1_11 = 0.1, ω_2_11 = 7 × 10^3 and ξ_2_11 = 0.1, as displayed in Fig. <ref> (c). Note that 𝒲_11(s) and 𝒲_22(s) provide a magnitude drop around the resonance frequency of the LCL (1100 Hz in this paper), which is used to reduce the control design emphasis around this frequency.

iv) Admittance performance: The transfer function matrix from [w_5 w_6]^⊤ to [z_9 z_10]^⊤ is G(s)Y(s). This transfer function matrix will significantly affect the system stability as it incorporates the admittance matrix Y(s). The choosing of G(s) will be elaborated in the following section to show how it enforces decentralized stability. We choose the weighting functions for this transfer function matrix, i.e., 𝒲_95(s), 𝒲_96(s), 𝒲_10,5(s) and 𝒲_10,6(s), as

    𝒲_95(s) = 𝒲_96(s) = 𝒲_10,5(s) = 𝒲_10,6(s) 
       = k_ Ad/(s^2/ω_1_95^2 + 2ξ_1_95 s/ω_1_95 + 1)^2 ,

where k_ Ad = 0.5, ω_1_95=1000 and ξ_1_95=0.6 in order to put more control effort in the frequency range of interest (0∼ 100  Hz in this paper). As displayed in Fig. <ref> (d), 𝒲_95(s) has low gain in the high-frequency range to put less emphasis on shaping the high-frequency characteristics of G(s)Y(s).

In addition to the above specified weighting functions in 𝒲(s), the other entries of 𝒲(s) are set as 0 such that the ℋ_∞ problem focuses on the above design objectives.



 §.§ Constant reactive power control mode



By employing the aforementioned setting, the converter regulates its active power and internal voltage magnitude (before the virtual impedance) to their reference values (P_ ref and V_d^ ref), referred to as “PV mode” in this paper. In this mode, the reactive power is indirectly controlled by changing the reference of the internal voltage magnitude. As mentioned in Remark <ref>, the reactive power can also be directly controlled by choosing different integrals in y such that the converter directly regulates the active and reactive power to their reference values, referred to as “PQ mode”.

To be specific, the following steps are needed to change the system configuration and design ℋ_∞-optimal controller for PQ mode: a) The output signal y_2 is changed to be the integral of reactive power tracking error (i.e., Q_ ref - Q_E + w_4) instead of the voltage magnitude tracking error (d-axis component). b) Let 𝒲_11(s) = 𝒲_31(s) = 0 be such that the ℋ_∞ design does not impose voltage magnitude tracking. c) Let z_3 = y_7, 𝒲_64(s) = 𝒲_53(s) and 𝒲_34(s) = 𝒲_83(s) in order to achieve reactive power tracking and disturbance rejection, similar to the configuration for active power regulation as discussed in the previous subsection.



§ DECENTRALIZED ℋ_∞ STABILITY CERTIFICATES



In this section we present a decentralized ℋ_∞ stability criterion and include it in the ℋ_∞-control framework to ensure the stability of multi-converter systems through local design.



 §.§ Multi-device system descriptions


Consider a Kron-reduced power network that interconnects n devices, wherein the interior nodes are eliminated by assuming that the loads are constant current sources, similar to that in <cit.>. Let Y_i(s) denote the 2×2 admittance matrix of the i th device (i ∈{1,...,n}). The terminal voltage and the current output of the i th device are respectively denoted by U_i and I_i, which satisfies I_i = Y_i(s)U_i. Let U, I∈ℝ^2n be respectively the stacked voltage and current vectors of the n devices, i.e., U = [U_1^⊤ ... U_n^⊤]^⊤ and I = [I_1^⊤ ... I_n^⊤]^⊤. The block diagonal admittance matrix Y(s) =  diag(Y_1(s),...,Y_i(s),...,Y_n(s)) represents the dynamics of the n devices, which satisfies

    I =  Y(s) U .


The transmission lines of the power network are assumed to be homogeneous with identical R/L ratio. For a transmission line that connects node i and node j (i,j ∈{1,...,n},i  j), the dynamic equation in the dq frame can be expressed as

    I_ij   = B_ijF(s)(U_i-U_j) ,
    F(s)    = 1/(s+τ)^2/ω _0 + ω _0[ [    s+τ   ω _0; - ω _0    s+τ ]] ,

where I_ij is the current vector from node i to node j, U_i is the voltage at node i, B_ij = 1/(L_ij×ω _0) is the susceptance between i and j,  τ is the identical R_ij/L_ij ratio of all the lines, and ω_0 denotes the nominal angular frequency <cit.>.

Let Q_ red∈ℝ^n × n be the Kron-reduced grounded Laplacian matrix of the power network that encodes the line topology and susceptances <cit.>, calculated by

    Q_ redij   =  - B_ij, i  j ,
    Q_ redii   = ∑_j = 1^nB_ij ,

and Q_ red⊗F(s) is the corresponding admittance matrix of the network (⊗ denotes the Kronecker product), which satisfies

    I = [Q_ red⊗F(s)] U ,

or equivalently,

    U = [Q_ red⊗F(s)]^-1 I .

Here B_ii is the susceptance between node i and the grounded node, i.e., self-loop. In this paper, the grounded node is the infinite bus (in small-signal modeling), similar to that in <cit.>.



To better illustrate how the devices interact with the power network, Fig. <ref> shows a three-converter-nine-bus system that is connected to an infinite bus through the point of common coupling (the main parameters are given in Table <ref>). By using Kron reduction to eliminate the interior nodes and simplify the system modeling, an equivalent three-node network with self-loops can be obtained which interconnects the three converters (Nodes 4, 6 and 8 are remained; Nodes 5, 7, 9 and the infinite bus are eliminated; Nodes 1, 2 and 3 are included in the device dynamics) <cit.>. Note that the loads are modeled as constant current sources such that the interior nodes can be directly eliminated to simplify the analysis.
It can be seen that the input-output models of the devices and the network can be respectively represented by (<ref>) and (<ref>), which together form the closed-loop system as depicted in Fig. <ref>. Therefore, the open-loop transfer function matrix of the multi-device system can be formulated by

    L(s) = -[Q_ red⊗F(s)]^-1 Y(s) .




It can be seen that the closed-loop diagram of the multi-device system has a particular structure, that is, the device dynamics can be described by a block-diagonal matrix Y(s), and the admittance matrix can be formulated as the Kronecker product of Q_ red and F(s). In the following, we will show that the block diagonal structure of Y(s) enables decentralized stability certificates for multi-device systems.





 §.§ Decentralized ℋ_∞ stability criterion


We now present a decentralized ℋ_∞ stability criterion for multi-device systems based on the small gain theorem.






	








	The multi-device system in (<ref>) is stable if
	
    max_i F^-1(s)Y_i(s)_∞ < λ_1 ,

	where ·_∞ denotes the ℋ_∞ norm, and λ_1>0 is the smallest eigenvalue of Q_ red.
	
	
		According to the small gain theorem <cit.>, the system in (<ref>) is stable if
		
    [Q_ red⊗ F(s)]^-1 Y(s)_∞ < 1 .

		
		Further, we have
		
    [Q_ red⊗ F(s)]^-1 Y(s)_∞
    
    		=   [Q_ red⊗ℐ_2]^-1[ℐ_n ⊗ F^-1(s)] Y(s)_∞
    ≤   [Q_ red⊗ℐ_2]^-1_∞×[ℐ_n ⊗ F^-1(s)] Y(s)_∞ ,

		where ℐ_n ∈ℝ^n × n denotes the n-dimensional identity matrix.
		
		Since
		
    [Q_ red⊗ℐ_2]^-1_∞ = σ̅(Q_ red^-1) = 1/λ_1

		due to the symmetry of Q_ red, where σ̅(·) denotes the largest singular value, and
		
    [ℐ_n ⊗ F^-1(s)] Y(s)_∞ = max_i F^-1(s)Y_i(s)_∞

		due to the block-diagonal structure of Y(s), it can then be deduced that condition (<ref>) is satisfied if (<ref>) holds, indicating that the system (<ref>) is stable, which concludes the proof.
	


Based on partitioning the system into two parts, which are the power network and the combination of all the devices as illustrated in Fig. <ref>, Proposition <ref> provides a convenient way to evaluate the overall system stability by simply looking at the network structure and the local dynamics of the devices. To be specific, the system is stable if F^-1(s)Y_i(s)_∞ is smaller than λ_1 for every device (i.e., ∀ i ∈{1,...,n}), while the system is more prone to instabilities if some devices present undesired dynamics characterized by high F^-1(s)Y_i(s)_∞.

On the one hand, the transfer function matrix F^-1(s)Y_i(s) in (<ref>) is solely related to the local frequency-domain dynamics of the i th device, as Y_i(s) is the device's admittance matrix and F(s) is a fixed transfer function matrix determined by the line R/L ratio. On the other hand, λ_1 is determined by the network structure, which in fact, can be seen as the connectivity strength of the network because λ_1 can only increase when the network becomes denser <cit.>. With Proposition <ref>, there are two ways to ensure the stability of the multi-device system. One is to make the power network dense enough (by building more transmission lines to make the grid strong enough) such that λ_1 is larger than F^-1(s)Y_i(s)_∞ for every device, which however, is not practical and needs lots of investments.

The other way is to decrease F^-1(s)Y_i(s)_∞ of the devices such that it is smaller than λ_1. Note that λ_1 is a constant if the network structure remains unchanged. Once F^-1(s)Y_i(s)_∞ < λ_1 is satisfied for every device, the overall system is stable, that is, by Proposition <ref>, the stability of the multi-device system can be guaranteed in a decentralized way.
We remark that by choosing G(s)=F^-1(s) in (<ref>), F^-1(s)Y(s)_∞ can be conveniently decreased via the ℋ_∞-control design proposed in Section <ref>, as F^-1(s)Y(s) is a submatrix of P(K)(s).














































 §.§ ℋ_∞-design with decentralized stability certificates










Recall that Proposition <ref> provides a sufficient condition to evaluate the system stability by partitioning the multi-device system into the network part and the device part. In order to reduce the conservativeness of this condition, we include the grid impedance L_g (L_g consists of grid-side inductor of the LCL and line inductance) in the device part when partitioning the system, which increases λ_1 of the network part.



































In this manner, the converter's grid impedance does not have to be known exactly as it contains the line inductance. Therefore, we consider the following ℋ_∞-optimal control problem to ensure the robustness against various grid inductances

    min_K max_L_g ∈𝕃𝒲(s) ∘ P(K)(s)_∞
    
    		=   min_K max_L_g ∈𝕃max_ωσ̅[𝒲(jω) ∘ P(K)(jω)] ,

where the set 𝕃 = {L|L=0.05x,x ∈{1,...,10}}.



Given the weighting functions in Section <ref> (PV mode) and the system parameters in Table <ref>, we use the hinfstruct routine in MATLAB to solve the ℋ_∞-optimal control problem in (<ref>), and the static gain matrix K is obtained as

    K = 
       [ [  -0.01  392.7    0.1  183.3      0   38.8   -2.1;  -3.01  -67.1  -0.09  484.6      0  -62.9    4.8;  -97.7   -1.9 -134.5    2.3   55.7   -0.4   0.04 ]].




The corresponding singular values of the transfer function matrix F^-1(jω)Y(jω) (denoted by σ_1(jω) and σ_2(jω), σ_1(jω) ≥σ_2(jω)) are plotted in Fig. <ref>. It can be seen that there is no unexpected resonance peak, and for the three choices of the grid inductance, it holds that σ_1(jω)<20 dB=10, ∀ω. Moreover, σ_1(jω) has a lower magnitude in the frequency range below 100 Hz due to the choice of the weighting function in (<ref>), which enhances the robustness of the system and prevents sub-synchronous oscillations <cit.>.

Additionally, an ℋ_∞-optimal controller for PQ mode can be obtained by carrying out the three steps in Section <ref> and then solving (<ref>), which leads to the  static gain matrix

    K = 
       [ [  -1.35  -61.8   0.66  361.8      0   13.5      0;  -0.77  -46.1  -0.22  -27.2      0  -14.9  -0.02;   -0.3   -9.3 -257.5   -8.3   61.6   -2.5   0.95 ]].








































































































§ SIMULATION RESULTS



 §.§ Single-converter system




To illustrate the effectiveness of the ℋ_∞-optimal design, we now provide detailed simulation studies based on the nonlinear model of the single-converter system in Fig. <ref>. The converter parameters are given in Table <ref>, and the ℋ_∞-optimal controller has been obtained in the previous section.

Fig. <ref> shows the time-domain responses with the ℋ_∞-optimal controller applied. At t=1  s, the active power reference steps from 0 to 1.0  p.u.. It can be seen that even with different grid inductances L_g, the active power has nearly the same response (fast dynamics and no overshoot). We note that the d-axis and q-axis voltage components have different steady-state values due to the virtual inductance, because the voltage controller regulates the virtual voltage behind the virtual inductor to the reference values. The internal frequency of the converter also has fast responses and the anticipated performance obtained through the weighting functions.



Instead of directly changing control parameters to achieve different dynamic performances, the ℋ_∞-design specifies different weighting functions to achieve different performances. For example, if we want to change the tracking speed of the active power, we need to correspondingly change the shapes of W_83(s) and W_53(s) in (<ref>) and (<ref>). Fig. <ref> plots W_83(s) and W_53(s) with different bandwidths (the blue ones have the lowest bandwidths and the yellow ones have the highest bandwidths). By solving the ℋ_∞-optimal control problem in (<ref>) one obtains different ℋ_∞-optimal controller K. Fig. <ref> shows the time-domain responses of the active power when the different weighting functions are adopted, and it can be seen that all the responses have the anticipated performance (no overshoot). Moreover, increasing the bandwidths of W_83(s) and W_53(s) leads to a faster response of the active power, that is, the ℋ_∞ design provides a convenient and systematic way to achieve expected system dynamics.

We also test the transient (large-signal) performance of the ℋ_∞-optimal controller, as shown in Fig. <ref>. The grid voltage magnitude drops from 1 p.u. to 0.5 p.u. at t = 1 s and recovers to 1 p.u. at t = 2 s. The converter is disconnected from the grid at t = 4 s and reconnected to the grid at t = 5 s (the converter is stilled attached to a local load (P_ Load = 0.5  p.u.) when disconnected from the grid). The internal voltage reference V_d^ ref (before the virtual inductance) steps from 1 p.u. to 1.45 p.u. at t = 7 s, which makes the converter's voltage magnitude (at the LCL's capacitor point) change to about 1.05 p.u. because the virtual inductance takes up most of the voltage drop from the internal voltage to the grid voltage.
	


Overall, the converter has acceptable transient performance under the aforementioned severe disturbances, even without additional control switching or auxiliary loops. Note that sometimes auxiliary loops are needed to help the converter ride through faults and large disturbances, e.g., in conventional droop-controlled converters <cit.>. 
It is in fact  also possible to fix the structure of K (by forcing some elements to be zeros) before solving (<ref>) in order to make the resulting controller share the same structure with some widely-used controllers, e.g., droop controller, such that the experience on transient behavior analysis and auxiliary loop design can be inherited.





 §.§ Three-converter system



In what follows, we test the performance of the ℋ_∞-optimal controller in the three-converter system in Fig. <ref>. The grid inductances of the converters are L_g1 = L_g2 = L_g3 = 0.2  p.u. (e.g., L_g1 includes the grid-side inductance of the LCL filter of Converter 1 and the inductance in Z_14).
The Kron-reduced Laplacian matrix of the network is

    Q_ red =
    [ [ 114.55    -10 -54.55;    -10     40     -5; -54.55     -5  59.55 ]] ,

whose smallest eigenvalue is λ_1 = 21.11. It can be deduced that the condition in (<ref>) is satisfied because F^-1(s)Y(s)_∞<λ_1 as shown in Fig. <ref>, indicating that the stability of the three-converter system is guaranteed when applying the ℋ_∞-optimal controller. Although the stability condition in (<ref>) is a sufficient one, its conservativeness in practice is acceptable. For example, with L_g1 = L_g2 = L_g3 = 0.5  p.u., the three-converter system is stable if λ_1 > 15.8 dB=6.17 (deduced from Fig. <ref>). By changing the network parameters, we observed from the simulation results that the real stability boundary is about λ_1 = 5, which is remarkably close to 6.17. Moreover, the condition for λ_1 (with regards to the results in Fig. <ref>) is reasonable because the long transmission line that interconnects the device (e.g., some remote renewable base) and the grid can be included in the device side and further robustified in (<ref>). In summary, our sufficient and decentralized small-gain condition (<ref>) is remarkably tight for the considered case study (which we attribute to optimal design incorporating the condition (<ref>)). Moreover, the minor conservativeness serves as a robustness margin to model uncertainties considering the fact that the system may still be stable even if the condition is not satisfied.

Fig. <ref> plots the time-domain responses of the three-converter system with the ℋ_∞-optimal controller, which shows that the system presents the anticipated performance (fast dynamics and no overshoot) to disturbances such as changes of power reference and grid topology.
For comparison, Fig. <ref> also shows the responses when the three-converter system applies the droop controller in (<ref>) and the PLL-based controller in (<ref>). It can be seen that the droop controller has overshoots and presents slower dynamics under the changes of grid topology. The PLL-based controller has no overshoot in the responses to the power reference steps, but the system becomes unstable when L_g1 steps from 0.2  p.u. to 0.4  p.u. at t=4  s (i.e., when the grid becomes weaker), consistent with the prevailing intuition that conventional grid-following converters are stable only in strong grids.
The above results demonstrate the superiority of the ℋ_∞-optimal controller over the conventional droop controller in (<ref>) and the PLL-based controller in (<ref>).



 §.§ HIL real-time simulation results


We now illustrate the effectiveness of the ℋ_∞-optimal controller using detailed and high-fidelity HIL real-time simulations. The HIL platform is shown in Fig. <ref>, which comprises an HIL simulator (OP5700), a digital controller (NI PXIe-1071), a host computer, and an oscilloscope. The HIL simulator has CPU and FPGA resources for real-time calculations, and is equipped with inputs/outputs to communicate with the digital controller. Here we use the FPGA resource to simulate the power part of the three-converter system in Fig. <ref> in real time, which allows small step size (less than 0.5 μs) and thus high accuracy. The ℋ_∞-optimal controller of Converter 1 is implemented in the digital controller (NI PXIe-1071) to test its performance in practice, and we independently separate two CPU cores in the HIL simulator to realistically implement the ℋ_∞-optimal controllers for Converter 2 and Converter 3. Here the three converters are operated in PQ mode by employing the controller obtained in (<ref>).







Fig. <ref> and Fig. <ref> display the time-domain responses of the three-converter system obtained from the HIL real-time simulations. It can be seen from Fig. <ref> that the converters track the active and reactive power references with fast dynamics (the rising time is less than 0.1 s) and no overshoot, which again confirms and validates the effectiveness of our earlier control design and theoretical analysis. The transient performance under control mode switching (between PQ mode and PV mode) is also fast and smooth. Fig. <ref> shows the three-phase voltage and current waveform of Converter 1 at around 1 s (active power reference step) and at around 8 s (control mode switching), which demonstrates that the three-phase voltages are very well maintained under the disturbances, and the currents have fast dynamics and acceptable transient performance.








§ CONCLUSIONS


This paper proposed an ℋ_∞-control design framework for grid-connected power converters to perform robust and optimal control. Instead of tuning parameters based on eigenvalue analysis or engineering experience, the proposed ℋ_∞-control is a systematic way to achieve optimal performance in terms of the multiple control objectives of power converters. We illustrated how the converter can be specified as grid-forming or grid-following type with regards to small-signal dynamics by properly choosing the weighting functions. Moreover, we first proposed a decentralized stability criterion for multi-device systems which demonstrates how to ensure the global stability of the entire system by local control design. We further presented how this decentralized stability certificate can be included in the ℋ_∞-control design to guarantee the stability of multi-converter systems. The obtained ℋ_∞-optimal controller was tested by detailed simulations and HIL implementation of a three-converter system, which showed that the ℋ_∞-optimal controller presents the anticipated dynamic performance and robust stability against variable grid conditions.



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