1

H∞-Control of Grid-Connected Converters: Design,Objectives and Decentralized Stability Certificates

Linbin Huang, Huanhai Xin, and Florian Dorfler

Abstract—The modern power system features high penetrationof power converters due to the development of renewables,HVDC, etc. Currently, the controller design and parametertuning of power converters heavily rely on rich engineering ex-perience and extrapolation from a single converter system, whichmay lead to inferior performance or even instabilities undervariable grid conditions. In this paper, we propose an H∞-controldesign framework to provide a systematic way for the robustand optimal control design of power converters. We discuss howto choose weighting functions to achieve anticipated and robustperformance 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 orgrid-following in terms of small-signal dynamics. Moreover, thispaper first proposes a decentralized stability criterion based onthe small gain theorem, which enables us to guarantee the globalsmall-signal stability of a multi-converter system through localcontrol design of the power converters. We provide high-fidelitynonlinear simulations and hardware-in-the-loop (HIL) real-timesimulations to illustrate the effectiveness of our method.

Index Terms—H-infinity control, admittance modeling, powerconverters, small gain theorem, stability, small-signal stability.

I. INTRODUCTION

The penetration rate of power-electronic converters in themodern power system is ever-increasing mainly because ofthe rapid development of renewables, energy storage systemsand high-voltage DC transmission (HVDC) systems. Thehigh controllability of power converters is making the powersystem more flexible, which allows auxiliary services such asfrequency support, voltage support and oscillation damping tobe provided for the power grid [1], [2].

Commonly, power converters facilitate multiple controlloops to achieve different control objectives. For example, ina typical grid-following converter, a phase-locked loop (PLL)is used for grid synchronization; a current control loop is usedfor current control and fast current limitation; an active powercontrol loop is used for active power tracking; and a voltagecontrol loop is used to regulate the terminal voltage [3].The corresponding control structure, i.e., how these loops are

L. Huang is with the College of Electrical Engineering at ZhejiangUniversity, Hangzhou, China, and the Department of Information Tech-nology 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. Dorfler is with the Department of Information Technology and ElectricalEngineering at ETH Zurich, Switzerland. (Email: dorfler@ethz.ch)

This research was supported in part by the 2020 Science and TechnologyProject of State Grid Power Company Limited (“Power Grid Strength Evalu-ation and Optimization to Accommodate High-penetration Renewables”) andin part by ETH Zurich Funds.

interconnected and tuned, is currently based on analysis of asingle converter together with rich engineering experience andphysical intuition. Compared with conventional synchronousgenerators, power converters generally present much morecomplex dynamics in a wide frequency range due to thesemultiple loops, thereby posing great challenges to the analysis,operation, and control of future low-inertia power grids [2].

It has been revealed that the control loops inside a con-verter are strongly coupled with each other, deteriorate theperformance and even lead to instabilities under variablegrid conditions [3]–[6]. The coupling among different loopscomplicates the parameter tuning of converters as properparameters can hardly be directly obtained from a tractablemodel. The most common method to deal with this issue isto model the whole converter system, analyze how certainranges of the parameters affect the stability of the converter(through eigenvalue loci, Nyquist diagrams, etc.), and thenpick some acceptable parameter sets to be tested in a realsystem [7]–[9]. However, this method may not lead to theoptimal parameter set in terms of damping ratio or stabilitymargin because eigenvalue loci or Nyquist diagrams can hardlydeal with multiple parameters (e.g., the converter may haveeight parameters to tune if it has four loops). It is even morechallenging to design optimal and stabilizing controller withregards to multi-converter systems, as the converters can bestrongly coupled and lead to complex interacting dynamics.

One convenient way to achieve optimal performance is touse H2/H∞-optimal control synthesis. In [10] and [11], H∞and gain scheduled H∞ controllers were used in microgridsto perform robust control. However, only a simplified modelof the converter (without current control loop) is considered,which may result in inferior performance for converters thathave a current control loop. An optimal voltage control prob-lem for islanded power converters was posed in [12] andsolved using H∞ synthesis. In [13], H∞ synthesis was used todesign the converter’s output current controller and improvethe robustness against variable grid impedance. In terms ofharmonic suppression, H∞ and repetitive control techniquescan be used for designing output voltage controllers to rejectharmonic disturbances from nonlinear loads or the public grid,as presented in [14].

The above methods will result in a dynamic controllerwhose order is the same as that of the system, which fur-ther complicates the system dynamics. It is also possibleto obtain fixed-structure fixed-order optimal controllers byH2/H∞ synthesis [15]. For example, a robust frequencycontrol was obtained in [16] through H∞ loop shaping designof a static control gain matrix to improve the grid frequency

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dynamics. A fixed-structure current controller based on H∞synthesis was developed in [17] to guarantee robust stabilityand performance for power converters. In [18], an H2-optimaltuning method was applied to optimize the PI gains in HVDCsystems. In [19], H2-optimal design is applied to allocate thevirtual inertia of grid-forming and grid-following convertersin low-inertia systems. It is noteworthy that grid-forming con-verters and grid-following converters, which present distinctdynamic behaviors, have been widely integrated in modernpower grids. It was shown in [20] that grid-forming and grid-following converters can be distinguished by their small-signalresponses to grid frequency disturbances in the frequencydomain. Based on this finding, we will analyze how to specifya converter as grid-forming or grid-following with regards tosmall-signal dynamics for the purpose of guiding the H∞-control design. Note that in this paper, akin to [20], we con-sider PLL-based converter as one prototypical grid-followingtype and droop-controlled converters as one prototypical grid-forming type.

BothH2 andH∞ optimal controllers are capable of stabiliz-ing the system. However, the existing H2/H∞-optimal designfor power converters generally considers a single-convertersystem, which can only ensure the system stability when theconverter is connected to an infinite bus, but provides noguarantee on the stability of a multi-converter system. More-over, the control objective of stable grid synchronization undervariable grid conditions was not included in the H2/H∞-optimal design in the existing literature, which may result ininferior synchronization performance and poor robustness.

In this paper, we propose a fixed-structure H∞-control de-sign framework to perform optimal, robust, and multivariablecontrol for grid-connected power converters. Our H∞-controldesign considers multiple control objectives (grid synchroniza-tion, active power and voltage regulations) simultaneously. Weelaborate on how to achieve the specified control objectives bychoosing proper weighting functions. Moreover, we show thatby choosing different weighting functions, the converter canbe conveniently specified as grid-forming or grid-followingin terms of small-signal dynamics. Furthermore, we proposea decentralized stability criterion based on the small gaintheorem, which enables to ensure the global stability of amulti-converter system through localH∞-control design of theconverters. We show that the resulting H∞-optimal controllerhas the anticipated dynamic performance, robustness againstvariable grid conditions, and it guarantees on the overallsystem stability. We illustrate our results by means of high-fidelity simulations and hardware-in-the-loop (HIL) real-timesimulations.

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

II. H∞-CONTROLLER FOR CONVERTERS

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

A. Converter system descriptions

Fig. 1 shows a three-phase power converter which is con-nected to a power grid via an LCL filter. In the controlscheme of this converter, the three-phase voltage and currentsignals are represented by two-dimensional vectors in the syn-chronously rotating dq frame (through Park transformation).Usually, the rotating frequency of this dq frame is generatedby a grid-synchronization unit, e.g., PLL, emulated swingequation or droop control. Here this frequency (denoted byω+ω0, and ω0 is the nominal value) is generated by the H∞-controller, as depicted in Fig. 1. A current control loop is usedto make the converter-side current (ICd and ICq) track theirreference values (Iref

Cd and IrefCq ) and implement fast current

limitation [21]. In the global dq frame (with constant rotatingfrequency ω0), the converter-side voltage vector is denoted byU ′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 thegrid-side voltage vector is U ′, as labeled in Fig. 1.

Let Y (s) be the converter’s admittance matrix (linearizedmodel), which is a 2×2 transfer function matrix reflecting howa perturbation from the terminal voltage affects the converter’scurrent output, i.e., I ′ = Y (s)U ′. It has been demonstrated thatthis admittance matrix dominates the stability of a convertersystem since it describes the input/output characteristics of theconverter as seen from the point of the AC grid, and for thedetailed derivation process we refer to [4], [7] or [22].

B. H∞-control design setup

The converter system in Fig. 1 (not including the H∞-controller K) can be modeled as[

zy

]=

[P11(s) P12(s)P21(s) P22(s)

] [wu

], (1)

where u is the control input of the system, y is the measuredoutput for the H∞-controller, w and z are the input/outputsignals chosen to quantify the performance of the system [23].Fig. 1 shows how the elements of w enter the system asdisturbances, and how z is chosen from the system variables.To quantify the power and voltage tracking performance, wechoose w1, w2, w3 and w4 to be the disturbance inputsimposed on the power and voltage reference signals, as shownin Fig.1. The disturbances w5 and w6 are used to incorporatethe admittance matrix in the H∞-controller for decentralizedstability certification, which will be elaborated upon in SectionIV. The disturbance signal w7 is used to specify the grid-synchronization performance of the system, as grid synchro-nization requires the angle output (i.e., θ) to reject disturbancesthat enter the closed loop.

3

AC GridThree-Phase

Power Converter

‐+PI

‐+PI

dq

abc

dq

abc

LCL

Power Part

Control Part

Current Control Loop

+

+

+

+

PWM

-

+

Power Calculation

++

+‐

+‐

+

+

+

+

+‐

+‐

+

+

+

+

Grid-Connected Converter System

+

‐

++

Fig. 1. One-line diagram of a three-phase power converter that is connected to the power grid via an LCL (LF , CF and Lg are the LCL parameters).

In this paper, u is chosen as u =[IrefCd Iref

Cq ω]>

, and y isthe vector of voltage/power signals and their integrals

y =

y1

y2

y3

y4

y5

y6

y7

=

V refd − Vd + IqXv + w1

1s (V ref

d − Vd + IqXv + w1)V refq − Vq − IdXv + w2

1s (V ref

q − Vq − IdXv + w2)Pref − PE + w3

1s (Pref − PE + w3)Qref −QE + w4

, (2)

where V refd and V ref

q are d-axis and q-axis voltage references,Vd and Vq are the d-axis and q-axis voltage components, Pref

and Qref are active and reactive power references, PE andQE are converter’s active and reactive power, and Xv is thevirtual reactance to enhance the system flexibility. In additionto the virtual impedance loop [1], other auxiliary control loops(e.g., for damping oscillations [24], improving power quality[25], etc.) can also be conveniently included in theH∞-controldesign framework by modifying the power/voltage referencesor the current control loop, which changes the model in (1)and possibly also the specification of the signals (u, y, w, z).

Remark 1 (Alternative outputs). We remark that the integralsin y determine the system’s steady state. To be specific, theintegrals of voltage and active power signals regulate thevoltage and active power to their references in steady state.The integral of reactive power is not included in y sincethe steady-state value of reactive power is determined by thepower flow when the voltage is regulated to the referencevalue. 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 someexisting voltage control loops (e.g., the ac voltage control loopin droop-controlled converters [1], [26]) in the sense that the

dq-axis components of V ′ in the controller’s rotating coor-dinate are regulated to their reference values. The frequencyof the controller’s coordinate (i.e., ω) is determined by theH∞-controller to achieve grid synchronization, which will beelaborated upon below.

The performance output vector z is chosen as

z = [z1 z2 z3 z4 z5 z6 z7 z8 z9 z10]>

=[Vd Vq y1 y3 PE QE θ y5 [I ′d I ′q]G(s)>

]>,

(3)

which quantifies the tracking and disturbance-rejection per-formance of the voltage, power and angle signals. Note thatG(s) is a 2×2 transfer matrix chosen to enable a decentralizedstability criterion, which will be elaborated in Section IV.

Note that the disturbance signals w1 ∼ w7 and the perfor-mance output vector z are chosen for specifying the controlobjectives and quantifying the system performance in the H∞control design. In other words, they are chosen in order toderive a proper transfer function matrix in (1) which describesthe response characteristics of the system. On this basis, wecan use weighting functions to shape the characteristics of thesystem and achieve certain control objectives, which will bedetailed in the next section.

Finally, we remark that the transfer function matrix in(1) can be conveniently obtained by first deriving the state-space model of the system with the inputs/outputs specifiedin Fig. 1 and then transforming this state-space model to theinput/output transfer function matrix. Another way is to derivethe transfer function matrix by incorporating the input/outputconfiguration (defined in Fig. 1) and then modifying theexisting frequency-domain models (e.g., those in [5] and [7])of power converters (including the dynamics of LCL, currentcontrol loop, etc.).

4

The H∞-controller can be formulated as

u = Ky , (4)

where K ∈ R3×7 is a static parameter matrix.

Two Noteworthy Cases

The controller in (4) can be regarded as a generalized formof PLL-based controller (grid-following type) and frequencydroop controller (grid-forming type). For example, we obtaina typical PLL-based controller [5] for the converter by setting

K=

[0 0 0 0 KPP KPI 0

−KVP −KVI 0 0 0 0 00 0 −KωP −KωI 0 0 0

], (5)

where KPP and KPI are the PI gains of the active powercontrol loop, KVP and KVI are the PI gains of the voltagecontrol loop, KωP and KωI are the PI gains of the PLL (V ref

q

can be set as zero to achieve voltage orientation).Moreover, a frequency droop controller [21], [27] (with a

cascaded voltage/current control structure) can be obtained by

K =

[KVP KVI 0 0 0 0 00 0 KVP KVI 0 0 00 0 0 0 Kf 0 0

], (6)

where Kf is the frequency droop coefficient.

C. H∞-control formulationIn the following, we discuss how K can be obtained opti-

mally by solving an H∞-optimization problem so as to makethe converter have optimal and multivariable performance.

By combining (1) and (4) we obtain the closed-loop transferfunction matrix of the system as

z ={P11(s) + P12(s)K [I − P22(s)K]

−1P21(s)

}w

∆= P (K)(s)w ,

(7)

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

The standard H∞-optimal control problem is to find astabilizing controller K by solving

minK‖W (s)P (K)(s)‖∞ = min

Kmaxω

σ [W (jω)P (K)(jω)] ,

(8)where ‖ · ‖∞ denotes the H∞ norm, W (s) is the user-defineddiagonal weighting transfer function matrix, and σ(·) denotesthe largest singular value.

However, in (8), the transfer functions from all the inputs toa particular output share the same weighting function, whichis not a suitable setting for the multi-objective design in thispaper. For example, sometimes we expect z1, which is the d-axis voltage Vd, to track a disturbance in the reference w1, andmeanwhile rejects the other disturbances in w. However, theseobjectives cannot be achieved simultaneously when z1 has thesame weighting function for all the inputs in w. In addition,standard algorithms to solve (8) result in a high-dimensionaldynamic controller with the same order as that of the system[28], which may complicate the system dynamics.

To deal with these problems, in this paper, we consider Kas a matrix of static gains to ensure the simplicity and thus

Fig. 2. Bode diagram of the sensitivity P77(K)(s). —– Droop controllerby choosing K as (6) and setting KVP = 2, KVI = 10 and Kf = 4π.—– PLL-based controller by choosing K as (5) and setting KPP = 0.5,KPI = 40, KVP = 0.5, KVI = 40, KωP = 171.8 and KωI = 14754.2.

implementability of the resulting controller. The correspondingH∞-optimal control problem can be solved with the algorithmin [29]. Moreover, in order to achieve multi-objective design,we specify the shape of P (K)(s) by solving

minK‖W(s) ◦ P (K)(s)‖∞

= minK

maxω

σ [W(jω) ◦ P (K)(jω)] ,(9)

where ◦ denotes the entrywise product of matrices, and W(s)is the weighting transfer function matrix (not necessarilydiagonal) which has the same dimension as P (K)(s). Theentrywise weighting functions in (9) enable us to specifythe shape of every entry of P (K)(s) and thus provide moreflexibility than (8). We will discuss the design of the weightingtransfer function matrix W(s) in next section.

D. Grid-forming and grid-following dynamics

The different settings for K will inevitably result in differentdynamic behaviors, e.g., making the converter behave like thegrid-following or grid-forming type in terms of small-signalfrequency/angle tracking dynamics. Fig. 2 plots the closed-loop Bode diagrams of P77(K)(s) under droop control (6)and under PLL-based control (5). Note that P77(K)(s) reflectshow 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 [23].

Observe that the PLL-based controller has a higher band-width than the droop controller, which indicates that the PLL-based controller has higher tracking speed but at the same timealso higher sensitivity to grid disturbances. These observationsare consistent with [20] analyzing the complementary sensitiv-ity and showing that grid-following converters generally havehigher control bandwidth to track the grid frequency.

III. CONTROL OBJECTIVES AND WEIGHTING FUNCTIONS

When solving the H∞-optimal control problem, the weight-ing transfer function matrixW(s) is used to specify the shapeof P (K)(s) and thus the performance objectives of the system.To be specific, the entry of W(s) in the ith row and jthcolumn, i.e., Wij(s), shapes how the ith output is affected bythe jth input, i.e., the transfer function Pij(K)(s).

5

Fig. 3. Bode diagrams of P53(K)(s) and P83(K)(s) with droop con-troller in (6). —– P53(K)(s) (i.e., complementary sensitivity function); —–P83(K)(s) (i.e., sensitivity function).

A. Brief review on weighting functions for H∞-control

The weighting functions W(s) in H∞-control design in (9)can be in fact considered as the expected upper bounds forshaping the transfer function matrix of the system P (K)(s).To illustrate this point, we explain how to choose a weightingfunction to specify reference tracking or disturbance rejectionfrom a particular input to a particular output.

To begin with, we recall the concepts of sensitivity functionand complementary sensitivity function [23]. The sensitivityfunction describes how the output rejects the disturbance fromthe input, while the complementary sensitivity function char-acterizes how the output tracks the input. Therefore, the Bodediagram of the sensitivity function is distinguished from thatof the complementary sensitivity function. For example, Fig. 3plots the Bode diagrams of P53(K)(s) and P83(K)(s) whenthe droop controller in (6) is applied. The transfer functionP53(K)(s) describes how the active power tracks its referencePref , i.e., a complementary sensitivity function. The transferfunction P83(K)(s) describes how the active power errorrejects the disturbance from Pref , i.e., a sensitivity function.It can be observed that the magnitude of the complementarysensitivity function approaches one (0dB) in the low-frequencyrange and approaches zero in the high-frequency range. Incontrast, the magnitude of the sensitivity function approachesone (0dB) in the high-frequency range and approaches zeroin the low-frequency range.

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

Step 1) Decide whether the output aims at tracking the inputor rejecting the disturbance from the input. For instance, ifthe output aims at tracking the input, then the inverse of theweighting function should be the shape of a complementarysensitivity function, i.e., the magnitude approaches one inthe low-frequency range and approaches zero in the high-frequency range.

Step 2) Decide the expected tracking/disturbance-rejectionspeed by assigning the bandwidth for the inverse of theweighting function.

For more details on choosing the weighting functions werefer to [23] and will further elaborate on the converter systembelow.

(c) (d)

(a) (b)

Fig. 4. Magnitudes of the weighting functions (only the frequency range from10−4Hz to 104Hz is plotted).

B. Specification of the weighting functions for converters

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

i) Grid synchronization: As displayed in Fig. 4 (a), wechoose W77(s) as

W77(s) =s+ s1 77

s+ s2 77(10)

with s1 77 = 0.2 and s2 77 = 0.0001 such that the internalphase θ (z7) has disturbance-rejection capability against w7.According to Fig. 2, the converter can conveniently emulategrid-forming or grid-following (small-signal) dynamics bychoosing different W77(s) to shape P77(K)(s). Note thatthe choice of W77(s) in Fig. 4 (a) will lead to grid-forming(small-signal) dynamics since the high gains appear only inthe low-frequency range (hence the bandwidth of P77(K)(s)is limited).

ii) Power regulation: The active power should track itsreference with fast dynamics, hence we choose W83(s) as

W83(s) =s+ s1 83

s+ s2 83, (11)

where s1 83 = 5 and s2 83 = 0.0005 such that the tracking er-ror is well eliminated in the low-frequency range, as displayedin Fig. 4 (b). In addition, we choose W53(s) as

W53(s) =s2/ω2

2 53 + 2ξ2 53s/ω2 53 + 1

s2/ω21 53 + 2ξ1 53s/ω1 53 + 1

(12)

so as to suppress high-frequency disturbances for the powerregulation. The magnitude ofW53(s) is displayed in Fig. 4 (b),with the parameters chosen as ω1 53 = 7× 105, ξ1 53 = 0.35,ω2 53 = 1000 and ξ2 53 = 0.8. Moreover, we choose W55(s)to be equal to W53(s) such that the active power control iscapable of rejecting disturbances from the power grid voltage.

Considering that when the terminal voltage remains con-stant, the change of active power flow also affects the steady-

6

state value of the reactive power, we choose W63(s) as

W63(s) = kQT ×s2/ω2

2 63 + 2ξ2 63s/ω2 63 + 1

s2/ω21 63 + 2ξ1 63s/ω1 63 + 1

(13)

with kQT = 16 , ω1 63 = 7× 105, ξ1 63 = 0.35, ω2 63 = 1000

and ξ2 63 = 0.8. As shown in Fig. 4 (b), W63(s) has a lowergain than W53(s) in the low-frequency range due to the factthat the change of active power reference has less impact onthe reactive power than on the active power, and the high gainof W63(s) in the high-frequency range is used to suppresshigh-frequency disturbances.

iii) Voltage regulation: The d-axis and q-axis voltage com-ponents should track their reference values with fast dynamics,hence we choose W31(s) and W42(s) to be

W31(s) =W42(s) =s+ s1 31

s+ s2 31(14)

with s1 31 = 20 and s2 31 = 10−5 in order to eliminate thetracking error in the low-frequency range, as shown in Fig. 4(c). Furthermore, W11(s) and W22(s) are chosen to be

W11(s) =W22(s) =s2/ω2

2 11 + 2ξ2 11s/ω2 11 + 1

s2/ω21 11 + 2ξ1 11s/ω1 11 + 1

(15)

with ω1 11 = 7 × 105, ξ1 11 = 0.1, ω2 11 = 7 × 103 andξ2 11 = 0.1, as displayed in Fig. 4 (c). Note that W11(s)and W22(s) provide a magnitude drop around the resonancefrequency of the LCL (1100Hz in this paper), which is usedto reduce the control design emphasis around this frequency.

iv) Admittance performance: The transfer function matrixfrom [w5 w6]> to [z9 z10]> is G(s)Y (s). This transferfunction matrix will significantly affect the system stabilityas it incorporates the admittance matrix Y (s). The choosingof G(s) will be elaborated in the following section to showhow it enforces decentralized stability. We choose the weight-ing functions for this transfer function matrix, i.e., W95(s),W96(s), W10,5(s) and W10,6(s), as

W95(s) =W96(s) =W10,5(s) =W10,6(s)

=kAd

(s2/ω21 95 + 2ξ1 95s/ω1 95 + 1)2

,(16)

where kAd = 0.5, ω1 95 = 1000 and ξ1 95 = 0.6 in orderto put more control effort in the frequency range of interest(0 ∼ 100Hz in this paper). As displayed in Fig. 4 (d), W95(s)has low gain in the high-frequency range to put less emphasison shaping the high-frequency characteristics of G(s)Y (s).

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

C. Constant reactive power control mode

By employing the aforementioned setting, the converterregulates its active power and internal voltage magnitude(before the virtual impedance) to their reference values (Pref

and V refd ), referred to as “PV mode” in this paper. In this mode,

the reactive power is indirectly controlled by changing thereference of the internal voltage magnitude. As mentioned inRemark 1, the reactive power can also be directly controlled by

choosing different integrals in y such that the converter directlyregulates the active and reactive power to their referencevalues, referred to as “PQ mode”.

To be specific, the following steps are needed to change thesystem configuration and designH∞-optimal controller for PQmode: a) The output signal y2 is changed to be the integral ofreactive power tracking error (i.e., Qref − QE + w4) insteadof the voltage magnitude tracking error (d-axis component).b) Let W11(s) = W31(s) = 0 be such that the H∞ designdoes not impose voltage magnitude tracking. c) Let z3 = y7,W64(s) =W53(s) and W34(s) =W83(s) in order to achievereactive power tracking and disturbance rejection, similar tothe configuration for active power regulation as discussed inthe previous subsection.

IV. DECENTRALIZED H∞ STABILITY CERTIFICATES

In this section we present a decentralized H∞ stabilitycriterion and include it in theH∞-control framework to ensurethe stability of multi-converter systems through local design.

A. Multi-device system descriptions

Consider a Kron-reduced power network that interconnectsn devices, wherein the interior nodes are eliminated by assum-ing that the loads are constant current sources, similar to thatin [22]. Let Yi(s) denote the 2×2 admittance matrix of the ithdevice (i ∈ {1, ..., n}). The terminal voltage and the currentoutput of the ith device are respectively denoted by Ui and Ii,which satisfies Ii = Yi(s)Ui. Let U, I ∈ R2n be respectivelythe 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(Y1(s), ..., Yi(s), ..., Yn(s))represents the dynamics of the n devices, which satisfies

I = Y(s)U . (17)

The transmission lines of the power network are assumed tobe homogeneous with identical R/L ratio. For a transmissionline that connects node i and node j (i, j ∈ {1, ..., n} , i 6= j),the dynamic equation in the dq frame can be expressed as

Iij = BijF (s)(Ui − Uj) ,

F (s) =1

(s+ τ)2/ω0 + ω0

[s+ τ ω0

−ω0 s+ τ

],

(18)

where Iij is the current vector from node i to node j, Ui isthe voltage at node i, Bij = 1/(Lij × ω0) is the susceptancebetween i and j, τ is the identical Rij/Lij ratio of all thelines, and ω0 denotes the nominal angular frequency [22].

Let Qred ∈ Rn×n be the Kron-reduced grounded Laplacianmatrix of the power network that encodes the line topologyand susceptances [30], [31], calculated by

Qredij = −Bij , i 6= j ,

Qredii =

n∑j=1

Bij ,(19)

and Qred⊗F (s) is the corresponding admittance matrix of thenetwork (⊗ denotes the Kronecker product), which satisfies

I = [Qred ⊗ F (s)]U , (20)

7

Device 1

Device 2

Device 3

Device dynamics Network dynamics

Closed‐loop dynamics

(Kron‐reduced)Frequency‐domain model

Load 1

Load 2Load 3

Converter 1 Converter 2

Converter 3

PCC

LCL LCL

LCL

1

3

24

5

6

7

89

Infinite busNetwork dynamics

Fig. 5. Closed-loop diagram of the multi-device system.

or equivalently,

U = [Qred ⊗ F (s)]−1

I . (21)

Here Bii is the susceptance between node i and the groundednode, i.e., self-loop. In this paper, the grounded node is theinfinite bus (in small-signal modeling), similar to that in [22].

To better illustrate how the devices interact with the powernetwork, Fig. 5 shows a three-converter-nine-bus system thatis connected to an infinite bus through the point of commoncoupling (the main parameters are given in Table I). By usingKron reduction to eliminate the interior nodes and simplify thesystem 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 infinitebus are eliminated; Nodes 1, 2 and 3 are included in thedevice dynamics) [22], [31]. Note that the loads are modeledas constant current sources such that the interior nodes can bedirectly eliminated to simplify the analysis. It can be seen thatthe input-output models of the devices and the network can berespectively represented by (17) and (21), which together formthe closed-loop system as depicted in Fig. 5. Therefore, theopen-loop transfer function matrix of the multi-device systemcan be formulated by

L(s) = − [Qred ⊗ F (s)]−1

Y(s) . (22)

It can be seen that the closed-loop diagram of the multi-device system has a particular structure, that is, the devicedynamics can be described by a block-diagonal matrix Y(s),and the admittance matrix can be formulated as the Kroneckerproduct of Qred and F (s). In the following, we will showthat the block diagonal structure of Y(s) enables decentralizedstability certificates for multi-device systems.

B. Decentralized H∞ stability criterion

We now present a decentralized H∞ stability criterion formulti-device systems based on the small gain theorem.

Proposition IV.1 (Stability of multi-device systems). Themulti-device system in (22) is stable if

maxi‖F−1(s)Yi(s)‖∞ < λ1 , (23)

where ‖·‖∞ denotes the H∞ norm, and λ1 > 0 is the smallesteigenvalue of Qred.

Proof. According to the small gain theorem [23], the systemin (22) is stable if

‖[Qred ⊗ F (s)]−1Y(s)‖∞ < 1 . (24)

Further, we have

‖[Qred ⊗ F (s)]−1Y(s)‖∞=‖[Qred ⊗ I2]−1[In ⊗ F−1(s)]Y(s)‖∞≤‖[Qred ⊗ I2]−1‖∞ × ‖[In ⊗ F−1(s)]Y(s)‖∞ ,

where In ∈ Rn×n denotes the n-dimensional identity matrix.Since

‖[Qred ⊗ I2]−1‖∞ = σ(Q−1red) = 1/λ1

due to the symmetry of Qred, where σ(·) denotes the largestsingular value, and

‖[In ⊗ F−1(s)]Y(s)‖∞ = maxi‖F−1(s)Yi(s)‖∞

due to the block-diagonal structure of Y(s), it can then bededuced that condition (24) is satisfied if (23) holds, indicatingthat the system (22) is stable, which concludes the proof.

Based on partitioning the system into two parts, which arethe power network and the combination of all the devices as il-lustrated in Fig. 5, Proposition IV.1 provides a convenient wayto evaluate the overall system stability by simply looking atthe network structure and the local dynamics of the devices. Tobe specific, the system is stable if ‖F−1(s)Yi(s)‖∞ is smallerthan λ1 for every device (i.e., ∀i ∈ {1, ..., n}), while thesystem is more prone to instabilities if some devices presentundesired dynamics characterized by high ‖F−1(s)Yi(s)‖∞.

On the one hand, the transfer function matrix F−1(s)Yi(s)in (23) is solely related to the local frequency-domain dynam-ics of the ith device, as Yi(s) is the device’s admittance matrixand F (s) is a fixed transfer function matrix determined by theline R/L ratio. On the other hand, λ1 is determined by thenetwork structure, which in fact, can be seen as the connectiv-ity strength of the network because λ1 can only increase whenthe network becomes denser [32]. With Proposition IV.1, there

8

TABLE IPARAMETERS OF THE POWER CONVERTER AND POWER NETWORK

Base values for per-unit calculationVoltage base value: Ub = 380 V Power base value: Sb = 50 kVA

Frequency base value: fb = 50 Hz

Parameters of the Power Part (per-unit values)Converter-side inductor: LF = 0.05 LCL capacitor: CF = 0.05

Grid-side inductor: Lg = 0.05 R/L ratio of the grid impedance: τ = 0.1

Parameters of the Control PartPI gains of the current control loop: 0.5 p.u., 10 p.u.

Voltage feedforward control: KVF = 1, TVF = 0.004 s

Virtual reactance: Xv = 0.3 p.u.

Network Parameters (per-unit values)Z14 = 0.015 + j0.15, Z49 = 0.00125 + j0.0125, Z89 = 0.0025 + j0.025

Z28 = 0.015 + j0.15, Z78 = 0.012 + j0.12, Z67 = 0.008 + j0.08

Z45 = 0.0012 + j0.012, Z56 = 0.0007 + j0.007, Z36 = 0.015 + j0.15

Z9 = 0.0005 + j0.005, PLoad1 = 0.12, PLoad2 = 0.2, PLoad3 = 0.18

are two ways to ensure the stability of the multi-device system.One is to make the power network dense enough (by buildingmore transmission lines to make the grid strong enough) suchthat λ1 is larger than ‖F−1(s)Yi(s)‖∞ for every device, whichhowever, is not practical and needs lots of investments.

The other way is to decrease ‖F−1(s)Yi(s)‖∞ of thedevices such that it is smaller than λ1. Note that λ1 is aconstant if the network structure remains unchanged. Once‖F−1(s)Yi(s)‖∞ < λ1 is satisfied for every device, the overallsystem is stable, that is, by Proposition IV.1, the stability ofthe multi-device system can be guaranteed in a decentralizedway. We remark that by choosing G(s) = F−1(s) in (3),‖F−1(s)Y (s)‖∞ can be conveniently decreased via the H∞-control design proposed in Section II, as F−1(s)Y (s) is asubmatrix of P (K)(s).

C. H∞-design with decentralized stability certificates

Recall that Proposition IV.1 provides a sufficient conditionto evaluate the system stability by partitioning the multi-devicesystem into the network part and the device part. In order toreduce the conservativeness of this condition, we include thegrid impedance Lg (Lg consists of grid-side inductor of theLCL and line inductance) in the device part when partitioningthe system, which increases λ1 of the network part.

In this manner, the converter’s grid impedance does not haveto be known exactly as it contains the line inductance. There-fore, we consider the following H∞-optimal control problemto ensure the robustness against various grid inductances

minK

maxLg∈L

‖W(s) ◦ P (K)(s)‖∞

= minK

maxLg∈L

maxω

σ [W(jω) ◦ P (K)(jω)] ,(25)

where the set L = {L|L = 0.05x, x ∈ {1, ..., 10}}.Given the weighting functions in Section III-B (PV mode)

and the system parameters in Table I, we use the hinfstructroutine in MATLAB to solve the H∞-optimal control problemin (25), 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

].

(26)

(c) (d)

(a) (b)

Fig. 6. Singular values of F−1(jω)Y (jω) with H∞-optimal controller in(26): —– Lg = 0.2 p.u., —– Lg = 0.35 p.u., —– Lg = 0.5 p.u..

0 1 2 3 4

49

50

51

52

53

0 1 2 3 4

0.00

0.25

0.50

0.75

1.00

1.00 1.02 1.04 1.06 1.08

0.75

0.80

0.85

0.90

0.95

0 1 2 3 4

-0.4

0.0

0.4

0.8

1.2

Time (s)

Active Power (p.u.)

Voltage (p.u.)

Freq

uency (Hz)

Time (s)

Time (s) Time (s)

0 1 2 3 4 5 6

0.00

0.25

0.50

0.75

1.00

0

50

100Active Power (p.u.)

Magnitude (dB)

Time (s)Frequency (Hz)

Fig. 7. Time-domain responses of the single-converter system with H∞-optimal controller: —– Lg = 0.05 p.u., —– Lg = 0.2 p.u., —– Lg =0.35 p.u., —– Lg = 0.5 p.u.

The corresponding singular values of the transfer func-tion matrix F−1(jω)Y (jω) (denoted by σ1(jω) and σ2(jω),σ1(jω) ≥ σ2(jω)) are plotted in Fig. 6. It can be seenthat there is no unexpected resonance peak, and for thethree choices of the grid inductance, it holds that σ1(jω) <20dB = 10,∀ω. Moreover, σ1(jω) has a lower magnitude inthe frequency range below 100Hz due to the choice of theweighting function in (16), which enhances the robustness ofthe system and prevents sub-synchronous oscillations [33].

Additionally, an H∞-optimal controller for PQ mode canbe obtained by carrying out the three steps in Section III-Cand then solving (25), 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

].

(27)

V. SIMULATION RESULTS

A. Single-converter system

To illustrate the effectiveness of the H∞-optimal design, wenow provide detailed simulation studies based on the nonlinearmodel of the single-converter system in Fig. 1. The converterparameters are given in Table I, and theH∞-optimal controllerhas been obtained in the previous section.

Fig. 7 shows the time-domain responses with the H∞-optimal controller applied. At t = 1 s, the active powerreference steps from 0 to 1.0 p.u.. It can be seen that even

9

0 1 2 3 4

49

50

51

52

53

0 1 2 3 4

0.00

0.25

0.50

0.75

1.00

1.00 1.02 1.04 1.06 1.08

0.75

0.80

0.85

0.90

0.95

0 1 2 3 4

-0.4

0.0

0.4

0.8

1.2

Time (s)

Active Power (p.u.)

Voltage (p.u.)

Freq

uency (Hz)

Time (s)

Time (s) Time (s)

0 1 2 3 4 5 6

0.00

0.25

0.50

0.75

1.00

0

50

100

Active Power (p.u.)

Magnitude (dB)

Time (s)Frequency (Hz)

Fig. 8. Time-domain responses of the active power with different weightingfunction designs in (11) and (12): —– s1 83 = 1.2, ω2 53 = 12.9, —–s1 83 = 5, ω2 53 = 31.6, —– s1 83 = 25, ω2 53 = 316.2.

with different grid inductances Lg , the active power has nearlythe same response (fast dynamics and no overshoot). We notethat the d-axis and q-axis voltage components have differentsteady-state values due to the virtual inductance, because thevoltage controller regulates the virtual voltage behind thevirtual inductor to the reference values. The internal frequencyof the converter also has fast responses and the anticipatedperformance obtained through the weighting functions.

Instead of directly changing control parameters to achievedifferent dynamic performances, the H∞-design specifies dif-ferent weighting functions to achieve different performances.For example, if we want to change the tracking speed of theactive power, we need to correspondingly change the shapesof W83(s) and W53(s) in (11) and (12). Fig. 8 plots W83(s)and W53(s) with different bandwidths (the blue ones havethe lowest bandwidths and the yellow ones have the highestbandwidths). By solving the H∞-optimal control problem in(25) one obtains different H∞-optimal controller K. Fig. 8shows the time-domain responses of the active power when thedifferent weighting functions are adopted, and it can be seenthat all the responses have the anticipated performance (noovershoot). Moreover, increasing the bandwidths of W83(s)and W53(s) leads to a faster response of the active power,that is, the H∞ design provides a convenient and systematicway to achieve expected system dynamics.

We also test the transient (large-signal) performance of theH∞-optimal controller, as shown in Fig. 9. The grid voltagemagnitude drops from 1 p.u. to 0.5 p.u. at t = 1s and recoversto 1 p.u. at t = 2s. The converter is disconnected from the gridat t = 4s and reconnected to the grid at t = 5s (the converteris stilled attached to a local load (PLoad = 0.5 p.u.) whendisconnected from the grid). The internal voltage referenceV refd (before the virtual inductance) steps from 1 p.u. to

1.45 p.u. at t = 7s, which makes the converter’s voltagemagnitude (at the LCL’s capacitor point) change to about1.05 p.u. because the virtual inductance takes up most of thevoltage drop from the internal voltage to the grid voltage.

Overall, the converter has acceptable transient performanceunder the aforementioned severe disturbances, even withoutadditional control switching or auxiliary loops. Note thatsometimes auxiliary loops are needed to help the converterride through faults and large disturbances, e.g., in conventionaldroop-controlled converters [26], [34]. It is in fact also possibleto fix the structure of K (by forcing some elements to be zeros)before solving (25) in order to make the resulting controllershare the same structure with some widely-used controllers,

0 1 2 3 4 5 6 7 8

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5 6 7 8

48

49

50

51

52

0 1 2 3 4 5 6 7 8

-0.5

0.0

0.5

1.0

1.5

0 1 2 3 4 5 6 7 8 9 10

48

49

50

51

52

0 1 2 3 4 5 6 7

-0.5

0.0

0.5

1.0

1.5

0 2 4 6

-0.5

0.0

0.5

1.0

1.5

0 1 2 3 4 5 6 7 8 9 10

-0.5

0.0

0.5

1.0

1.5

0 1 2 3 4 5 6 7 8 9 10

0.4

0.6

0.8

1.0

1.2

Time (s)

Time (s)

Time (s)

Active Power (p.u.)

Freq

uen

cy (Hz)

Voltage Magnitude (p.u.)

3.9 4.0 4.1 4.2 4.3 4.4 4.5

-0.5

0.0

0.5

1.0

1.5

Active Power (p.u.)

Voltage (p.u.)

0 1 2 3 4 5 6 7

-0.5

0.0

0.5

1.0

1.5

0 2 4 6

-0.5

0.0

0.5

1.0

1.5

0 1 2 3 4

-0.5

0.0

0.5

1.0

1.5

0 1 2 3 4

-0.5

0.0

0.5

1.0

1.5

3.9 4.0 4.1 4.2 4.3 4.4 4.5

0.4

0.8

1.2

1.6

Active Power (p.u.)

Active Power (p.u.)

Active Power (p.u.)

Voltage (p.u.)

Voltage (p.u.)

Voltage (p.u.)

Time (s) Time (s)

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

Time (s)

Time (s)

Time (s)

Active Power (p.u.)

Freq

uen

cy (Hz)

Voltage Magnitude (p.u.)

Fig. 9. Responses of the H∞-optimal controller under disturbances (thecurrent reference IrefCd is limited within ±1.1 p.u. by a saturation link, andIrefCq is limited within ±0.5 p.u. such that voltage support can be providedunder disturbances and the current magnitude is limited within about 1.2 p.u.).

0 1 2 3 4 5 6 7

-0.5

0.0

0.5

1.0

1.5

0 2 4 6

-0.5

0.0

0.5

1.0

1.5

0 1 2 3 4 5 6 7 8 9 10

-0.5

0.0

0.5

1.0

1.5

0 1 2 3 4 5 6 7 8 9 10

46

48

50

52

54

0 1 2 3 4 5 6 7 8 9 10

0.4

0.6

0.8

1.0

1.2

Time (s)

Time (s)

Time (s)

Active Power (p.u.)

Freq

uen

cy (Hz)

Voltage Magnitude (p.u.)

3.9 4.0 4.1 4.2 4.3 4.4 4.5

-0.5

0.0

0.5

1.0

1.5

Active Power (p.u.)

Voltage (p.u.)

0 1 2 3 4 5 6 7

-0.5

0.0

0.5

1.0

1.5

0 2 4 6

-0.5

0.0

0.5

1.0

1.5

0 1 2 3 4

-0.5

0.0

0.5

1.0

1.5

0 1 2 3 4

-0.5

0.0

0.5

1.0

1.5

3.9 4.0 4.1 4.2 4.3 4.4 4.5

0.4

0.8

1.2

1.6

Active Power (p.u.)

Active Power (p.u.)

Active Power (p.u.)

Voltage (p.u.)

Voltage (p.u.)

Voltage (p.u.)

Time (s) Time (s)

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

Fig. 10. Responses of the three-converter system: — Converter 1, —Converter 2, — Converter 3. The active power references of the three

converters step from 0 to 1.0 p.u. at t = 1 s, t = 2 s and t = 3 s,respectively. To simulate the effects of changes of grid topology, e.g., lineoutages, the inductance Lg1 steps from 0.2 p.u. to 0.4 p.u. at t = 4 s, Lg2

steps from 0.2 p.u. to 0.3 p.u. at t = 5 s, and Lg3 steps from 0.2 p.u. to0.5 p.u. at t = 6 s.

e.g., droop controller, such that the experience on transientbehavior analysis and auxiliary loop design can be inherited.

10

B. Three-converter system

In what follows, we test the performance of theH∞-optimalcontroller in the three-converter system in Fig. 5. The gridinductances of the converters are Lg1 = Lg2 = Lg3 = 0.2 p.u.(e.g., Lg1 includes the grid-side inductance of the LCL filterof Converter 1 and the inductance in Z14). The Kron-reducedLaplacian matrix of the network is

Qred =

[114.55 −10 −54.55

−10 40 −5

−54.55 −5 59.55

],

whose smallest eigenvalue is λ1 = 21.11. It can bededuced that the condition in (23) is satisfied because‖F−1(s)Y (s)‖∞ < λ1 as shown in Fig. 6, indicating thatthe stability of the three-converter system is guaranteed whenapplying the H∞-optimal controller. Although the stabilitycondition in (23) is a sufficient one, its conservativeness inpractice is acceptable. For example, with Lg1 = Lg2 =Lg3 = 0.5 p.u., the three-converter system is stable ifλ1 > 15.8dB = 6.17 (deduced from Fig. 6). By changingthe network parameters, we observed from the simulationresults that the real stability boundary is about λ1 = 5, whichis remarkably close to 6.17. Moreover, the condition for λ1

(with regards to the results in Fig. 6) is reasonable becausethe long transmission line that interconnects the device (e.g.,some remote renewable base) and the grid can be included inthe device side and further robustified in (25). In summary,our sufficient and decentralized small-gain condition (23) isremarkably tight for the considered case study (which weattribute to optimal design incorporating the condition (23)).Moreover, the minor conservativeness serves as a robustnessmargin to model uncertainties considering the fact that thesystem may still be stable even if the condition is not satisfied.

Fig. 10 plots the time-domain responses of the three-converter system with the H∞-optimal controller, whichshows that the system presents the anticipated performance(fast dynamics and no overshoot) to disturbances such aschanges of power reference and grid topology. For comparison,Fig. 10 also shows the responses when the three-convertersystem applies the droop controller in (6) and the PLL-basedcontroller in (5). It can be seen that the droop controller hasovershoots and presents slower dynamics under the changesof grid topology. The PLL-based controller has no overshootin the responses to the power reference steps, but the systembecomes unstable when Lg1 steps from 0.2 p.u. to 0.4 p.u.at t = 4 s (i.e., when the grid becomes weaker), consistentwith the prevailing intuition that conventional grid-followingconverters are stable only in strong grids. The above resultsdemonstrate the superiority of the H∞-optimal controller overthe conventional droop controller in (6) and the PLL-basedcontroller in (5).

C. HIL real-time simulation results

We now illustrate the effectiveness of the H∞-optimal con-troller using detailed and high-fidelity HIL real-time simula-tions. The HIL platform is shown in Fig. 11, which comprisesan HIL simulator (OP5700), a digital controller (NI PXIe-1071), a host computer, and an oscilloscope. The HIL simu-

Fig. 11. HIL real-time simulation platform.

Fig. 12. Time-domain responses of the three-converter system. The activepower reference of Converter 1 (2, 3) steps from 1 p.u. to 0 at t = 1 s(t = 2 s, t = 3 s) and back to 1 p.u. at t = 5 s (t = 6 s, t = 7 s). Thereactive power reference of Converter 1 steps from 0 to 0.5 p.u. at t = 2 sand back to 0 at t = 4 s. Converter 1 is switched from PQ mode to PV modeat t = 8 s and back to PQ mode at t = 9 s.

lator has CPU and FPGA resources for real-time calculations,and is equipped with inputs/outputs to communicate with thedigital controller. Here we use the FPGA resource to simulatethe power part of the three-converter system in Fig. 5 in realtime, which allows small step size (less than 0.5 µs) and thushigh accuracy. The H∞-optimal controller of Converter 1 isimplemented in the digital controller (NI PXIe-1071) to test itsperformance in practice, and we independently separate twoCPU cores in the HIL simulator to realistically implement theH∞-optimal controllers for Converter 2 and Converter 3. Herethe three converters are operated in PQ mode by employingthe controller obtained in (27).

Fig. 12 and Fig. 13 display the time-domain responses ofthe three-converter system obtained from the HIL real-timesimulations. It can be seen from Fig. 12 that the converterstrack the active and reactive power references with fast dy-namics (the rising time is less than 0.1 s) and no overshoot,which again confirms and validates the effectiveness of ourearlier control design and theoretical analysis. The transientperformance under control mode switching (between PQ modeand PV mode) is also fast and smooth. Fig. 13 shows the three-phase voltage and current waveform of Converter 1 at around1 s (active power reference step) and at around 8 s (controlmode switching), which demonstrates that the three-phasevoltages are very well maintained under the disturbances,and the currents have fast dynamics and acceptable transientperformance.

11

Fig. 13. Three-phase voltage and current waveform of Converter 1. (a) Voltagewaveform at around 1 s. (b) Current waveform at around 1 s. (c) Voltagewaveform at around 8 s. (d) Current waveform at around 8 s.

VI. CONCLUSIONS

This paper proposed an H∞-control design framework forgrid-connected power converters to perform robust and optimalcontrol. Instead of tuning parameters based on eigenvalueanalysis or engineering experience, the proposed H∞-controlis a systematic way to achieve optimal performance in termsof the multiple control objectives of power converters. Weillustrated how the converter can be specified as grid-formingor grid-following type with regards to small-signal dynamicsby properly choosing the weighting functions. Moreover, wefirst proposed a decentralized stability criterion for multi-device systems which demonstrates how to ensure the globalstability of the entire system by local control design. We fur-ther presented how this decentralized stability certificate can beincluded in the H∞-control design to guarantee the stability ofmulti-converter systems. The obtained H∞-optimal controllerwas tested by detailed simulations and HIL implementation ofa three-converter system, which showed that the H∞-optimalcontroller presents the anticipated dynamic performance androbust stability against variable grid conditions.

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