A generalized descriptor-system robust H∞ control of autonomous microgrids to improve small and large signal stability considering communication delays and load nonlinearities

Electrical Power and Energy Systems 92 (2017) 63–82

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

A generalized descriptor-system robust H1 control of autonomous microgrids to improve small and large signal stability considering communication delays and load nonlinearities Hamid Reza Baghaee, Mojtaba Mirsalim ⇑, Gevork B. Gharehpetian, Heidar Ali Talebi Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 26 December 2016 Received in revised form 14 March 2017 Accepted 17 April 2017

Keywords: Microgrid Descriptor-system robust H1 control Distributed energy resources Linear matrix inequality Time-delay systems Iterative algorithm Stability

a b s t r a c t This paper presents a descriptor system H1 approach to enhance performance of robust H1 controller and previously-reported decentralized robust servo-mechanism (DRSM) control scheme for autonomous voltage sourced converter (VSC)-based microgrids including multiple distributed energy resources (DERs). The power management system specifies voltage set points for local controllers and the frequency of each DER unit is specified by a hierarchical droop-based control structure. A descriptor system H1 robust controller is designed based on a closed-loop representation of microgrid either with H1 or DRSM controller for set point tracking, disturbance rejection and improving performance of microgrid for small/large-signal disturbances and nonlinear loads. Here, unlike some of the previous researches, the load current is modeled as disturbance and also communication time-delay is considered. The theoretical concepts of proposed control strategy, including mathematical modeling of microgrid, basic theorems, and design procedure are outlines. Then, design problem is formulated by a set of linear/bilinear matrix inequalities and then solved using a new iterative algorithm in the form of convex optimization problem. To demonstrate effectiveness of the proposed control scheme, offline time-domain simulation studies are performed on a multi-DER microgrid in MATLAB/Simulink environment and also the results are experimentally verified by OPAL-RT real time digital simulator. Ó 2017 Published by Elsevier Ltd.

1. Introduction 1.1. Motivation and incitement Due to technical, environmental, and economic reasons, an increasing interest is shown by energy sector in adopting micro and smart grid technologies, such as novel control and protection techniques and advanced communications to enhance the efficiency and reliability of future electricity grids [1–3]. Microgrid is a small-scale power grid in the low voltage that must be able to locally solve energy issues and enhance the flexibility and can operate either in grid-connected or islanded (autonomous) mode of operation [4–7]. Particularly, the DER units are forming building blocks for the smart microgrids that can include renewable energy sources (RESs) and energy storage systems (ESS) [8–10]. However, sever concerns over stability, control, and efficiency of microgrids have been raised with the coexistence of multiple DER units with different dynamic features [10,11]. In this regard, several issue ⇑ Corresponding author. E-mail address: [email protected] (M. Mirsalim). http://dx.doi.org/10.1016/j.ijepes.2017.04.007 0142-0615/Ó 2017 Published by Elsevier Ltd.

regarding microgrid resiliency are concerned such as fault ride through (FRT) capability, inclusion of nonlinear and unbalanced loads, communication time-delay (CTD), power sharing and advanced microgrid power-flow methods [6–7,10–12]. 1.2. Literature review Several strategies have been reported to control the VSCs in the microgrids [6–7,10–24]. Droop-control method uses small-signal modeling, works mainly based on local measurements and almost obviates need for high-bandwidth communication [4,6,7,10–23]. However, it has several limitations such as lack of stability when load dynamics are considered, weak transient performance, and lack of black start capability [11,18,24]. Centralized control demands high-bandwidth communication and thus, failure in communication system lead to system collapse [25,26]. Also, Master-Slave control strategy has been reported which is flexible for connection/disconnection of DER unis; but, it requires dominant DER unit for suitable operation [27,28]. Robust servomechanism and multivariable controllers provides some improvements for robustness against microgrid uncertainties.

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However, it still depends on dominant DER unit and still cannot be used practically [28]. In the decentralized control that is preferred over the centralized one from reliability and redundancy points of view, each DER unit has its own local controller (LC), which is not necessarily aware of the whole microgrid system. Hierarchical control is a tradeoff between centralized and decentralized control methods. The hierarchical droop-based control (HDRC) scheme of microgrid is overally organized in primary, secondary and tertiary levels [6,7,11,13,14]. The primary control level exploits the droop control to provide power sharing between DERs and regulate voltage and frequency deviations according to the load demand. In the secondary control, the frequency and amplitude deviations are restored. The tertiary control performs power management between microgrid and upstream grid and regulates the power sharing between the microgrid and main grid at the point of common coupling (PCC) [6,7,11,13,14,24]. The secondary control can perform cooperative characteristics to restore the microgrid voltage and frequency, so that each DER unit acts as an agent, which operates together with other agents to achieve a common goal [13–15,29]. When the communication latencies are considered, the distributed secondary control performs better in contrast with the conventional central one [30,31]. The impact of CTD on secondary frequency control in a proportional integral (PI)-based centralized control structure has been studied [21]. Also, a centralized robust secondary control scheme has been developed for frequency restoration, considering variable and unknown CTD and using a phase-locked loop (PLL) to obtain frequency at bus loading [22]. Later, a distributed control structure, including primary and secondary control levels, has presented based on small-signal modeling of islanded microgrid using delay differential equations (DDEs), and by applying consensus algorithm for secondary level that exploits a data network to present CTD based on graph theory [23]. Also, by combining decentralized droop-based PI control with distributed averaging, a distributed averaging proportional integral (DAPI) controller has been proposed for secondary frequency and voltage control in islanded microgrids [32]. In the reported literature, microgrids have been studied based on small-signal modeling for normal condition or small-signal events; however, no solution has been presented to maintain the stability of the system in large-signal disturbances. Moreover, in the electrical distribution systems and especially in microgrids, load is parametrically uncertain and topologically unknown and so, it is a source of dynamics that cannot be modeled exactly [27]. However, it is assumed that load current is measurable and bounded. More importantly, in most of the reported researches [15,17,18,27,28], the communication time delay (CTD) has been ignored or at least have not been discussed in detailed. Practically, the microgrid stability margins are decreased by CTD which may lead to instability [27,33]. 1.3. Contribution and paper structure This paper presents a new add-on feature for two kinds of robust control schemes namely H1 controller and the previouslyreported DRSM controller based on a new descriptor-system H1 approach not only with considering CTD, but also by modeling the load current as disturbance (unlike the previous methods that easily modeled a constant RLC load [18]). The presented control scheme aims to enhance microgrid dynamic performance for small and large-signal disturbances, improve FRT capability and guarantee the desired power sharing, under unbalanced and nonlinear load conditions. Also, unlike [18] that uses an open-loop frequency control and synchronization scheme, here, for all LCs of DER units, a HDRC scheme provides amplitude and frequency of the sinusoidal reference signal to provide desired proper power sharing. The controller design problem based on descriptor-system H1 approach is expressed based on a delay-dependent bounded real

lemma (BRL) and then, the basic theorem is outlined to express the design problem as a bilinear matrix inequality (BMI). After that, a new iterative algorithm is proposed based on convex optimization to convert the BMI problem to a linear matrix inequality (LMI) for the sake of finding controller parameters. Finally, the effectiveness of the proposed control scheme is evaluated by offline time-domain simulation studies performed in MATLAB/SIMULINK environment and experimental real-time verification by OPAL-RT real time digital simulator (RTDS). The rest of this paper is organized as follows. Section 2 describes the structures of the microgrid, proposed control strategy and power management system (PMS). Dynamic model of the microgrid is presented in Section 3. Section 4 elaborates on the proposed controller based on the descriptor-system H1 control of microgrid system with robust H1 and modified DRSM control schemes considering CTD and presents basic lemma, controller design procedure, theorems for closed loop stability analysis of multi-DER robust H1-controlled and DRSM-controlled microgrids. Verification of the performance and viability of the proposed method, based on offline time-domain simulations and experimental real-time verifications, are presented in Section 5. Finally, conclusions and discussions are stated in Sections 6 and 7, respectively. 2. Study system In [18], a control strategy has been proposed for a typical multiDER radial microgrid system, including a general PMS, DRSMbased LC and frequency control/synchronization scheme. The independent oscillator of LC controls the microgrid frequency in an open-loop manner and all oscillators are synchronized by a common time-reference signal based on global positioning system (GPS). However, using GPS-based synchronization/frequency controller is sometimes impossible or at least cumbersome. Moreover, decentralized HDRC schemes can resolve previously drawbacks of the centralized control and conventional droop-based controllers and require only a low-bandwidth communication system. Anyway, the CTD may result in microgrid instability. Even though the open loop frequency control and synchronization scheme of [18] is used, the effect of CTD is considerable. 2.1. Structure of proposed control strategy Fig. 1 illustrates the structure of the proposed control strategy. The author aim to design controller K(s) (that is derived from robust H1-based and DRSM-based controllers of [27,33] and [18], respectively) to achieve to the desired performance, enhance system stability and moreover, minimize the effect of disturbance on the output voltages considering the effect of CTD. In [27], a two-degree of freedom (2DOF) control scheme has been presented with two different controllers for plants G and Gd while in this paper, the goal is to realize the abovementioned objectives with one controller shown in Fig. 1(a), based on a descriptor system H1 controller for an autonomous multi-DER microgrid. Here, the problem is to propose a descriptor-system H1 solution for robust H1 and DRSM controllers considering CTD, not only for robust stabilization and disturbance attenuation, but also for reference tracking and improve microgrid performance for small and large-signal disturbances and nonlinear loads. The proposed descriptor-system H1 controller is used in the inner control loop of HDRC structure to enhance the stability and track the references values (Fig. 1(b)) [11,24]. More details for the HDRC structure and its parameters are provided in [6,7,11,24]. As shown in Fig. 1(b), active and reactive powers of DER unit are passed through low-pass filters (LPFs) with cut off frequency of 10 Hz to suppress high frequency ripples.

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Proposed Robust H Controller

d

Droop-based Power Sharing Control Unit

Gd(s)

f

PDER

pDER

Reference Generaon

QDER

r

e

K(s)

G(s)

+

Droop Control Scheme

-

qDER

+

LPF fc=10 Hz

yG

|V| Communicaon Delay

a) Secondary Control E*MG

(Islanded Mode)

-

+

Gf sec s

kiF s

kpF

Terary Control

E*MGter

(Grid Connected Mode)

Gd s

1 1 Ts

Emax

GEter s

kiQg

k pQg

s

Emin

(Grid Connected Mode) (Islanded Mode)

Gfter s

s

-

+

Gf sec s

kpF

kiF s

Gd s

1 1 Ts

Gst s

+

h 1, 5, 7,11

s2

P&Q Calculaons [10]

P*

kist s

k pst

ω*

oh

2

+ -

Gv s

k rhV s

k pV h 1 , 5 , 7 , 11

i*

s2

o

h

2

Proposed Robust Control Scheme

+ - -

Three Phase Reference Generator E.sin(ωt)

abc

vviαβ

GP s

-

E

+

+

p

vc

io

vc

io

q

vc

io

vc

io io

abc

kiP s

GQ s kpQ

-

-+ P

+

c

s

Q

c

c

s

-

c

Q*fu

ioαβ

abc

k pP

E*

vcαβ

ilαβ X ÷

PG

-

-+ k rhI s

k pI

+

Complementary Control Loop

Primary/Inner Control GI s

s

v vi

R vi

io

L vi

io

v vi

R vi

io

L vi

io

abc

PWM

Measurements from Microgrid

ωsync

abc

kiPg

kpPg

ωmmin

P*G

1 1

ωmax

ω*MG

QG

Q*G

Low-Bandwidth Communicaon System

ω*MGter

+

b) Fig. 1. (a) Structure of the proposed robust controller. (b) Hierarchical control block diagram of a micro-grid consisting of two DER units and a common load [8,20].

The reference signals are produced by droop-based controllers using these average values and then, the droop controller of LC in each DER unit provides magnitude and frequency of the reference signal (r) for the voltage controller. The proposed robust controller is designed to meet following criterion [33,34]: 1. For all uncertainties in microgrid uncertain variables, the closed loop system should be asymptotically stable. 2. The impact of disturbance signal on the performance of voltage controller should be minimized. 3. The bandwidth of the closed-loop system should be sufficiently smaller than the switching frequency of VSC.

active/reactive powers of DER units and loads are transferred to the PMS and LCs via a low-bandwidth communication system to provide desired power sharing based on either DER unit cost functions and/or market signal, and moreover, control magnitude and phase angle of DERi connecting bus PCCi, respectively. Depends on the demand for active and reactive power from microgrid and predefined load sharing strategy, PDER,i and QDER,i and in addition, magnitude and angle set points of voltage for the connecting bus of each DER unit can be obtained from classical load-flow analysis as [4,7,24,36]:

PDER;i ¼

N X jV i jjV j jjY ij j cosðdi  dk  hij Þ

ð1Þ

j¼1

2.2. Power management system Because the DER units have almost three-phase three-wire systems, the zero-sequence components of the currents are zero. Hence, the state space equations of the system are obtained in the stationary reference frame (STRF) (ab coordinates) using the Clarke transformation [35]. It should be noted that instantaneous

Q DER;i ¼

N X jV i jjV j jjY ij j sinðdi  dk  hij Þ

ð2Þ

j¼1

where PDER,i, QDER,i are Active and reactive power set point of DERi Anytime that the microgrid operating conditions change, the power flow analysis shall be performed to update set point of LCs. Time

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H.R. Baghaee et al. / Electrical Power and Energy Systems 92 (2017) 63–82

interval between two consecutive update depends on variation interval of microgrid operating condition and communication media. In [7,24], a power calculation method based on radial basis function neural networks (RBFNNs) has been used in a HDRC scheme of multi-DER microgrid for calculation of the active/reactive power set points and perform as the backup of communication system (in case of any failure/interruption). In fact, the RBFNN has two complementary functionalities. The first and the main function of RBFNN is solving power-flow problem based on the algorithm mentioned in [7,24]. The second and not the least-important functionality is that when communication system is failed or interrupted, the RBFNN can approximate the related variables and produce estimated set points for the proposed control system. This function realizes a decentralized HDRC scheme that is able to provide desired power sharing. The mentioned RBFNN-based power-flow algorithm has been generalized by the authors in [4] to solve three-phase AC/DC for balanced/unbalanced microgrids including wind/solar, droop-controlled and electronically-coupled DER units.

2 3 2 A þ BK 1 C x_ 6_7 6 4g5 ¼ 4C K5C b_ y ¼ ½C

BK 2

BK 3

0

0

0

K1

32 3 2 3 0 x 76 7 6 7 54 g 5 þ 4 I 5yref 0 b

ð7Þ

0 0 v

where v = [x, g, b]T. Based on single line diagram (SLD) of microgrid shown in Fig. 2(a), when CTD is considered and also the load current is modeled as disturbance signal, dynamic equations of the study system can be written in natural reference frame (NARF) (abc coordinates) and then, transformed to ab coordinates to develop the microgrid state space realization as:

_  xðtÞ

m m X X _  giÞ ¼ F i xðt Ai xðt  hi Þ þ Bu þ Bd d i¼1

ð8Þ

i¼0

y ¼ Cx where m is number of DER units and,

xT ¼ ðV 1a ; V 1b ; I1a ; I1b ; It1a ; It1b ; V 2a ; V 2b ; I2a ; I2b ; It2a ; It2b ; V 3a ; V 3b ; I3a ; I3b Þ;

3. Mathematical model of microgrid 3.1. DRSM control In this paper, the proposed control scheme is developed based on the linearized model of the microgrid that is shown in Fig. 2 (a), in STRF. However, it can be developed in a synchronously rotating frame (SYRF) (dq coordinates) [35]. The system includes dispatchable electronically-coupled DER units used in 0.6 kV voltage level and power rating of 1.6, 1.2 and 0.8 MVA each have local load in a 13.8 kV radial microgrid. Each DER unit has 1.5 kV direct current (DC) voltage source, VSC and related series RL filter and connected to the main grid via a 0.6 kV/13.8 kV step up transformer. The parameters of the system are presented in [18]. Practically, there are not ideal communications between DER units with zero CTD. Moreover, the load is topologically unknown and parametrically uncertain that should be modeled as a disturbance. In this paper, it is assumed that the load current is measurable. Here, there are two main differences between the microgrid model developed in [18] and the mathematical model of microgrid in this paper. First, in this paper the load current is modeled as disturbance signal and second, the descriptor-system H1 controller design is performed for robust H1 and DRSM control schemes considering CTD, unlike the previous methods mentioned in [18,27,28]. In [18], the state space representation of (3) has been given for the overall multiDER microgrid system shown in Fig. 2(a).

x_ ¼ Ax þ Bi ui yi ¼ C i x

ð9Þ

uT ¼ ðV t1a ; V t1b ; V t2a ; V t2b ; V t3a ; V t3b Þ;

i ¼ 1; 2; 3 i ¼ 1; 2; 3

y ¼ ðV 1a ; V 1b ; V 2a ; V 2b ; V 3a ; V 3b Þ T

where F i and Ai  R1616 , B  R166 , Bd  R166 and C  R616 are the system state space matrices and given in Appendix A (Here, Fi = 0). When DRSM control scheme of (4)–(6) is augmented with time-delayed microgrid (8) (considering CTD), the closed loop representation is given by:

3 3 2 32 _ A0 þ BK 1 C BK 2 BK 3 xðtÞ xðtÞ 7 6_ 7 6 76 0 0 54 gðtÞ 5 4 gðtÞ 5 ¼ 4 C _ K5C bðtÞ 0 K4 bðtÞ 3 2 3 2 32 32 A2 0 0 A1 0 0 xðt  h1 Þ xðt  h2 Þ 7 6 7 6 76 76 þ 4 0 0 0 54 gðt  h1 Þ 5 þ 4 0 0 0 54 gðt  h2 Þ 5 2

0 0 bðt  h1 Þ 3 Bd   6 7 yref þ 4 I 0 5 d 0 0 2

0

0

bðt  h2 Þ

0 0

0

3 0 Bd   6 7 yref ¼ A0;cl vðtÞ þ A1;cl vðt  h1 Þ þ A2;cl vðt  h2 Þ þ 4 I 0 5 d 0 0

ð3Þ

2

y ¼ Cx ð10Þ

Because the load current has not been used as disturbance signal in (3), Ai  R2222 , B1 and B2  R86 , B3  R66 , C 1 and C 2  R68 , and C 3  R66 [18]. Also, A DRSM controller with the following structure has been proposed [18]:

b_ ¼ K 4 b þ K 5 y

ð4Þ

u ¼ K1y þ K2g þ K3b

ð5Þ

g_ ¼ y  yref

ð6Þ 1

T

d ¼ ðIL1a ; IL1b ; IL2a ; IL2b ; IL3a ; IL3b Þ;

2

blockdiag{Ki , Ki , Ki3},

where Ki = i = 1, 2, . . . , 5. In fact, (4) and (5) construct the decentralized compensator that is designed to stabilize the closed-loop system and (6) is the decentralized servo-compensator that ensures set point tracking with zero steady-state error. Augmenting (4)–(6) with plant (3) yields the DRSM-controlled closed-loop system (without CTD and also without assuming the load current as disturbance) [18].

where v(t) = [x(t), g(t), b(t)] . T

3.2. Robust H1 and mixed H2/H1 control Different control strategies have been reported for robust control of microgrid. The reported methods includes robust 2DOF [27], robust H1 [33], and robust mixed H2/H1 [37] controllers. In these types of controllers, the microgrid robust controller is designed based on the following general state-space representation:

( KðsÞ :

f_ ¼ Ak f þ Bk y u ¼ C k f þ Dk y

ð11Þ

Similar to the analysis developed in Section 3.1, when this control structure is integrated with the time-delay state-space representation of microgrid system as:

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H.R. Baghaee et al. / Electrical Power and Energy Systems 92 (2017) 63–82

Fig. 2. (a) Schematic diagram of study microgrid system [18]. (b) The OPAL-RT eMEGAsim real-time digital simulator in the lab.

_  gÞ ¼ A0 xðtÞ þ A1 xðt  h1 Þ þ A2 xðt  h2 Þ _  F xðt xðtÞ þ½ Bx1 Bx2 xðtÞ þ BuðtÞ ðtÞ ¼ C 0 xðtÞ þ ½ Dx1 Dx2 xðtÞ y z ¼ colfC 1 x; D12 ug;

4. Problem formulation

ð12Þ

xðtÞ ¼ 0 8t 6 0 T

where Bd = [Bx1 Bx2] and x = [r d] . Then, the closed loop system is given by:

       _ A1 0 xðt  h1 Þ A0 þ BDk C BC k xðtÞ xðtÞ þ ¼ _ fðtÞ 0 0 fðt  h2 Þ Bk C Ak fðtÞ      Bx1 þ BDk Dx1 Bx2 þ BDk Dx2 A2 0 xðt  h2 Þ þ xðtÞ þ Bk Dx1 Bk Dx2 0 0 fðt  h2 Þ ¼ A0;cl vðtÞ þ A1;cl vðt  h1 Þ þ A2;cl vðt  h2 Þ þ Bx xðtÞ 

ðtÞ ¼ C 0 xðtÞ þ ½ Dx1 Dx2 xðtÞ y where v(t) = [x(t), f(t)]T.

4.1. L2 Gain analysis of linear time-delay systems Lemma 1. Delay-Dependent Bounded Real Lemma (BRL) [38]: Consider the following linear time-delay system,

_  xðtÞ

m m X X _  gi Þ ¼ F i xðt Ai xðt  hi Þ þ Bd xðtÞ i¼1

i¼0

xðtÞ ¼ 0; t 2 ½h; 0 ð13Þ

zðtÞ ¼ colfC 0 xðtÞ þ DxðtÞ; C 1 xðt  h1 Þ; . . . ; C m xðt  hm Þ; C mþ1 xðt  g 1 Þ; . . . ; C 2m xðt  g m Þg n

xðtÞ 2 R ; xðtÞ 2

Lq2 ½0; 1Þ;

zðtÞ 2 R

p

h0 ¼ 0; hi > 0; g i > 0; i ¼ 1; . . . ; m h ¼ max fhi ; g i g i¼1;...;m

ð14Þ

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H.R. Baghaee et al. / Electrical Power and Energy Systems 92 (2017) 63–82

min c

For a prescribed c > 0, the cost function (performance index),

Z

1

JðxÞ ¼

ðzT z  c2 xT xÞds

ð15Þ

0

achieves for all nonzero - 2 Lq2 ½0; 1Þ and for all positive delays g1, . . . , gm, if there exist n  n matrices 0 < P1, P2, P3, Si = STi , Ui = UTi , Wii, Wi2, Wi3, Wi4 and Ri = RTii, Ri2, Ri3 = Ri3T, i = 1, . . . , m satisfying following LMI, as shown in (16), min c st: 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

W1 W2 P T2 B1 h1 U11  hm Um1 W T13 A1  W Tm3 Am P T2 F 1  PT2 F m Ce T C T0  W

T 3 P 3 B1



 c2 I













h1 U12  hm Um2

W T14 A1



0

h1 R1 

0



W Tm4 Am

P T3 F

0



0

0

0



:







: 0



















 hm Rm













































 



PT3 F m

0

0



0

0

0

0



0

0

:

:



:

:

3

7 07 7 7 DT 7 7 7 07 7 7 : 7 7 7 07 7 7 07 7 07 70 7 : 7 7 7 07 7 7 07 7 7 : 7 7 7 07 7 07 5

0



0

0



0

0

0



0

0



0

0



S1



0

0



0

0

:

:



:

:



:

:









Sm

0



0

0













U 1 

0

0





:

:



:

:



:

:

















 U m 0



:

:

















I



:

:

















 I

ð16Þ

where

! m m m m X X X X T Ai P2 þ PT2 ð Ai Þ þ ðW Ti3 Ai þ ATi W i3 Þ þ Si

W1 ¼

i¼0

i¼0

1¼1

i¼1

m m X X W2 ¼ P1  PT2 þ ð ATi ÞP3 þ ATi W i4

W3 ¼ P3  PT3 þ

i¼0 m X



Ui1 ¼ W Ti1 þ P1 W Ti3 þ PT2 ; 

Ri ¼

Ri1

Ri2

RTi2

Ri3



;

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

W1 W2 P T2 B1 h1 U11  hm Um1 W T13 A1;cl  W Tm3 Am;cl Ce T C T0  W3 P T3 B1 h1 U12  hm Um2 W T14 A1;cl  W Tm4 Am;cl 0 

 c2 I

i ¼ 1; . . . ; m;

0



0

0



0

0 0







h1 R1



0

0



0

:

:

:

:



:

:



:

:

:

:

:

:



0

0



0

0









 hm Rm

0



0

0





:

:





S1



0

0

:

:

:

:



:

:



:

0





:

:









Sm

0



:

:











I





:

:











 I ð18Þ

where (note that here, Fi = 0): m X

W1 ¼

! T

Ai;cl P 2 þ P T2

i¼0



Ui2 ¼ W Ti2 W Ti4 þ PT3



2m X K e¼ eT C C C Ti C i i¼1

4.2. Controller design: instantaneous measurement 4.2.1. DRSM control A dynamic DRSM-based controller is considered with the state space representation given by (4)–(6). In [38], using a descriptor model transformation of the system, the output-feedback controller design problem has been solved for both instantaneous and delayed measurements for continuous-time, linear, retarded and neutral type systems in terms of linear matrix inequalities (LMIs). The main focus of this paper is to propose a dynamic controller based on the state space representation of (14). By applying delay dependent BRL, the problem of finding controller K(s) is solved based on the following theorem.  2 Rr is the measureTheorem 1. Consider the system (12), where y T T e e ment vector and x = [r d] . We define R ¼ D D12 and assume that R 12

is not singular and B1 DT21 ¼ 0: For a prescribed c>0 and objective function J(x) defined by (14), the controller K(s) with the state-space representation given by (4)–(6) Achieves J(x) < 0 and for all positive delays g1, . . . , gm, and makes the closed loop system to be asymptotically stable if and only if there exist N  N matrices 0 < P1, P2, P3, Si = STi , Ui = UTi , Wi1, Wi2, Wi3, Wi4 and Ri = Ri1T, Ri2, Ri3 = Ri3T, i = 1, . . . , m satisfying the BMI problem (17),

m X Ai;cl

!

m X

þ

i¼0

T

ðW Ti3 Ai;cl þ Ai;cl W i3 Þ

i¼1

3 3 2X m m X T T T Ai þ ðBK 1 CÞ C T ðK 5 CÞ 7 Ai þ BK 1 C BK 2 BK 3 7 6 6 m 7 6 i¼0 X 7 6 i¼0 7 6 T 7 þ Si ¼ 6 7P 2 þ P 2 6 6 C T 6 0 0 7 0 0 7 5 4 i¼1 5 4 ðBK 2 Þ 0 K4 K5C 0 ðK 4 ÞT ðBK 3 ÞT 3 2X m T T ðW A þ A W Þ 0 0 i i3 i i3 7 X 6 m 7 6 i¼1 7þ þ6 Si 7 6 0 0 0 5 i¼1 4 2

0

ð17Þ

3

7 07 7 7 7 DT 7 7 7 07 7 7 : 7 7 7 07 70 07 7 7 07 7 7 07 7 7 07 7 7 07 5



i¼1

ðU i þ hi ATi Ri3 Ai Þ

i¼1



st: 2

m X

W2 ¼ P1  P T2 þ

0 0 ! ATi;cl P 3 þ

i¼0

m X

ATi;cl W i4 ¼ P 1  P T2

i¼1

3 3 2X m m X T ATi þ ðBK 1 CÞT C T ðK 5 CÞT 7 ðA W Þ 0 0 6 i4 i 7 6 7 6 i¼0 7 6 i¼1 7 6 7 þ6 7P 3 þ 6 7 6 T 7 6 0 0 0 ðBK Þ 0 0 5 4 2 5 4 T T 0 0 0 0 ðK 4 Þ ðBK 3 Þ 2

W3 ¼ P3  P T3 þ

m X i¼1



ðU i þ hi ATi Ri3 Ai Þ 



Ui1 ¼ W Ti1 þ P1 W Ti3 þ P T2 ; Ui2 ¼ W Ti2 W Ti4 þ P T3 "

Ri ¼

Ri1 Ri2 T Ri2

Ri3

#

K e¼ eT C ; i ¼ 1; . . . ; m; C

2m X



C Ti C i

1¼1

ð19Þ

Proof. For the system (12), it has been proved in [38] that, if the LMI (16) is satisfied with the same conditions mentioned in the above theorem and Lemma 1 (delay-dependent BRL), for the closed loop system including controller K(s) (with the state-space representation given by (4)–(6)), the condition J(x) < 0 is achieved for all nonzero x 2 Lq2 ½0; 1Þ and for any delay g > 0. Thus, the closed loop system (10) is used and the related closed loop matrices Ai,cl are substituted in (16) and after rearranging, the BMI (18) is easily obtained. h As it can be seen, there are some terms in w1⁄ and w2⁄ (for example the terms including the products of matrices Ki and partitions of P2 and P3) that cause (18) to be a BMI. Thus, the Section 4.4 introduces an iterative numerical algorithm to solve the BMI given by (18) by converting it to LMI.

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H.R. Baghaee et al. / Electrical Power and Energy Systems 92 (2017) 63–82

4.2.2. Robust H1 and mixed H2/H1 control For the dynamic robust controller with the state-space representation of (11) that is obtained by robust H1 and/or mixed H2/H1 control [27,33,37], by applying delay dependent BRL [38], the problem of finding controller K(s) is solved based on the following theorem.  2 Rr is the measureTheorem 2. Consider the system (13), where y T T e e ment vector and x = [r d] . We define R ¼ D D12 and assume that R 12

is not singular and B1 DT21 ¼ 0. For a prescribed c > 0 and objective function J(x) defined by (15), the controller K(s) with the state-space representation given by (11) Achieves J(x) < 0 and for all positive delays g1, . . . , gm, and makes the closed loop system to be asymptotically stable if and only if there exist N  N matrices 0 < P1, P2, P3, Si = STi , Ui = UTi , Wi1, Wi2, Wi3, Wi4 and Ri = Ri1T, Ri2, Ri3 = Ri3T, i = 1, . . . , m satisfying the BMI problem (18), where (note that again, Fi = 0): m X

W1 ¼

!

ATi;cl P 2 þ PT2

i¼0

! m m X X Ai;cl þ ðW Ti3 Ai;cl þ ATi;cl W i3 Þ i¼0

i¼1

2X 3 2X 3 m m T T T m X A þ ðBD C Þ ðB C Þ A þ BD C BC k 0 k 0 i k 0 k7 i 6 7 T6 þ Si ¼ 4 i¼0 5P2 þ P2 4 i¼0 5 i¼1

ðBC k ÞT AT 3 k m X m ðW Ti3 Ai þ ATi W i3 Þ 0 7 X 6 þ4 i¼1 Si 5þ

Bk C 0

2

Ak

0 0 ! m m X X W2 ¼ P 1  PT2 þ ATi;cl P3 þ ATi;cl W i4 ¼ P1  PT2

Bk C

Ak m X W3 ¼ P3  PT3 þ ðU i þ hi ATi Ri3 Ai Þ

U

 i1

Ri



¼ W Ti1 þ P1 "  # Ri1 Ri2

¼

T

Ri2 Ri3

i¼1 W Ti3

þ PT2



 i2

;U ¼



W Ti2

0

W Ti4

0

T

ðK 4 Þ

i¼0 3 2 T 3 Ai þ BK i C i 0 0 Ai þ ðBK i C i ÞT C Ti K 5 C Ti 6 7 6 7 Ci 0 05 þ 4 þ ðW Tj3 4 0 0 0 5W i3 Þ j¼1 K5Ci 0 0 0 0 0 m m m X X X T T  T T Ai;cl P 3 þ Ai;cl W i4 ¼ P 1  P 2 þ Si W2 ¼ P 1  P 2 þ

2

þ PT3

i¼1

ð22Þ

i¼1

m X

02 þ



m X B6 @4 j¼1

ðBK 3 Þ ATi

h

T

"

1¼1

Again, there are some terms in w1⁄ and w2⁄ (for example the terms including the products of matrices Ak, Bk, Ck, Dk and partitions of P2 and P3) that cause (18) to be a BMI that should be solved by proposed iterative numerical algorithm (presented in Section 4.4) by converting it to LMI. 4.3. Controller design: the case of delayed measurement 4.3.1. DRSM control The method of Theorem 1 can be applied to the case that the measurements in (3) and (12) are delayed as follows:

ð21Þ

Ri

¼

Ri1 Ri2 T Ri2

0

þ ðBK i C i Þ 0 0

Ri3

T

C Ti

K 5 C Ti

T

3

ðK 4 Þ 1

7 C 0 5W i4 A 0 i h i W Ti3 þ P T2 ; Ui2 ¼ W Ti2 W Ti4 þ P T3

Ui1 ¼ W Ti1 þ P1

Proof. Similar to the proof of Theorem 1, if the LMI (16) is satisfied for the system (12) with the same conditions mentioned in the above theorem and Lemma 1 (delay-dependent BRL), for the closed loop system including controller K(s) (with the state-space representation given by (13), the condition J(x)<0 is achieved for all nonzero x 2 Lq2 ½0; 1Þ and for any delay g > 0. Thus, the closed loop system (13) is used and the related closed loop matrices Ai,cl are substituted in (15) and after rearranging, the BMI (18) is easily obtained. h

i¼0

T

T T3 m m m X X X ATi þ ðBK 1 C i Þ C Ti ðK 5 C i Þ 7 6 7 6 1¼0 i¼0 i¼0 i¼0 7P 3 þ6 7 6 T 5 4 ðBK 2 Þ 0 0

2m X e T C¼ eK ; i ¼ 1; . . . ; m; C C Ti C i

m X C i xðt  hi Þ

i¼1

ðBK 3 Þ 0 2X 3 m m X A þ BK ð C Þ BK BK i 1 i 2 37 6 6 1¼0 7 i¼0 6 7 m 6 7 X T6 Ci 0 0 7 þP 2 6 7 6 7 i¼0 6 7 6 7 m X 4 5 K5ð CiÞ 0 K4

i¼1

ð20Þ

ðtÞ ¼ y

i¼0

T T3 m m m m X X X X T T A þ ðBK 1 C i Þ C i ðK 5 C i Þ 7 6 m X 7 6 1¼0 i i¼0 i¼0 i¼0 7P 2 þ Si ¼ 6 7 6 T 5 4 ðBK 2 Þ 0 0 i¼1

2

2

i¼1

2 m 3 3 m X X T T A þ BD C BC ðA W Þ 0 k k i4 6 7 6 7 i þ4 i¼0 i 5 5P3 þ 4 i¼1 2

i¼0

m X

i¼1

i¼0

Theorem 3. Consider the system (12) with the delayed measurement (21) and cost function of (15). For a prescribed c > 0 there exist a DRSM-based controller K(s) with the state-space representation given by (4)–(6) that achieves J(x) < 0 and for all positive delays g1, . . . , gm, and makes the closed loop system to be asymptotically stable if and only if there exist N  N matrices 0 < P1, P2, P3, Si = STi , Ui = UTi , Wi1, Wi2, Wi3, Wi4 and Ri = Ri1T, Ri2, Ri3 = Ri3T, i = 1, . . . , m satisfying the BMI problem (18) where (note that again, Fi = 0): ! ! m m m X X X T T  W1 ¼ Ai;cl P 2 þ P 2 Ai;cl þ ðW Ti3 Ai;cl þ ATi;cl W i3 Þ

#

0 0

e T C¼ eK ; i ¼ 1; . . . ; m; C

2m X

C Ti C i

1¼1

Proof. In this case, by replacing the delayed measurement (21) with the instantaneous measurement defined in (8), the closed loop (10) is changed as:

3 3 2 32 _ xðtÞ xðtÞ A0 þ BK 1 C 0 BK 2 BK 3 7 6_ 7 6 76 4 gðtÞ 5 ¼ 4 C0 0 0 54 gðtÞ 5 _ bðtÞ bðtÞ K 5C0 0 K4 3 2 32 xðt  h1 Þ A1 þ BK 1 C 1 0 0 7 6 76 þ4 C1 0 0 54 gðt  h1 Þ 5 2

K 5 C1

0 0

A2 þ BK 1 C 2

0 0

32

bðt  h1 Þ

3 2 0 xðt  h2 Þ 7 6 76 þ I g ðt  h Þ 4 0 0 54 2 5 0 bðt  h2 Þ 0 0

3 Bd   7 yref 0 5 C2 d 0 K 5 C1 2 3 0 Bd   6 7 yref ¼ A0;cl vðtÞ þ A1;cl vðt  h1 Þ þ A2;cl vðt  h2 Þ þ 4 I 0 5 d 0 0 m X C i xðt  hi Þ y¼ 2

6 þ4

i¼0

ð23Þ

70

H.R. Baghaee et al. / Electrical Power and Energy Systems 92 (2017) 63–82

where v(t) = [x(t), g(t), b(t)]T. Similar to the proof of the Theorem 1, for the system (12) with the delayed measurement (21), if the LMI (16) is satisfied with the same conditions mentioned in Theorems 1 and 2 and Lemma 1 (delay-dependent BRL), for the closed loop system including controller K(s) (with the state-space representation given by (4)–(6)), the condition J(x) < 0 is achieved for all nonzero x 2 Lq2 ½0; 1Þ and for any delay g > 0. Thus, the closed loop system (23) is used and the related closed loop matrices Ai,cl are substituted in (16) and after rearranging, the related BMI is easily obtained. Aain, we have some terms in w1⁄ and w2⁄ (for example the terms including the products of matrices Ki and partitions of P2 and P3) that cause (18) to be a BMI. Thus, again the iterative algorithm described in Section 4.4 is used to covert the BMI to LMI and solve it. h 4.3.2. Robust H1 and mixed H2/H1 control For Robust H1 and mixed H2/H1 control, the method of Theorem 1 is applied again to the case that the measurements in (12) are delayed as (21). Theorem 4. Consider the system (12) with the delayed measurement (21) and cost function of (15). For a prescribed c>0 there exist a Robust H1 or mixed H2/H1 control controller K(s) with the state-space representation given by (11) that achieves J(x)<0 and for all positive delays g1, . . . , gm, and makes the closed loop system to be asymptotically stable if and only if there exist N  N matrices 0 < P1, P2, P3, Si = STi , Ui = UTi , Wi1, Wi2, Wi3, Wi4 and Ri = Ri1T, Ri2, Ri3 = Ri3T, i = 1, . . . , m satisfying the BMI problem (18) where (note that again, Fi = 0): ! m m m X X X ATi;cl P 2 þ P T2 ð Ai;cl Þ þ ðW Ti3 Ai;cl þ ATi;cl W i3 Þ

W1 ¼

i¼0

i¼0

2 m 6 X 6 þ Si ¼ 6 4 i¼1

i¼1

2

T m m X X ATi þ ðBDk C 0 Þ i¼0

i¼0

ðBC k Þ

T

3 m m X X T3 m X Ai þ BDk C 0 BC k 7 6 6 i¼0 7 ðBk C 0 Þ 7 i¼0 6 7 7 T 7 i¼0 7P 2 þ P 2 6 6 7 5 m 6 7 X 4 T Bk C 0 Ak 5 Ak i¼0

2

3

m X

ðW Ti3 Ai þ ATi W i3 Þ 0 7 X m 6 6 7 þ6 i¼1 S 7þ 4 5 i¼1 i 0 0

W2 ¼ P1  PT2 þ

! m m X X ATi;cl P3 þ ATi;cl W i4 ¼ P 1  P T2 i¼0

2 6 6 6 þ6 6 6 4

i¼1

m X

m X

i¼0

i¼0

ATi þ BDk

Bk

m X C0

3

2 m 3 X T C 0 BC k 7 7 ðA W Þ 0 6 7 i4 i 7 7 7P 3 þ 6 6 i¼1 7 7 4 5 7 5 Ak 0 0

i¼0

W3 ¼ P3  PT3 þ

m X ðU i þ hi ATi Ri3 Ai Þ i¼1

 i1

U ¼

½W Ti1

2 Ri

¼4

þ P1 W Ti3 þ PT2 ; Ui2 ¼ ½W Ti2 W Ti4 þ P T3 

Ri1 Ri2 T

Ri2 Ri3

3 K eT C e¼ 5; i ¼ 1; . . . ; m; C

2m X C Ti C i 1¼1

ð24Þ

Proof. By substituting the delayed measurement (21) with the instantaneous measurement defined in (12), the closed loop (13) is changed as:

# #   "  " _ xðtÞ A0 þ BDk C 0 BC k xðtÞ A1 þ BDk C 1 0 xðt  h1 Þ ¼ þ _ fðtÞ fðtÞ Bk C 0 Ak 0 fðt  h2 Þ Bk C 1 " #    Bx1 þ BDk Dx1 Bx2 þ BDk Dx2 A þ BDk C 2 0 xðt  h2 Þ þ 2 xðtÞ þ Bk Dx1 Bk Dx2 0 fðt  h2 Þ Bk C 2 

¼ A0;cl vðtÞ þ A1;cl vðt  h1 Þ þ A2;cl vðt  h2 Þ þ Bx xðtÞ m X ðtÞ ¼ y C i xðt  hi Þ þ ½ Dx1 Dx2 xðtÞ i¼0

ð25Þ T

where v(t) = [x(t), f(t)] . Similar to the proof of the Theorem 3, again for the system (12) with the delayed measurement (21), if the LMI (16) is satisfied with the same conditions mentioned in Theorems 1 and 2 and Lemma 1 (delay-dependent BRL), for the closed loop system including controller K(s) (with the state-space representation given by (13), the condition J(x) < 0 is achieved for all nonzero x 2 Lq2 ½0; 1Þ and for any delay g > 0. Thus, the closed loop system (25) is used and the related closed loop matrices Ai,cl are substituted in (16) and after rearranging, the related BMI is easily obtained. Aain, we have some terms in w1⁄ and w2⁄ (for example the terms including the products of matrices Ki and partitions of P2 and P3) that cause (18) to be a BMI. Thus, again the iterative algorithm described in Section 4.4 is used to covert the BMI to LMI and solve it. h 4.4. Proposed iterative algorithm The proposed iterative algorithm can be easily implemented to convert the BMI (18) to LMI and solve it with a sufficient degree of accuracy. The proposed iterative algorithm is summarized in the following steps (Algorithm 1): Step 1: As an initial guess, some values are chosen for Ki matrices in DRSM controller or Ak, Bk, Ck, Dk matrices in robust H1 and/or mixed H2/H1 controller. Here, the initial values of these matrices have been chosen with the same values obtained in [18] for DRSM controller and in [27,33,37] for robust H1 and/ or mixed H2/H1 controller. Step II: Now, assuming known values for Ki matrices in DRSM controller or Ak, Bk, Ck, Dk matrices in robust H1 and/or mixed H2/H1 controller, the matrix inequality (18) becomes LMI to obtain the matrices Pi, Si, Wij, and Ri and can be solved with the efficient software tools available in MATLAB LMI toolbox or other commercial available solvers such as CVX and YALMIP [38–41]. Detailed explanation for LMIs have been provided in [42]. Step III: The obtained values for matrices Pi, Si, Wij, and Ri are substituted in (18). Now, the LMI is solved with assuming Ki matrices in DRSM controller or Ak, Bk, Ck, Dk matrices in robust H1 and/or mixed H2/H1 controller as unknowns. Step IV: The obtained values for Ki matrices in DRSM controller or Ak, Bk, Ck, Dk matrices in robust H1 and/or mixed H2/H1 controller are compared to their values in the previous step. If the Euclidean norm of the difference between values of Ki matrices in DRSM controller or Ak, Bk, Ck, Dk matrices in robust H1 and/or mixed H2/H1 controller (i = 1, . . . , 5) in two consecutive iterations is greater than the predefined value (here, e = 103), the algorithm is repeated at Step II; otherwise, the obtained values for Ki matrices in DRSM controller or Ak, Bk, Ck, Dk matrices in robust H1 and/or mixed H2/H1 controller are selected as solution matrices. Solution of the optimization problem by abovementioned algorithm (Algorithm 1), yields the following controller parameters for DRSM and robust H1 and/or mixed H2/H1 (after reduction to order 10) controllers, respectively.

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H.R. Baghaee et al. / Electrical Power and Energy Systems 92 (2017) 63–82

 K 11 ¼ K 31 ¼

8:54 0:72  4:14



14:12 6834

K 22

¼

K 13

¼ ½ 63:34

57:34





 0:47 51:29 6:25 ; K 21 ¼ ; 1:24 36:74 252:36    0:98 572:63 1763:1 ; K 12 ¼ ; 72:03 1634:4 72:64    1263 482:31 2293 ; K 32 ¼ 7412 2984 898:14  T T 2 194:91  ; K 3 ¼ 6288 1:493  105 ; T

K 33 ¼ ½ 34:96 511:03  ; K 34 ¼ 1463;

K 14 ¼ 89512;

K 15 ¼ ½ 1 806 ;

invariant systems that ‘‘The time delay adds exponential transcendentality to the characteristic equation”. It has also been established that the delayed linear system is asymptotically stable if and only if all the roots of the characteristic equation are on the left-half complex s-plane. Theoretically, there are many roots for this characteristic equation that may result in huge computation burden [43]. To analyze the impact of CTD on the microgrid control architecture, first, if the microgrid is assumed to be asymptotically stable (without delay) namely matrices A = A0 + A1 + A2 is Hurwitz matrix. The characteristic equation of (14) is then given by:

K 24 ¼ 6263;

K 25 ¼ ½ 1 9:57 ;

  Dðs; h1 ; h2 Þ ¼ det sI  A0  A1 eh1 s  A2 eh2 s

K 35 ¼ ½ 1 172 ; ð26Þ

2

306534 6 21956 6 6 6 63:071 6 6 0:155574 6 6 6 8264 Ak ¼ 6 6 0:04648 6 6 6 362:28 6 6 5965 6 6 4 823:76

21958 33:166 0:09204 9:4372 0:22411 0:00043

8270:8 35:631

0:06861 0:00048

361:6 2:3801

0:45197

0:05019

0:00002

0:09360

0

0:09621

0:18692

0:00107

0

0:05867

0:00028

0

0:00027

0:00084

0:00122

35:326

2:2489

0:00528

311:29

0:00313

63:147

335:53

52:778 0:0051

5964 770:18 30:461 3:19540 0:52360

0:00026

0:00002

0

0:00269

0:06103

0:00055

0:00325

2:4545

0:15082

0:00037

62:927

0:00066

1:9156

1214:7

21:53

30:445

1:7477

0:00419

334:55

0:00323

1214:7

422:91

59:91

79:649 100:84

19:692 22:403

5:2628 0:27832 0:00067 49:234 0:00056 21:768 692:22 3:67:9 0:21507 0:00051 59:942 0:00052 7:3302 3 2 0 2059:2 0 7:7792 0 52:344 6 0 10:441 0 1:9776 0 4:3214 7 7 6 7 6 6 0:00072 0:1442 0:00016 0:1599 0:00041 0:19832 7 7 6 6 0:32778 0:00036 0:07489 0:00035 0:09123 0:00052 7 7 6 7 6 6 0 23:9237 0 1:3639 0 3:3352 7 7 Bk ¼ 6 6 0:07670 0:00021 0:20384 0:00001 0:72345 0:00041 7 7 6 7 6 7 6 0 1:2984 0 0:07379 0 0:98765 7 6 7 6 0 18:817 0 0:02852 0 2:3213 7 6 7 6 4 0 2:7637 0 1:56697 0 3:7843 5 0 2:2696 0 0:58687 0 1:4322 2 0 0 0:0074 3:33623 0 0:2178 0 0 6 2062:4 10:609 0:00447 0:00003 23:914 0:00022 1:2813 18:828 6 6 6 0 0 0:0051 8:1524 0 0:7892 0 0 Ck ¼ 6 6 1980:2 9:5406 0:00332 0:00004 53:092 0:00028 2:0065 22:934 6 6 4 0 0 0:0053 6:15314 0 0:7542 0 0 1889:6 8:8093 0:00332

0:00005

61:921

0:00036

If these exist a controller for the mentioned robust H1 or mixed H2/H1 [27,33] and DRSM [18] control structures namely these exist matrices Ak, Bk, Ck, Dk for robust H1 or mixed H2/H1 and Kji for DRSM controllers (the existence condition for these controllers have been explained in [18,27,33]), these matrices can be used as the initial guess for the proposed iterative algorithm. If there exist such these matrices, the initial guess can converge the algorithm to a feasible solution. 5. Eigenvalue and time-delay analysis 5.1. Time-delay margin This section presents an analysis on the basis of mathematically-proved proved fact in time-delayed linear

3:0765

20:005

ð28Þ

3 262:13 2:32878 7 7 7 0:50483 7 7 0:00110 7 7 7 12:8092 7 7; 0:00012 7 7 7 3:3557 7 7 16:427 7 7 7 17:322 5 34:523

ð27Þ

0

0

3

1:10 7 7 7 0 0 7 7 3:4567 3:930 7 7 7 0 0 5

2:2389

5:0086 2:988

The system is asymptotically stable if and only if all roots of (28) are in open left-half complex plan [44]. Then, the locationof the roots of the characteristic equation is studied using the so-called Rekasius substitution [45], which is given by ess ¼ 1Ts ; s 2 Rþ ; T 2 R and 1þTs it is defined only on the imaginary axis i.e. s ¼ jx; x 2 R. This exact transformation holds if and only if

sðx; TÞ ¼

2

x

tan1 ðxTÞ þ l

p 2

;

l ¼ 0; 1; 2; . . .

ð29Þ

This equation describes an asymmetric mapping, where T is mapped into infinities s‘s for a given x On the other hand, a given and T and x correspond to a unique s. The fundamental property of this substitution is that it transforms the transcendental characteristic equation into an algebraic equation. The problem can be restated by recasting it into a simpler form as:

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H.R. Baghaee et al. / Electrical Power and Energy Systems 92 (2017) 63–82

Dðs; T 1 ; T 2 Þ ¼ detðsI  A0  A1

1  T 1s 1  T2s  A2 Þ ¼ 0: 1 þ T 1s 1 þ T2s

On the other hand, replacing s = jx and rewriting (28), we have

  Dðs; s1 ; s2 Þ ¼ det sI  A0  As1 es1 Jx  As2 es2 Jx

ð30Þ

By replacing s = jx, we obtain Dðjx; T 1 ; T 2 Þ ¼ detðjxI  A0  1 Jx 2 Jx  A2 1T Þ ¼ 0 which can be simplified as: A1 1T 1þT 1 J x 1þT 2 J x

Dðjx; T 1 ; T 2 Þ ¼ det ððjxI  A0 Þð1 þ T 1 J xÞð1 þ T 2 J xÞ  A1 ð1  T 1 JxÞð1 þ T 2 J xÞ  A2 ð1  T 2 JxÞð1 þ T 2 J xÞÞ ¼ 0

ð31Þ

for T 2 ½0; 1Þ. The corresponding relation between T and the CTD is given by (29). This transformation is useful, since on the imaginary axis (and only on the imaginary axis) (28) has a root on it if and only if (31) has a root on it [46]. Eq. (31) allows to use a Routh-Hurwitz approach to analyze the stability under time-delays. The RouthHurwitz array is then obtained using Eq. (30). Expressing D(s,T) in P k a compact form, we obtain Dðs; TÞ ¼ 2n k¼0 bk s ¼ 0, where n is the basic system order. The main idea is either to find a maximum time-delay smax such that the system is asymptotically stable for all s 2 ½0; smax , or to conclude that there is not such a time-delay. We have found that some simple power systems are very sensitive to time-delays. Through a line search optimization algorithm, we find a Tmax which is the maximum T that satisfies the constraint

jDðs; TÞj > 0

ð32Þ

which holds for all T 2 ½0; T max , for all 2

½0; 1. Also, for all

x 2 ½0; 1Þ, and for all T 2 ½0; 1Þ. A computationally efficient analysis algorithm using Rekasius’s substitution and sum of squares (RSOS) has been proposed in [46,47]. This method is briefly explained as follows. The basic idea consists in writing the problem (32) in terms of a polynomial inequality. To ensure strict positivity of jDðjx; TÞj, the strict positivity of jDðjx; TÞj2 is checked. The second step is to incorporate the T max constraints of problem (32), with the substitution T ¼ 1þu 2 and

x ¼ xmin þ v 2 , where u and v are dummy variables that are used

to incorporate the constraints and to rewrite the problem as an sum of squares (SOS) problem. Therefore, these substitutions map from u ! T : ð1; 1Þ ! ½0; T max Þ, v ! ½xmin ; 1Þ respectively. The first substitution turns the characteristic Eq. (31) into a rational polynomial. However, by multiplying again with n

ð1 þ u2 Þ we obtain a new polynomial that is a necessary condition to use SOSTOOL. The test stability analysis algorithm used, can be written as [46] (Algorithm 2): Step 1: Check using SOS techniques if



 2 ð1 þ u2 Þn D jx; T max P e 2 1þu

ð33Þ

where e is a small positive constant and holds for all x; u 2 R . Step 2: Check using SOS techniques if

jDðjðxmin þ v 2 ; T 2 ÞÞj2 P e

ð34Þ

where e is a small constant and holds for all T; v 2 R Step 3: Compute a lower bound of smax by

smax ¼

2

xmin

tan1 ðxmin T max Þ

selection of bound produces convergence of line search algorithm. This parameter is adjusted via trial and error or by using singleobjective / multi-objective optimization (SOO/MOO) algorithms [48]. In this procedure we can identify that the magnitude of depends on complexity of dynamic model (e.g. order of matrix A). In the next section we show an application of the methodology and its validation via simulation results. In this paper, the proposed descriptor system H1 controller has been designed based on a constant time delay. Practically, the delay may not be constant. However, the controller is designed based on worse case condition. The design of the descriptor H1 controller based on the variable communication time delay is the subject of future research of the authors. 5.2. Small-signal and eigenvalue analysis for droop-based control structure The main function of the proposed descriptor system H1 control scheme and also DRSM [18] and robust H1 and/or mixed H2/H1 controllers [27,33,37] is enhancing performance of the microgrid for small and large-signal disturbances. However, the small-signal modeling of the microgrid have been presented in the previous researches [15,49–52]. In [52], an integrated framework for the small-signal modeling of the islanded microgrid including DER units with the hierarchical droop-based control systems considering CTD. Here, the analytical method presented in [52] is extended for the microgrid model described in Appendix A and then the eigenvalue/small-signal analysis is provided in the next section. To avoid over-explanation, the further details is referred to [52]. However, the extension of the model presented in [52] for unbalanced and hybrid AC/DC microgrids is the subject of the future research of the authors. The small-signal model can be expressed as (36), where u(t) is the initial history function. Equation (36) belongs to the class of delay differential equation (DDE) [53]:

(

_ ¼ ADXðtÞ þ A1 DXðt  h1 Þ þ A2 DXðt  h2 Þ; t 2 ½td ; 0 DXðtÞ DXðtÞ ¼ uðtÞ; t>0

ð36Þ

where the characteristic equation for the system described in (36) is given by(28). When the system has no CTD, the abovementioned DDE given by (36) is changed to an ordinary differential equation as:

8 m X > < DXðtÞ _ ¼ ð Ai ÞDXðtÞ; t > 0 i¼1 > : DXðtÞ ¼ uðtÞ; t¼0

ð37Þ

where the u(t0) is the initial condition and the historical function is no longer necessary. 6. Simulation results In this section, a multi-DER microgrid, is simulated in the MATLAB/Simulink software environment and the results are experimentally verified by OPAL-RT RTDS (Fig. 2(b)) [53]. Parameters of the microgrid have been obtained from [18]. As shown in Fig. 2, the LCs and the rest of microgrid systems are simulated by using CPU1 and CPU2, respectively. 6.1. Case 1: Eigenvalue analysis

ð35Þ

Using the proposed algorithm (Algorithm 2), it is possible to obtain the lower bound smax as large as possible. We implement a complementary line search algorithm to obtain Tmax and xmin. Using this algorithm, Tmax is maximized and xmin is minimized. The proper

Fig. 3 shows the root locus corresponding o the numerical approximation of DDE (36) for the hierarchical droop-based control structure of [11,24] (the exterior control framework of the presented control scheme) as it is an arduous task to determine the exact values of eigenvalues in DDE systems, mainly in the case of the presented model where Ai matrices do not commute, that is,

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they are not simultaneously triangularizable. An error analysis for this numerical approach is presented in [54] for a system with an analytical solution, then it is expected that the root locus presented in Fig. 3 corresponds to a well-defined accuracy. It is noted that the system maintains stability in spite of the variation of the timedelay over the considered range. As the large time delay in communication implies a low exponential decay in the system’s answer, the low frequency modes move toward imaginary axis on the root locus graph, but they do not cross it. Fig. 4 compares the effect of the communication delay in the proposed descriptor system H1 control scheme with the case of a DRSM scheme of [18] ((without proposed descriptor H1 solution given by (26) and with the same Kji matrices mentioned in [18])). Fig. 4(a) shows the proposed descriptor system H1 control scheme eigenvalue spectrum when the communication delay is assumed to be as big as 1000 ms. As can be seen, the proposed scheme remains stable in presence of high CTD and the dominant oscillatory modes are not affected. Note that this was well expected considering that in the proposed structure, only the reference values (sent from PMS) can be affected by the CTD and thus will not compromise system stability. The advantage of the proposed descriptor system H1 control scheme over the DRSM scheme of [18] (without proposed descriptor H1 solution given by (26) and with the same Kji matrices mentioned in [18]) is better reflected in Fig. 4(b) where it is shown that instability can occur with the delay values as small as 24 ms. It

should be noted that typical communication delays can be in the order of 100–300 ms [55]; therefore, robustness against CTD is crucial to facilitate effective and reliable networked controlled systems. 6.2. Case 2: Time-delay margins Based on a model of the MG described in the previous sections, we obtain a time-delay model given by (14). For the proposed descriptor system H1 control scheme, using the previous Ai matrices and the RSOS algorithm [46–47], with e = 1  103 we obtain Tmax and xmin that correspond to the critical values. Hence, we find that Tmax = 0.924 and xmin = 0.614. These values lead to the system margin delay, which is smax = 1.682 s. This values are slightly less for the robust H1 control scheme of [27] i.e. smax = 0.354 s, i.e. smax = 0.354 s. This result is verified in simulation results that are presented in the next sections. 6.3. Case 3: Time-delay impact on the control system Fig. 5(a) and (b) shows the active and reactive power responses of the DRSM scheme [18] (without proposed descriptor H1 solution given by (26) and with the same Kji matrices mentioned in [18]) in presence of 24 ms of CTD. Note that in the DRSM scheme [18] the voltage and amplitude and angle references are directly

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generated by the PSM. As can be seen from Fig. 5, the CTD yields instability as suggested by the eigenvalue analysis shown in Fig. 4(b). However, by using the proposed descriptor system H1 control scheme as suggested Section 4, the system stability can be greatly increased. Once again note that this is due to the fact that the feedback power signals used to regulate power in each LC is directly adopted from the local measurements, and therefore unlike fully centralized methods or master-slave methods [25–28], the feedback signals can be assumed to be delay-free and therefore only power reference values will be affected by any communication delays. This implies that achieving the desired power set points will be held up for the delay period without compromising its stability. In this case, the presence of CTD would only appear as a timescale shift in the system responses as suggested by Fig. 6(a), which depicts the delayed system responses to changes in the power-sharing ratios in PMS at t = 3 s. The active power sharing ratios decided by the PMS are updated at t = 3 s. However, the reactive power-sharing ratios are not changed. As can be seen from Fig. 6(a) and (b), the CTD between PMS and the LCs result in delayed achievement of the updated objectives. Note that unlike [56], the stability of the system is not compromised due to the fact that the delay does not affect the local feedback measurements

(used by the LCs) and will only reflect on the reference signals; thus shifting the response times.

6.4. Case 4: Islanded mode – step load change Fig. 7 shows the frequency changes and power sharing curve when microgrid supplied local loads in the islanded mode and is influenced by load stepping increase and the LCs are governed by the proposed descriptor H1 solution given by (27). This load increase is happened at t = 0.8 s, so that the amounts of 0.8 pu is added to DER2, and it can be observed that this load increscent is well covered by DER units. As can be observed from Fig. 7(a), voltage magnitude and phase angle have stable and desirable situation after heavy load change in DER2 the proposed descriptor H1 solution given by (27). Also, the active and reactive power sharing curves of the microgrid in addition to the modulation index of the DER2 VSC the proposed descriptor H1 solution given by (27) and the descriptor H1 solution of DRSM given by (26) are respectively illustrated in Fig. 7(b). After a smooth transient, the power sharing curves will reach to the final value.

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6.5. Case 5: Islanded mode – motor starting In this part of the analysis, a three-phase 132-kW, 4-pole, 400-V, 50-Hz induction motor with DOL starting method under

nominal load is started in t = 0.8 s at bus 2. This issue can be studied and simulated for two different cases: 1- with the DRSM control scheme of [18] that are designed using the proposed descriptor H1 solution given by (26) and, 2-with the proposed

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c) Fig. 8. Time-domain simulation for motor starting in islanded microgrid respectively with the descriptor H1 solution of DRSM given by (26), and with the proposed descriptor H1 solution given by (27): (a) microgrid voltages, (b) microgrid active powers, and (c) microgrid reactive powers.

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descriptor system H1 robust controller given by (27). Fig. 8 illustrates the time-domain simulation results of voltages magnitude, active and reactive powers for the mentioned control schemes for motor starting in islanded microgrid. As shown in Fig. 8, using the proposed robust controller, the microgrid is stable and the voltage drops is in the standard range. In fact, the microgrid has better performance with this robust control scheme in contrast with the descriptor H1 solution of DRSM control scheme of [18] and the microgrid can face all grid traditional disturbances. It can be observed from Fig. 8(a) that after a momentarily drop, the voltage immediately returns to its nominal value. 6.6. Case 6: Islanded mode – inclusion of nonlinear load In this section, the performance of the microgrid system with respect to the inclusion of a highly nonlinear load is analyzed. At t = 0.8 s, a three-phase six-pulse diode-bridge rectifier feeding a 200 HP, 500 v, 1750 rpm shunt DC moto is connected to the bus DER2 via a step down 13.8/0.6 kV transformer while Load2 is still

connected. Fig. 9(a) and (b) shows the microgrid active and reactive power sharing curves respectively with the DRSM control scheme of [18] that are designed using the proposed descriptor H1 solution given by (26) and, with the proposed descriptor system H1 robust controller given by (27). The voltage magnitude of Bus 2 using the mentioned control structures have been illustrated in Fig. 9(c). Also, Fig. 9(d) illustrates the total harmonic distortion (THD) for voltage waveforms of buses in the microgrid when respectively the DRSM control scheme of [18] that are designed using the proposed descriptor H1 solution given by (26) and, the proposed descriptor system H1 robust controller given by (27) are exploited. It can be seen from Fig. 9(d) that using the mentioned control structures, the THD has desirable value and below 5% which is acceptable according to the IEEE standard [57]. However, it has poor transients. This issue is well covered by exploiting the descriptor system H1 robust controller given by (27). It can be seen that although the rectifier input currents are highly distorted (THD of 31%), as shown in Fig. 9(d), the control strategy provides the load with high quality voltages.

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6.7. Case 7: Islanded mode - real time verification using RTDS: three phase and singlephase to ground faults and improvement of FRT capability In the next study, although this experimental real-time verification can be done for other cases, here a large disturbance namely three-phase occurs in the middle of the line between DER1 and DER2 buses (Fig. 2(a)) and eMEGAsim OPAL-RT RTDS (Fig. 2(b)) is used to verify the results [53]. The initial conditions of microgrid are obtained by the RBFNN-based power-flow algorithm of [4,24] that has been implemented on a Xilinx Virtex 7 FPGA. The microgrid control system is simulated by the CPU1 and the rest of the microgrid is simulated by CPU2 of RTDS. The Measured microgrid operational signals are downloaded back into MATLAB for immediate plotting and analysis. The fault happen at the moment t = 0.8 s and it clears after six cycles (Fig. 10). It is seen that after fault clearing voltage become stable and power sharing continues without any problem the DRSM control scheme of [18] that are

designed using the proposed descriptor H1 solution given by (26) and, the proposed descriptor system H1 robust controller given by (27). However, using the current limiting strategy of [12] can lead to have less voltage droop based better power sharing curves.

7. Discussions The objective of this paper was to introduce a descriptor system robust H1 control strategy for the robust H1 and/or robust mixed Hi/H1 and DRSM-controlled multi-DER microgrid and evaluate their performance based on offline time-domain simulations in MATLAB/Simulink environment and experimentally validate using real-time digital verifications by OPAL-RT RTDS. Although this verification is performed for three phase to ground faults, but this verifications could be performed for other cases to evaluate performance of proposed system for small and large signal disturbances.

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8. Conclusion In this paper, a control strategy was proposed based on a descriptor system H1 robust H1 and/or robust mixed Hi/H1 and DRSM-based controller for a multi-DER VSC-based autonomous microgrid grid. This control strategy employs power management system of the overall microgrid based on classical power-flow analysis to determine the terminal voltage set points for DER units, hierarchical droop-based frequency control and synchronization for each DER unit and a local, descriptor system H1 solution of robust H1 and/or robust mixed Hi/H1 and DRSM-based controller for each DER unit and finally, global synchronization signals of DER units. A descriptor system H1 solution for the robust H1 and/or robust mixed H2/H1 and DRSM-based control system was designed for set point tracking, disturbance rejection and moreover, improve performance of microgrid for small and large signal disturbances, enhance fault ride through capability and guarantee the desired power sharing. The problem is formulated by a set of bilinear matrix inequalities and then converted to linear matrix

inequality using a new iterative algorithm to be solved as convex optimization problem. The theoretical concepts including mathematical modeling of microgrid, basic lemma, theorems and design procedure of the proposed descriptor system H1 controller was presented and finally, performance the controller was evaluated for small and large signal disturbances and nonlinear loads, based on offline time-domain simulations in the MATLAB/Simulink environment and experimentally verified in OPAL-RT real-time digital simulation. As was observed and indicated by the results, by using the proposed robust control scheme, the microgrid maintained its stability when faced with small and large signal disturbances its performance was improved for small and large signals and nonlinear loads.

Appendix A. System equations and state matrices The A matrix of (20), A 2 R1616 , is A = blockdiag{AI,AII,AIII}, where AI, AII and AIII are defined as:

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3

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1 0 x0 C11 0 0 C1 7 6 1 1 7 6 x0 0 0 0 C1 C1 7 6 7 6 Rf 1 7 6 1 0 x0 0 0 7 6 Lf 1 Lf 1 7 6 AI ¼ 6 7 1 6 0 x0 RLf 1f 1 0 0 7 7 6 Lf 1 7 6 7 6 1 Rt1 6 Lt1 0 0 0 x0 7 Lt1 5 4 Rt1 1 0 0 0 x 0 Lt1 Lt1 3 2 1 0 x0 C12 0 0 C2 7 6 1 1 7 6 x0 0 0 0 C2 C2 7 6 7 6 Rf 2 7 6 1 0 x0 0 0 7 6 Lf 2 Lf 2 7 6 AII ¼ 6 7 Rf 2 1 7 6 0 x 0 0 0 7 6 Lf 2 Lf 2 7 6 6 1 Rt2 7 6 Lt2 0 0 0 x0 7 Lt2 5 4 Rt2 1 0 0 0 x 0 Lt2 Lt2 3 2 x0 C13 0 0 7 6 6 x 1 7 0 0 0 6 C3 7 7 6 AIII ¼ 6 1 7 Rf 3 6 L 0 x 0 7 L f3 7 6 f3 5 4 Rf 3 1 0 x 0 L L f3

Non-zero

elements

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are:

A5,7 = A6,8 = 1/Lt1,

A7,5 = A8,6 = 1/C2. Non-zero elements of A2 2 R1616 are: ‘A11,13 = A12,14 = 1/Lt2 and A13,11 = A14,12 = 1/C3. Nonzero elements of B 2 R166 are B3,1 = B4,2 = 1/Lf1, B9,3 = B10,4 = 1/Lf2 and B15,5 = B16,6 = 1/Lf3. Nonzero elements of Bd 2 R166 are Bd1,1 = Bd2,2 = 1/C1, Bd 7,3 = Bd 8,4 = 1/C2 and Bd 13,5 = Bd14,6 = 1/C2. Nonzero elements of C 2 R616 that are C1,1, C2,2, C3,7, C4,8, C5,13 and C6,14 are unity. In this paper, matrices F i 2 R1616 (i = 1,2,3) are all zero. References [1] Baghaee HR, Mirsalim M, Gharehpetian GB, Talebi HA. Reliability/cost based multi-objective Pareto optimal design of stand-alone wind/PV/FC generation microgrid system. Energy 2016;115(1):1022–41. http://dx.doi.org/10.1016/j. energy.2016.09.007. [2] Olivares NDE, Mehrizi-Sani A, Etemadi AE, Canizares CA, Iravani R, Kazerani M, et al. Trends in microgrid control. IEEE Trans Smart Grid 2014;5(4):1905–19. [3] Kashefi-Kaviani A, Baghaee HR, Riahy GH. Optimal sizing of a stand-alone wind/photovoltaic generation unit using particle swarm optimization. Simulation. Int Trans Soc Model Simul 2009;85(2):89–99. http://dx.doi.org/ 10.1177/0037549708101181. [4] Baghaee HR, Mirsalim M, Gharehpetian GB, Talebi HA. Three phase AC/DC power-flow for balanced/unbalanced microgrids including wind/solar, droop controlled and electronically-coupled distributed energy resources using RBF neural networks. IET Power Electron 2017;10(3):313–28. http://dx.doi.org/ 10.1049/iet-pel.2016.0010. [5] Hatziargyriou ND, Asano H, Iravani R, Marnay C. Microgrids. IEEE Power Energ Mag 2007;5(4):78–94. [6] Baghaee HR, Mirsalim M, Gharehpetian GB. Performance Improvement of multi-DER microgrid for small and large-signal disturbances and nonlinear loads: novel complementary control loop and fuzzy controller in a hierarchical droop-based control scheme. IEEE Syst J 2016(99):1–8. http://dx.doi.org/ 10.1109/JSYST.2016.2580617. [7] Baghaee HR, Mirsalim M, Gharehpetian GB, Talebi HA. Nonlinear load sharing and voltage compensation of microgrids based on harmonic power-flow calculations using radial basis function neural networks. IEEE Syst J 2017;PP (99):1–11. http://dx.doi.org/10.1109/JSYST.2016.2645165. [8] Baghaee HR, Mirsalim M, Gharehpetian GB. Multi-objective optimal power management and sizing of a reliable wind/PV microgrid with hydrogen energy storage using MOPSO. J Intell Fuzzy Syst 2017;32(3):1753–73. http://dx.doi. org/10.3233/JIFS-152372. [9] Baghaee HR, Mirsalim M, Gharehpetian GB, Talebi HA. Application of RBF neural networks and unscented transformation in probabilistic power-flow of microgrids including correlated wind/PV units and plug-in hybrid electric vehicles. Simul Model Pract Theory 2017;72(C):51–68.

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