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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 4, OCTOBER 2012 1843

A Decentralized Robust Control Strategy forMulti-DER MicrogridsPart I:

Fundamental ConceptsAmir H. Etemadi, Student Member, IEEE, Edward J. Davison, Life Fellow, IEEE, and Reza Iravani, Fellow, IEEE

AbstractThis paper presents fundamental concepts of acentral power-management system (PMS) and a decentralized,robust control strategy for autonomous mode of operation of a mi-

crogrid that includes multiple distributed energy resource (DER)

units. The DER units are interfaced to the utility grid throughvoltage-sourced converters (VSCs). The frequency of each DERunit is specified by its independent internal oscillator and alloscillators are synchronized by a common time-reference signalreceived from a global positioning system. The PMS specifies

the voltage set points for the local controllers. A linear, time-in-

variant, multivariable, robust, decentralized, servomechanismcontrol system is designed to track the set points. Each controlagent guarantees fast tracking, zero steady-state error, and robustperformance despite uncertainties of the microgrid parameter,

topology, and the operating point. The theoretical concept of theproposed control strategy, including the existence conditions,

design of the controller, robust stability analysis of the closed-loopsystem, time-delay tolerance, tolerance to high-frequency effectsand its gain-margins, are presented in this Part I paper. Part IIreports on the performance of the control strategy based on digital

time-domain simulation and hardware-in-the-loop case studies.

Index TermsAutonomous mode of operation, decentralizedcontrol, microgrid, robust control.

I. INTRODUCTION

T ECHNICAL and economical viability of the distributedenergy resource (DER) technologies for distributionvoltage class applications have resulted in the emergence of

the microgrid concept [1], [2]. Impacts of the DER units on the

host microgrid and their control, protection, and management

requirements for successful operation of the microgrid have

been extensively reported in the technical literature [3][7].

However, the anticipated high-depth of penetration of DERunits in the microgrid necessitates a systematic and compre-

hensive approach to their integration. This stems from the need

to 1) enable the microgrid to operate in the grid-connected

mode, the islanded mode, and the virtual power plant (VPP)

mode; 2) respond to the external commands (e.g., market

Manuscript received June 06, 2011; revised November 04, 2011, February29, 2012; accepted May 27, 2012. Date of publication July 13, 2012; date ofcurrent version September 19, 2012. Paper no. TPWRD-00490-2011.

The authors are with the Department of Electrical and Computer Engi-neering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TPWRD.2012.2202920

signals); 3) accommodate the microgrid inherent unbalanced

conditions, uncertainties in parameters and topology, and

frequent load/generation changes; and 4) make provisions for

demand-side integration.

The existing/reported DER control strategies of a microgrid

are as follows.

Droop-based methods [7][19]: The main advantage of

the droop-based approach is that it obviates the need for

communication and operates based on local measurements.However, it presents several limitations [20][22]: 1) poor

transient performance; 2) lack of robustness due to in-

ability to account for load dynamics; and 3) inherent lack

of black-start capability.

Centralized-control methods [23], [24]: This approach re-

lies on fast and high-bandwidth communication, and the

communication failure can lead to the system collapse.

Master/slave control method [25]: This method is flexible

in terms of connection and disconnection of DER units;

however, it requires a dominant DER unit for satisfactory

operation.

Robust servomechanism control method [26]: Althoughthis method is robust to microgrid parametric uncertain-

ties, it is not readily applicable to multiple DER units.

This paper presents a power-management system (PMS) and

a control strategy for an islanded multi-DER microgrid to over-

come the drawbacks of the existing approaches. Based on the

proposed strategy: 1) the PMS specifies voltage set points for

each voltage-controlled bus, based on a fast power flow anal-

ysis; 2) local voltage controllers (LCs) provide tracking of the

voltage set points; and 3) an open-loop frequency control and

synchronization scheme maintains system frequency.

The prominent features of the proposed strategy are the fol-

lowing: 1) The PMS precisely controls powerflow of thesystem

and achieves a prescribed load sharing among the DER units;

2) LCs track corresponding voltage set points and rapidly re-

ject disturbances; 3) LCs are highly robust to parametric, topo-

logical, and unmodeled uncertainties of the microgrid; 4) LCs

are implemented in a decentralized manner; this obviates the

need for a high-bandwidth communication medium to feed the

systems information to a central authority and makes it scal-

able for larger number of DER units; 5) LCs enable the system

to sudden connection/disconnection of DER units; and 6) fre-

quency of the system is fixed and cannot deviate due to tran-

sients. The proposed approach requires low-bandwidth commu-

nication for both synchronization and PMS data transmission.

Furthermore, temporary failure of communication will not lead

0885-8977/$31.00 2012 IEEE

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ETEMADI et al.: DECENTRALIZED ROBUST CONTROL STRATEGY FOR MULTI-DER MICROGRIDSPART I 1845

Fig. 2. Angle waveform generated by a crystal oscillator.

reference signal provided by a GPS, Fig. 1(b). These entities are

further described in Sections II-B1, II-B2, and II-B3.

1) Power-Management System: The main function of the

PMS is to provide load sharing among DER units based on ei-

ther a cost function associated with each DER unit or a market

signal. In the proposed method, voltage angle and magnitude

of each DER PC bus are directly controlled; thus, the ac-

tive/reactive power injection by at the th PC-bus is

(1)

(2)

Equations (1) and (2) indicate that powerflow of the system

is determined based on the voltage magnitude and angle of

, and [Fig. 1(a)]. The power-management

process should be performed frequently to maintain the pre-

specified load sharing scheme among the DER units as the

microgrid operating point changes (e.g., due to load changes).

The time-interval between setpoint updates depends on the rate

of microgrid operating point variation.2) Frequency Control and Synchronization: The microgrid

frequency is controlled in an open-loop manner. The LC of

each DER unit includes a crystal oscillator which generates

, where and is the nominal power

frequency of the microgrid. Fig. 2 illustrates the angle waveform

deduced from the oscillator of which is usedfor the

to transformation of the mathematical model.

Based on the proposed control strategy, all DERunits are syn-

chronized by a global synchronization signal that is communi-

cated to the crystal oscillators of DER units [Fig. 1(b)] through

a GPS [27]. The global synchronization signal is communicated

at relatively large time intervals (e.g., once per second) to 1) pre-vent drift among local oscillators and 2) to initialize incoming

DER units. Crystal oscillators with high accuracies (e.g., an

error of to seconds per year) and rela-

tively low costs are currently available [28]. All LCs can be syn-

chronized with a high degree of reliability of a common timing

source of the GPS radio clock (e.g., with a theoretical accuracy

of higher than 1 s [29]). Although there are 610 satellites vis-

ible to each area at all times, one can rely on the accuracy of

crystal oscillators in case of the unavailability of the synchro-

nizing signal.

3) Local Controllers: The LC of each DER unit tracks the

setpoints specified by the PMS and rejects disturbances. Each

LC measures the voltage of thecorresponding PC bus, and trans-

forms the three-phase voltage to the frame based on the

Fig. 3. Block diagram of the th local control agent.

angle signal generated by its internal oscillator and syn-

chronized with the time-reference signal received from a GPS.The voltage magnitude and angle setpoints received from the

PMS are also transformed to the frame to generate

-based reference values (i.e., and

). The measured and reference values are

provided to the LC to determine the voltage components of

the terminal of the corresponding DER unit (i.e., and )

and generate the terminal voltage at the high-voltage

side of the transformer. is divided by the turn ratio

of the transformer and shifted by 30 to obtain corre-

sponding to the low-voltage side of the transformer.

is then fed to the PWM signal generator of the interface VSC

of the DER unit. It should be noted that although the realisticand theoretical turn ratio and phase shift of the transformer are

slightly different, the controller can compensate for the mis-

match. Fig. 3 illustrates a block diagram of where the mea-

sured and reference voltages are transformed to the frame of

reference. After performing the control action, the outputs are

transformed to the frame to generate the PWM switching

signals of the th interface VSC. Section IV describes the de-

sign steps of the controller based on the mathematical model of

the system. It should be noted that although the controllers are

designed based on RLC load models, other types of loads (e.g.,

motor loads) can be handled by the proposed controllers due to

their robustness.The control strategy is developed based on a decentralized

robust servomechanism approach to devise a decentralized con-

troller so that outputs of the system asymptotically track con-

stant reference inputs independent of 1) constant disturbances

which the microgrid is subjected to and 2) variations in the plant

parameters and gains of the control system [30]. The robustness

and the decentralized nature of the controller are highly desir-

able for a microgrid since:

a centralized controller which requires all inputs to be com-

municated to a common center is uneconomical due to the

complexity and cost of the required high-bandwidth com-

munication system;

a robust controller overcomes the uncertainty issues of the

plant structure and/or parameters.

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1846 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 4, OCTOBER 2012

Fig. 4. Single-line diagram of the microgrid of Fig. 1 used to derive state-space equations.

The conditions for the existence of a centralized system so-lution based on the servomechanism problem [31] differ from

those of a decentralized system [30], [32]. In this case, a plant

often has a number of control agents and each control agent has

a number of local control inputs and outputs, and a separate and

distinct controller is applied to each control agent. In particular,

there exists a solution to the problem of stabilizing a plant based

on a decentralized control system if and only if the plant has

no unstable decentralized fixed modes (DFMs) and that certain

rank conditions of the plant data hold true [33].

The notion of robustness leads to the problem of constructing

a controller for a plant such that the resultant closed-loop system

satisfies a given robustness constraint (e.g., having a compa-rable so-called real stability radius for both open-loop and

closed-loop systems [34], [35]). Thiscan be done by a decentral-

ized controller design based on the approach of [30], [32], [36],

[37], subject to a robustness constraint. The constraints depend

on the problem under consideration. The principles of applica-

tion of this decentralized control scheme to conventional power

systems are provided in [38][42].

In this paper, the decentralized control strategy is applied to

a microgrid system which is generally more susceptible to os-

cillations caused by transients compared with the conventional

power system. To ensure robust performance of the controller,

the decentralized controller optimization is carried out subjectto a specified robustness constraint [35]. To solve the robust ser-

vomechanism problem and to achieve further improvements for

the microgrid performance, the normal optimal control perfor-

mance index , which is used for

obtaining optimal controllers to reject impulse disturbances, is

replaced with the performance index ,

where is a small weighting parameter. The resulting op-

timal controller is now an optimal servomechanism controller

[43]. The microgrid model, existence conditions of the con-

troller, design procedure, and the properties of the closed-loop

system (e.g., robustness, gain margin, and tolerance to time-

delay and unmodelled high-frequency effects) are discussed in

the following sections.

III. MATHEMATICAL MODEL OF THE MICROGRID

The proposed decentralized control is developed based on a

linearized model of the microgrid of Fig. 1(a) in a synchronously

rotating frame. Fig. 4 shows a one-line diagram of the micro-

grid. The controller is designed based on the fundamental fre-

quency component of the system of Fig. 4. Each DER unit is

represented by a three-phase controlled voltage source and a se-

ries RL branch. Each load is modeled by an equivalent parallel

RLC network (Fig. 4). Each distribution line is represented by

lumped series RL elements.

The microgrid of Fig. 4 is virtually partitioned into three sub-

systems. The mathematical model of Subsystem in the

frame is

(3)

where is a 3 1 vector. Assuming a three-wire system,

(3) is transformed to the synchronously rotating frame of

reference, as described in Section II.B.2 by [44]

(4)

where is the phase angle generated by the crystal oscillator

internal to DER . Based on (3) and (4), the mathematical model

of Subsystem in the frame is

(5)

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Similarly, the frame-based models of Subsystem and

Subsystem , are also developed and used to construct the

state-space model of the overall system

(6)

where

, and are the state

matrices as given in the Appendix.

The system (6) can alternatively be written as

(7)

where

and a decentralized controller

(8)

is to be found, where denotes the system error, de-

notes the input, and the controller transfer function in (8) is re-

stricted to being a proper transfer function .

IV. DECENTRALIZED CONTROLLER STRATEGY

This section elaborates on the properties of the open-loop

system, and characterizes its robust properties. Then, a pro-

posed robust decentralized servomechanism controller is de-

signed with a real stability radius constraint imposed [35]. Fi-

nally, a robust stability analysis and performance evaluation

of the closed-loop system is provided. The decentralized con-

troller will be found so that the real stability radius of the final

closed-loop system is approximately the same as the open-loop

system (i.e., the closed-loop system should have a robustness

index which is not worse than, say, 50% of the robustness index

of the open-loop system).

Throughout this paper, the norm is assumed to be the

spectral norm and a square real matrix is said to be asymptoti-

cally stable if the eigenvalues of the matrix are contained in the

open left-half part of the complex plane.

An extended LTI model of the microgrid based on (6) is given

by

(9)

TABLE IIOPEN-LOOP PLANT EIGENVALUES AND TRANSMISSION ZEROS

where is the state, is the input, is

the output, is an unmeasurable constant disturbance,

is the desired constant set point for the system, and

is the error in the system.

We impose the condition that the system must contain

control agents, each corresponding to one of the three virtual

subsystems of Fig. 4, and we rewrite (9) as

(10)

where and are the inputs and outputs of control agent , and

is the error in control agent . The open-loop

eigenvalues and transmission zeros [45] of (9), corresponding

to the system of Fig. 4, are given in Table II, which indicates

that the system is stable and minimum phase.

A. Controller Design Requirements for the Decentralized

Robust Servomechanism Problem (DRSP)

A decentralized controller for the plant (10) includes the fol-

lowing desired features.

1) The closed-loop system is asymptotically stable.

2) Steady-state asymptotic tracking and disturbance regula-

tion occurs for 1) all constant setpoints and

2) all constant disturbances (i.e.,

) for all constant disturbances and setpoints.

3) The controller is robust. That is, condition 2) should holdfor any perturbations of the plant model (10), including dy-

namic perturbations which do not destabilize the perturbed

closed-loop system.

4) The controller should be fast with smooth non-peaking

transients (e.g., it should respond to constant setpoints and

constant disturbance changes within about three cycles of

60 Hz).

5) Low interaction should occur among the output channels

of the control agents, and among the outputs contained

in each of the control agents, for tracking and regulation

[46].

6) It is required that the aforementioned conditions should be

satisfied for as wide as possible a range of the load param-

eters , and of each of the control agents.

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These conditions will be achieved by a decentralized controller

based on the solution of the DRSP [30], [32].

B. Existence Conditions

The following existence conditions for a solution to the

DRSP, so that conditions 1), 2), 3) from before all hold, are

obtained from [30]. Given plant (10), let

(11)

where

...

and let . and .

Theorem 1 [30]: Given the system (10), then there exists a

solution to the DRSP such that conditions 1), 2), and 3) all hold,

if and only if the following conditions are all satisfied:

1) The system (10) has no unstable decentralized fixed modes

(DFM).

2) , where is the number of outputs of the th con-

trol agent and is the numberof outputs of Subsystem

.

3) The system has no .

Remark: If , the condition 3) becomes

For the microgrid of Fig. 4,it can beverified that the existence

conditions of Theorem 1 are all satisfied. In particular, 1) plant

(10) has no decentralized fixed modes; 2) it has

; and 3) the rank of the matrix , given by

(6), is equal to .

Remark: In the plant model (9), the three load parameters

, and of each subsystem (Fig. 4) can vary and result in

structural uncertainty in the plants nominal model. It is also ob-

served in (9) that the load parameters affect neither the outputs

gain matrix nor the input control matrix . Therefore, only

matrix of (9) is affected by changing load parameters. We use

this observation to design a controller with the desirable robust-

ness properties.

The following analysis shows that condition 3) of Theorem

1 always holds true for the study microgrid regardless of its

numerical values. The determinant of the matrix is

given by where

Clearly, the determinant is always nonzero which implies

for any numerical value of microgrid

parameters and, thus, the third condition of Theorem 1 always

holds.

C. Real Stability Radius Constraints

To evaluate the robustness of a control scheme, the following

definition is used [35].

Given a real matrix which is asymptotically stable,

assume that is subject to a real perturbation ,

where and are given real matrices, and is a real matrix

of uncertain parameters. Then, it is desired to find ,

so that 1) is asymptotically stable for all real per-

turbations with the property that and 2) there

exists a perturbation with the property that ,

so that is unstable. In this case, rstab is called the

real stability radius of .

D. Controller Design Procedure

Given the plant (10) with , to solve the DRSP, it is

necessary [30] that the decentralized controller should include

the decentralized servo-compensator

(12)

where , together with a decentralized stabi-

lizing compensator, which will be assumed to have the structure

(13)

where

so that the controlled closed-loop system is described by

(14)

In this case, the controller parameters (13) are obtained by ap-

plying the optimal controller design method [47], [48] to min-

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imize the expected value of the performance index

given by

(15)

where is obtained by solving the Lyapunov matrix equa-

tion corresponding to

(16)

where , subject to the conditions that

1) the resultant closed-loop system (14) is asymptotically

stable;

2) the real stability radius of the closed-loop system

should be greater than or equal

to the real stability radius of the open-loop system, where

Substituting for and from (14) in (16) provides a closed-

form expression for (16)

(17)

where [47]. The control design problem

becomes, therefore, a constrained parameter optimization

problem whose solution yields the control parameters

, and .

The reason to impose the robust constraint above is that the

load parameters , may vary, but since they

affect only the matrix and not the , and

matrices of (10), then it is desired to minimize the effect of

load parameter perturbations by keeping the closed-loop system

robustness index, as measured by the real stability radius, as

close as possible to the open-loop robustness index. In this

case, the constraint imposed is such that the closed-loop robust-

ness index is not worse than the open-loop robustnessindex.

E. Obtained Decentralized Controller

On carrying out the optimization of the controller (13) as

measured by the performance index (15), the following decen-

tralized controller is obtained for the microgrid of Fig. 4

(18)

whose variables are defined in the equation at the bottom of the

next page.

F. Properties of the Closed-Loop System

A transfer function representation of the control agents of the

decentralized controller (18) is given by

where and are the outputs

and errors of controllers, and plants outputs, respectively. Poles

of the decentralized controller are 0 (repeated 6 times),

, and . The closed-loop eigenvalues of the plant,

using the decentralized controller (18), are listed in Table III.

G. Robustness Properties of the Closed-Loop System

The closed-loop system of plant (10) and controller (18) is

highly robust with respect to changes in the load parameters

of the microgrid. In particular assume

that in the open-loop system (9) the matrix is subject to un-certainty (e.g., due to changes in the load parameters and per-

turbations in the elements of the matrix ). Also assume that

, where is an unknown real perturbation ma-

trix, and where is an asymptotically stable matrix. The largest

perturbations for which the system remains stable are deter-

mined from the real stability radius of [35].

In this case, the open-loop perturbed plant

remains asymptotically stable for all real matrix pertur-

bations with if and only if , where

. This implies that the open-loop plant

remains asymptotically stable for any real pertur-

bation of size 0.15% or less.

Consider now the closed-loop system obtained by applying

the controller (18) to (10), and assume that the elements of the

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TABLE IIICLOSED-LOOP EIGENVALUES OF THE PLANT OBTAINED

USING THE DECENTRALIZED CONTROLLER(18)

controller are fixed and that the and el-

ements of the plant are fixed, but that due to load changes in

(and other perturbations), the elements

of are allowed to perturb. We also assume in the closed-loop

plant (14) that , where is an unknown per-

turbation matrix. In this case, we obtain from [35] that the per-

turbed closed-loop system remains asymptotically stable for all

Fig. 5. Bode plot of the closed-loop system for control agent No. 1 (associatedwith input and output ).

real matrix perturbations with if and only if

, where .

Thus, the stability robustness index of the closed-loop system

with respect to perturbations of the

matrix is larger than the open-loop system given by

, and so there is no deterioration in robustness of

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the closed-loop system compared to the open-loop system with

respect to perturbations of the elements of . A bode plot of the

closed-loop system is illustrated in Fig. 5. Fig. 5 shows that the

closed-loop transfer function from to of the first control

agent has a gain margin of 29.5 dB at the frequency of 372 Hz

(as denoted bya dotted line) and a phase margin of which

clearly highlights the robust stability of the closed-loop system.

H. Other Robust Measures of the Closed-Loop System

1) Input and Output Gain Margins: The following definition

which is an extension of the classical SISO gain margin to mul-

tivariable systems [49] is provided.Definition: Given a plant controlled

by a controller , assume that the

closed-loop system is asymptotically stable. Let be

replaced by a , where is a constant gain

matrix; then if the closed-loop system remains stable with

, the system has an output gain margin of .

Also let be replaced by ; then if the closed-loop

system remains stable with , the system is said

to have an input gain margin of of . The gain margin can

be calculated from the real stability radius of the system, and the

closed-loop system (10), (18) has output and input gain margins

of 1.38 and 1.28, respectively, which are quite satisfactory [50].

2) Input Time-Delay Tolerance: Often a system may have

time-delays which are ignored in the modeling of a system, and

it is important that the controlled system be robust to such un-

modelled effects. The following definition is used to describe

such a robust property [51].

Definition: Given a plant con-

trolled by the controller assume

that the closed-loop system is asymptotically stable, and let

be replaced by , corresponding to a time-delay of s.

Then, if there exists such that the closed-loop system re-

mains stable , the closed-loop system has an input

time-delay tolerance of [51]. In this case, the closed-loop

system (10), (18) has an input time-delay tolerance of

s, which is acceptable considering the switching frequencyof the VSCs.

3) Tolerance to Unmodelled High-Frequency Effects: It is

of interest to determine the extent that the closed-loop system is

tolerant to unmodelled high-frequency effects in the plant model

(9). This is captured by including the term in the model of

the plant as follows:

(19)

where are given in (9), and is a real per-

turbation matrix. In this case, on applying the same

controller (18) to plant (19), we obtain that the resul-

tant closed-loop system remains asymptotically stable

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if and only if , where

[35]. This implies that the per-

turbed closed-loop system remains asymptotically stable

for all perturbations in the gain matrix of (9) with the

property that % %

(where ), which satisfactorily meets the

requirements.

V. CONCLUSION

This paper presents a power-management and control

strategy for an autonomous, multi-DER microgrid. The envi-

sioned strategy provides: 1) power management of the overall

microgrid; 2) open-loop frequency control and synchroniza-

tion; and 3) a local, decentralized control for each DER unit.

The power-management system, based on classical power-flow

analysis, determines the terminal voltage setpoints for DER

units. The frequency of the system is controlled in an open-loop

manner by utilizing an internal crystal oscillator for each

DER unit that also generates the angle waveform requiredfor transformations. Synchronization of DER

units is achieved by exploiting a GPS-based time-reference

signal. The local control of each DER unit, which is the main

focus of this paper, is designed based on a new multivariable

decentralized robust servomechanism approach which utilizes

a linear state-space model of the microgrid. Various attributes

of the controller (i.e., the existence conditions, gain margins,

robustness, and tolerance to delays and high-frequency effects)

are analytically discussed, and the design procedures are out-

lined. The performance evaluation of the controller, based on

offline digital time-domain simulation studies and real-time

hardware-in-the-loop implementation, is presented in Part II of

this paper.

APPENDIX

SYSTEM EQUATIONS AND STATE MATRICES

The -matrix of (6), , is

, where , , and are de-

fined at the top of the previous page. Other entries of are

zero except for the following:

, and

. Nonzero entries of are

, and. Nonzero entries of , that is,

, and are unity.

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