Small Signal Stability Of A Microgrid With Parallel Connected Distributed Generation

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Small Signal Stability Of A Microgrid With ParallelConnected Distributed GenerationMahesh Illindala a & Giri Venkataramanan ba Power & Motion Components Research, Technical Center Building ‘G’, 41/TC-G 855Caterpillar Inc., Peoria, IL, 61656 E-mail:b Department of Electrical and Computer Engineering, University of Wisconsin-Madison, 1415 Engineering Dr, Madison, WI, 53706 E-mail:Published online: 01 Mar 2013.

To cite this article: Mahesh Illindala & Giri Venkataramanan (2010) Small Signal Stability Of A Microgrid WithParallel Connected Distributed Generation, Intelligent Automation & Soft Computing, 16:2, 235-254, DOI:10.1080/10798587.2010.10643079

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Intelligent Automation and Soft Computing, Vol. 16, No. 2, pp. 235-254, 2010 Copyright © 2010, TSI® Press

Printed in the USA. All rights reserved

235

SMALL SIGNAL STABILITY OF A MICROGRID WITH PARALLEL

CONNECTED DISTRIBUTED GENERATION

MAHESH ILLINDALA1, AND GIRI VENKATARAMANAN2

1Power & Motion Components Research, Technical Center Building ‘G’ 41/TC-G 855

Caterpillar Inc. Peoria, IL 61656

E-mail: [email protected]

2Department of Electrical and Computer Engineering University of Wisconsin-Madison

1415 Engineering Dr Madison, WI 53706

E-mail: [email protected]

ABSTRACT—Distributed Generation, or DG, involves utilization of small generators that are distributed in a power network, to supply the electric power demands of utility customers. This paper presents the small signal analysis of droop based generation control schemes for parallel connected DG inverters comprising of active power-frequency and reactive power-voltage controllers. Small-signal models are developed for microgrids consisting of several DGs connected in a parallel configuration. Mathematical propositions that develop sufficiency conditions for stability of the system are developed. Key Words: Distributed generation, dynamic stability, microgrids

1. INTRODUCTION Distributed generation (DG) using a variety of technologies are being considered as a

supplementary source of electric power in addition to the predominantly centralized generation in the emerging electrical utility infrastructure [1]-[7]. While some of the DG technologies make use of rotating electric machines, others employ inverters to derive utility grade ac power from the primary energy source. Since inverter based DG technologies do not involve heavy rotating masses they do not have the inertia that would source or absorb energy during transients in local load demand, particularly in stand-alone operation. Therefore, the dc link of the inverter may include an energy storage device to absorb load transients while the primary source availability and/or dynamics precludes load following [8, 9]. In order to increase the operational flexibility and improve reliability of the power supplied to loads, several DG devices may be interconnected in a local or dispersed manner to form a microgrid.

Various publications have discussed the dynamic behavior of the generation systems in the microgrid [8, 9]. However, none of them have addressed it especially related to a microgrid

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236 Intelligent Automation and Soft Computing

consisting of parallel connected DGs, although such a configuration is among the most common primitive building blocks for scaling the system to higher power levels [10]. Reference [11] provides a framework for performing detailed small signal eigenvalue analysis for such a system. The objective of this paper is to study the small-signal dynamics of a distributed generation control system involving active and reactive power controllers that allow operation of parallel connected DGs in a microgrid, with specific outcome of developing broad and generalized stability conditions. Section 2 presents a short review of local generation control scheme consisting of active power-frequency and reactive power-voltage controllers as applied to a inverter interfaced DG. The dynamic (small-signal) behavior analysis for microgrids containing several DGs connected at single point of common coupling (PCC) is developed in Section 3. The results are extended to study the stability of the microgrid under the grid connected mode in Section 4. Sufficient conditions are developed by means of eigenvalue analysis to guarantee small-signal stability of microgrids with a number of DG units. Section 5 covers the special case of grid-interfaced mode of operation of the parallel connected DGs. A brief discussion of the results is presented in Section 6. Section 7 illustrates a practical scenario of DG interconnections with tie-lines of a finite non-zero R/X ratio and its design recommendations, which is followed by a concluding summary.

2. DG CONFIGURATION AND MODELING A voltage source representation of a typical DG system is illustrated in Figure 1. The three

phase output voltage of the inverter is represented in the form of a complex rotating vector at a nominal frequency, say ω60. Even though the inverter may involve switching ripple filter and/or transformers, they invariably incorporate internal voltage regulator that is capable of maintaining the terminal voltage to a certain magnitude and frequency. It is therefore reasonable to assume that the internal dynamics of the inverter may be ignored, particularly if the external power generation controls are designed to have a bandwidth lower by at least one order of magnitude as compared to the internal voltage regulators [12].

vt = V ejω60t

PL, QL

Figure 1. Simplified representation of an inverter based DG system in a microgrid

The generation controller for the DG system has the task of generating the set point for the complex ac voltage vector at the point of load. The reference voltage space-vector can be set according to the nominal system specifications under stand-alone operation. However, when several such DGs are interconnected in a microgrid configuration, it is necessary to device a methodology to determine the magnitude and phase of voltage vector such that DGs can be dispatched to seamlessly share the overall load in a predetermined manner, using only locally measurable information.

As may be observed from Figure 1, the voltage space-vector vt specification at each DG bus comprises the magnitude (V) and frequency (ω60) information. Such a specification constitutes the voltage command that needs to be provided to each inverter based DG in the microgrid. The

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Small Signal Stability of a Microgrid with Parallel Connected Distributed Generation 237

active power-frequency and reactive power-voltage droop based controllers, first presented in the context of UPS systems in [12] and further applied to microgrids [4] are described further.

The active power frequency control of a DG illustrated in Figure 2(a) is a simplified representation comparable to the generation controller in a rotating machine generator that employs a speed governor [4]. It has a frequency droop with a proportional gain transfer function (b > 0), which provides the necessary load governing functionality that is beneficial for paralleling DG units. Furthermore, a frequency restoration segment is present where Kbβ represents the extent of frequency restoration in a DG upon load transient, and 0 < Kbβ < 1 so that a very low value Kbβ → 0 denotes almost no frequency restoration whereas Kbβ → 1 denotes total frequency restoration.

Since DGs in a microgrid employ semiconductor devices in the power electronic converters, their current capacity is limited by the device ratings. It is therefore necessary to control the apparent power drawn from a DG in a microgrid. This can be achieved by having a controller for regulating reactive power drawn from the DG along with the active power regulator shown in Figure 2(b).

PL

Δω 11 + s/ωG

b PL-ref

Kbβ1

1 + s/ωb Δωh

(a)

Voltage droop QL

QL-ref

ΔV 11 + s/ωq

1

Dq

(b)

Figure 2. Block diagrams of (a) active power and (b) reactive power controllers of a DG unit

Reactive power-voltage controller exploits the dependency of the reactive power supplied by the DG on the voltage magnitude at the load bus. As may be observed, a simple feedback controller containing first-order lag is employed for the deviation in the reactive power. Unlike the active power-frequency controller, the reactive power-voltage controller does not contain a voltage restoring loop. Moreover, it does not achieve a zero steady-state error but provides a voltage droop upon an increase in the reactive load. As seen in the input labeled ‘QL-ref’ is the load ref. set-point that is the control input to shift the DG’s voltage regulator characteristic in order to give the reference voltage (magnitude) at any desired reactive power output.

The dynamic behavior of a parallel-configured microgrid consisting of several such DGs is investigated in the following section, where in each inverter based DG is equipped with the reactive power-voltage (magnitude) controller, and the active power-frequency controller

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presented in this section. Furthermore, the following assumptions are made in the analysis of dynamic behavior of the microgrid —

i. The analysis is based on linearized small-signal models of the power system ii. Inverter based DGs operate within their maximum capacity limits iii. Dynamics of the inverter internal controls can be neglected in the study of generation

controller dynamics iv. Tie-lines between any two sources in the microgrid are purely inductive in nature. The

case of resistive networks is discussed in Section 7. It should be noted that the linearized model is developed at the operating point of the system,

and sensitivity effect of load variations are not discussed in this paper. To be sure, small and large transient variations in the load will nevertheless have a strong impact on the dynamic response of the microgrid. While the framework presented in this paper may be extended to study small variations in load conditions, large variations in load are beyond the scope of the current study.

3. DYNAMIC BEHAVIOR OF MULTIPLE DGS CONNECTED IN PARALLEL This section develops a small signal model for examining the stability for the microgrid

architecture shown in Figure 3, designated a parallel configuration. As seen in the figure, the kth DG is connected to the Point of Common Coupling (PCC) by an inductive tie-line of reactance Xk. The small signal stability model for the microgrid model is developed in the form of several propositions outlined further in the next section.

DGn

(V2, ∠δ2) (V1,∠δ1) (Vn, ∠δn)

X1 X2 Xn

DG2 DG1

(Vp, ∠δp) PCC

Figure 3. Single-line diagram of a microgrid parallel structure consisting of several DGs

Proposition 1: (i) The incremental change in voltage magnitude ΔVp at the PCC in a parallel microgrid is a

weighted average of the incremental changes in voltage magnitudes ΔVk (k = 1, 2, …, n) of n DG units connected to the PCC. Or mathematically,

ΔVp = c1 ΔV1 + c2 ΔV2 + … + cn ΔVn (1)

where ck (k = 1, 2, …, n) is a constant such that ck ≥ 0 and ∑ck = 1. (ii) The incremental change in phase angle Δδp at the PCC in a parallel microgrid is a

weighted average of the incremental changes in phase angles Δδk (k = 1, 2, …, n) of n DG units connected to the PCC. Or mathematically,

Δδp = d1 Δδ1 + d2 Δδ2 + … + dn Δδn (2)

where dk (k = 1, 2, …, n) is a constant such that dk ≥ 0 and ∑dk = 1.

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(iii) Also, the incremental change in frequency Δωp at the PCC in a parallel microgrid is a weighted average of the incremental changes in frequencies Δωk (k = 1, 2, …, n) of n DG units connected to the PCC. Or mathematically,

Δωp = d1 Δω1 + d2 Δω2 + … + dn Δωn (3)

Proof: (i) As seen in Figure 3, n DGs are connected to the infinite bus through a star point, known as

point of common coupling (PCC). The voltage at the PCC is dependent on the voltages at all the sources connected to it by a weighted average. This PCC voltage can be determined by the phasor relationship

~Vp = ⎝

⎛⎠⎞Xprll

X1

~V1 + ⎝

⎛⎠⎞Xprll

X2

~V2 + … + ⎝

⎛⎠⎞Xprll

Xn

~Vn, (4)

which is the ac phasor form of Millman’s theorem [13]. In the above equation, the expression for equivalent parallel inductance Xprll is obtained from

1

Xprll =

1X1

+ 1

X2 + … +

1Xn

(5)

Equation (4) for phasor voltage PCC can be rewritten in terms of complex exponential quantities as

Vp ejδp = ⎝

⎛⎠⎞Xprll

X1 V1 e

jδ1 + ⎝⎛

⎠⎞Xprll

X2 V2 e

jδ2 + … + ⎝⎛

⎠⎞Xprll

Xn Vn e

jδn (6)

Expansion of complex exponential functions into real and imaginary parts gives

Vp ( )cos(δp) + j sin(δp) = ⎝⎛

⎠⎞Xprll

X1 V1 ( )cos(δ1) + j sin(δ1) + ⎝

⎛⎠⎞Xprll

X2 V2 ( )cos(δ2) + j sin(δ2)

+ … + ⎝⎛

⎠⎞Xprll

Xn Vn ( )cos(δn) + j sin(δn) (7)

Practically, the difference between the phase angles of any two voltage sources connected by an inductive tie-line is a very small value; and a phase angle reference can be chosen such that at every voltage node can be approximated as

⎭⎪⎬⎪⎫cos(δk) = 1

sin(δk) = δk ∀ k ∈ 1, 2, …, n (8)

As a particular case, if the infinite bus is one of the sources connected to the PCC, it can be chosen as the phase angle reference and therefore its phase angle is zero. The expression for voltage phasor at the PCC would then become

Vp ( )1 + j δp = ⎝⎛

⎠⎞Xprll

X1 V1 ( )1 + j δ1 + ⎝

⎛⎠⎞Xprll

X2 V2 ( )1 + j δ2 + … + ⎝

⎛⎠⎞Xprll

Xn Vn ( )1 + j δn (9)

Equating the real parts on left- and right-hand sides of (9), the magnitude condition is determined as

Vp = c1 V1 + c2 V2 + … + cn Vn (10)

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where ck = ⎝⎛

⎠⎞Xprll

Xk ≥ 0 and ∑ck = 1.

If incremental changes are allowed in magnitudes of voltages, the relationship between these incremental changes in voltages is given by

ΔVp = ∑k=1

n

⎝⎛

⎠⎞∂Vp

∂Vk ΔVk or ΔVp = c1 ΔV1 + c2 ΔV2 + … + cn ΔVn (11)

where ck = ⎝⎛

⎠⎞Xprll

Xk ≥ 0 and ∑ck = 1. This is valid under the assumption that the incremental change

in reactance ΔXk ≈ 0 (for k = 1, 2, …, n) so that Δck ≈ 0 (for k = 1, 2, …, n). (ii) Equating the imaginary parts on left- and right-hand sides of (9), the phase angle

condition is determined as

δp = d1' δ1 + d2' δ2 + … + dn' δn (12)

where dk' = ⎝⎛

⎠⎞Vk

Vp ck ≥ 0 and ∑dk' = 1.

If incremental changes are allowed in phase angles of voltages, since δk << 1 (for k = 1, 2, …, n), the relationship between the incremental changes in phase angles is given by

Δδp = ∑k=1

n

⎝⎛

⎠⎞

⎝⎛

⎠⎞∂δp

∂δk Δδk + ⎝

⎛⎠⎞∂δp

∂dk' Δdk' or Δδp = d1 Δδ1 + d2 Δδ2 + … + dn Δδn (13)

where dk = ⎝⎛

⎠⎞Vko

Vpo ck ≥ 0 and ∑dk = 1. This is valid under the assumption

⎝⎛

⎠⎞∂δp

∂dk' Δdk' = ck δk ⎝

⎛⎠⎞Vpo ΔVk - Vko ΔVp

Vpo2 ≈ 0 since δk << 1.

(iii) Since the coefficients ck and dk (for k = 1, 2, …, n) are independent of time, the expression for incremental change in frequency Δωp at the PCC, after differentiating with respect to time on both sides of the phase angle relationship given above in (13), is determined as

Δωp = ddt (Δδp) = d1 Δω1 + d2 Δω2 + … + dn Δωn (14)

Therefore, the incremental change in frequency at the PCC is a weighted average of the incremental changes in frequencies of the sources connected to it. This weighting factor is determined from the tie-line inductance and voltage magnitudes on either side of the tie-line.

4. SMALL SIGNAL STABILITY ANALYSIS With the help of the Proposition 1 stated in the previous section, it is now possible to study

the system matrix of the parallel microgrid. The state variable schematic of the parallel microgrid illustrating the kth DG connected to the PCC by an inductive tie-line of reactance Xk is given in Figure 4. The following definitions and observations are made on Figure4:

i. bkPok represents the amount of p.u. droop in the frequency (say ωdtk) of kth DG at 1 p.u.

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power immediately following a transient event (before any frequency restoration becomes active).

ii. The gain of the frequency restoration integrator (ωbk) may be related to the time constant of the frequency restoration transient response by τGk =Kbβ/[ωbk(1+ Kbβ)]=1/ωGk

Furthermore, the state vector x can be chosen such that its transpose is given by xT = [Δωh1 Δδ1p Δωh2 Δδ2p Δωh3 … Δδn-1.p Δωhn ΔV1 ΔV2 … ΔVn].

The system matrix, A, from which the stability of the interconnected parallel microgrid can be determined. The matrix A for the microgrid in illustrated in Figure 4 is given by

A = ⎣⎢⎡

⎦⎥⎤[A11]2n-1x2n-1 [0]2n-1xn

[0]nx2n-1 [A22]nxn

(15)

Matrix A can be represented as a direct sum of its principle submatrices as

A = [A11]2n-1x2n-1 ⊕ [A22]nxn , (16)

where A11 and A22 represent the active and reactive power flows, respectively. Their eigenvalues, λ11 (of A11) and λ22 (of A22), can be determined independently.

Moreover, the sum of real powers flowing from all DGs to the PCC is equal to zero.

i.e., ΔP1+ΔP2+ … +ΔPn = 0 (17)

or Po1pΔδ1p+ Po2pΔδ2p + … + PonpΔδnp = 0 (18)

The partitioned submatrices of A refer to the active and reactive power flows, respectively, and are determined from Figure 4 as

A11 =

⎣⎢⎢⎢⎡

⎦⎥⎥⎥⎤

-ωb1 -b1(ωb1-ωG1)Po1p 0 … 0 0

-(1-d1) -[b1(1-d1)+bndn]Po1p d2 … (bn-1dn-1-bndn)Po.n-1.p dn

0 0 -ωb2 … 0 0

d1 (b1d1-bndn)Po1p -(1-d2) … (bn-1dn-1-bndn)Po.n-1.p dn

. . . … . .

0 bn(ωbn-ωGn)Po1p 0 … bn(ωbn-ωGn)Po.n-1.p -ωbn

(19)

and

A22 =

⎣⎢⎢⎢⎡

⎦⎥⎥⎥⎤

-ωq1⎝⎛

⎠⎞1+

Dqtie.1.pDq1

+ c1ωq1V1o

Dq1X1

c2ωq1V1oDq1X1

…cnωq1V1o

Dq1X1

c1ωq2V2oDq2X2

-ωq2⎝⎛

⎠⎞1+

Dqtie.2.pDq2

+c2ωq2V2o

Dq2X2…

cnωq2V2oDq2X2

. . … .

c1ωqnVnoDqnXn

c2ωqnVnoDqnXn

… -ωqn⎝⎛

⎠⎞1+

Dqtie.n.pDqn

+cnωqnVno

DqnXn

(20)

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Vko

Xk,p

bk

Δωhk(s)

Δωk(s)

Kbβk

ck

Pok,p

1/Dqk

ωbk

s

ωqk

s

Dqtiek,p

ΔVk(s)

ΔVp(s)

1s

Δδk,p(s)

dk

Δωp(s)

Figure 4. Small-signal state-variable schematic under zero input conditions of a microgrid having DGs in parallel

The properties of the eigenvalues of A11 and A22 are determined separately below, beginning with A22 as it is easily solvable with known properties.

Proposition 2: The sufficient conditions for all eigenvalues of Α22 of a parallel microgrid to have negative real parts are ΔVk_max < Vko and ωqk, Dqk > 0 (k = 1, 2, …, n), where ΔVk_max is the maximum incremental change in voltage of kth DG unit when operated in stand-alone mode and Vko is its nominal voltage.

Proof: By applying Gerschgorin theorem [14] on Α22 = (a22.i,j), it is well known that the

eigenvalues of A22 lie in disks with centers a22.i.i and radii ρ22.i = ∑j = 1

n |a22.i.j| (where i = 1, 2, …, n,

and j ≠ i). With all of its diagonal elements strictly negative, A22 has only eigenvalues with negative real parts if it is diagonally dominant under the following condition (for i = 1, 2, …, n):

|a22.i.i| > ρ22.i (21)

i.e., ωqi⎝⎛

⎠⎞1+

Dqtie.i.pDqi

> ωqiVioDqiXi

∑j = 1

n cj =

ωqiVioDqiXi

(22)

i.e., ( )Dqi + Dqtie.i.p > VioXi

(23)

i.e., Dqi + 2Vio - Vpo

Xi >

VioXi

(24)

i.e., Vio ⎝⎛

⎠⎞Vio - Vpo

Xi > -Dqi Vio (25)

The expression in the left hand side of above inequality is a measure of total reactive power

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generated by the ith DG under nominal conditions. If Qio denotes this nominal value of generated reactive power, matrix A22 becomes diagonally dominant

i.e., - QioDqi

< Vio (26)

or ΔVi_max < Vio (27)

The above inequality signifies that the condition that diagonal dominance of A22 is when the (maximum) stand-alone droop in voltage at every DG terminal is strictly less than its nominal value. This is generally true as practically the controller gains are never designed to result in a voltage droop larger than the nominal value. Consequently, under these conditions A22 has only negative real eigenvalues λ22.

Proposition 3: The sufficient conditions for all eigenvalues of A11 of a parallel microgrid to have negative real parts are

⎭⎬⎫ωbk + bkPokp = C1

bkωGkPokp = C2

(for k = 1, 2, …, n) (28)

where C1 and C2 are some positive constants. Proof: The dense matrix A11 is characteristic of the parallel microgrid where every DG unit

has some form of coupling to all remaining units. This coupling obscures and makes it difficult to evaluate the properties of matrix A11. However, the matrix A11 can be made sparse by the similar transformation, E11 = P11 A11 P11

-1 where

E11 =

⎣⎢⎢⎢⎡

⎦⎥⎥⎥⎤

-(ωb1+b1Po1p) -b1ωG1Po1p 0 0 … b1Po1p

1 0 0 0 … -1

0 0 -(ωb2+b2Po2p) -b2ωG2Po2p … b2Po2p

0 0 1 0 … -1

. . . . … .

e11.2n-1.1 e11.2n-1.2 e11.2n-1.3 e11.2n-1.4 … e11.2n-1.2n-1

(29)

and the transformation matrix,

P11 =

⎣⎢⎢⎢⎡

⎦⎥⎥⎥⎤

-1 -b1Po1p 0 0 … 0

0 1 0 0 … 0

0 0 -1 -b2Po2p … 0

0 0 0 1 … 0

. . . . … .

-d1 (dnbn-d1b1)Po1p -d2 (dnbn-d2b2)Po2p … -dn

(30)

In the matrix E11 = (e11.i,j), the elements of the 2n-1th row are determined as

⎭⎬⎫e11.2n-1.2k-1 = dk(ωbn - ωbk + bnPonp - bkPokp)

e11.2n-1.2k = dk(bnωGnPonp - bkωGkPokp) (for k = 1, 2, …, n-1) (31)

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and e11.2n-1.2n-1 = - ⎝⎜⎛

⎠⎟⎞

ωbn + bnPonp - ∑h = 1

n dhbhPohp (32)

As seen in (29), all rows/columns except one in E11 = (e11.i,j) already have a distinctive sign pattern determined as

sign(E11) =

⎣⎢⎢⎡

⎦⎥⎥⎤

- - 0 0 … +

+ 0 0 0 … -

0 0 - - … +

0 0 + 0 … -

. . . . … .

x x x x … x

(33)

The entries labeled ‘x’ in sign(E11) are not known a priori. Nevertheless, once these entries are known the sign pattern of matrix E11 is adequate to demonstrate qualitative stability known as sign stability. A matrix E11 is called sign stable if each matrix F11 of the same qualitative (or sign) pattern as E11 (sign e11.i,j = sign f11.i,j for all i,j) is stable regardless of the magnitudes of e11.i,j [15]. The stability of A11 can be judged from that of E11 as the eigenvalues of similar matrices A11 and E11 are identical.

According to Jeffries et al [15, Theorem 2], the 2n-1 x 2n-1 real matrix E11 = (e11.i,j) is sign stable if and only if it satisfies the following five conditions:

i. e11.ii ≤ 0 for all i. ii. All the diagonal entries of E11 are non-positive. iii. e11.ij e11.ji ≤ 0, for all i ≠ j. iv. The off-diagonal entries e11.ij and e11.ji must not be of same sign. v. The directed graph DE11 has no k-cycle for k ≥ 3. In the directed graph DE11 — the vertex set consists of 2n-1 elements V = {1, 2, 3, …, 2n-1}

and the edge set comprises the non-zero off-diagonal ordered pairs of E11, ED = {(i,j): i ≠ j and e11.i,j ≠ 0} with arrows starting at i and pointing towards j. Thereafter, the directed graph DE11 must not contain cycles with more than 3 edges.

In every RE11-coloring of the undirected graph GE11, all vertices are black. In the undirected graph GE11 — the vertex set consists of 2n-1 elements V = {1, 2, 3, …, 2n-

1}, the edge set comprises the ordered pairs with non-zero product among off-diagonal entries of E11, EG = {(i,j): i ≠ j and e11.i,j ≠ 0 ≠ e11.j,i} and rows having non-zero diagonal entries form the set RE11 = {i: e11.i,i ≠ 0}. According to the definition [15], elements of RE11 are painted black and no black vertex is allowed to have precisely one white neighbor, and each white vertex must have atleast one white neighbor. Thereafter, the undirected graph GE11 must have RE11-coloring with all vertices in black.

The undirected graph GE11 admits a (V ∼ RE11)-complete matching. According to [15], if condition (iii) is satisfied then condition (v) is equivalent to some term

in the expansion of det(E11) being different from zero. It is observed that the unique solution that satisfies all the above five conditions, (i) through

(v), is the last row off-diagonal entries of E11 being equal to zero; i.e. e11.2n-1,j = 0 (for j = 1, 2, …, 2n-2).

In this case,

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⎭⎬⎫dk(ωbn - ωbk + bnPonp - bkPokp) = 0

dk(bnωGnPonp - bkωGkPokp) = 0 (for k = 1, 2, …, n-1) (34)

and therefore, condition (i) is satisfied for the last row element also as shown by

e11.2n-1.2n-1 = - ⎝⎜⎛

⎠⎟⎞

ωbn + bnPonp - ∑h = 1

n dhbhPohp = - ∑

h = 1

n ωbk (35)

i.e., e11.2n-1.2n-1 < 0 (36)

Condition (ii) is verified by inspection of (33). For condition (iii), the directed graph DE11 is illustrated in Figure 5. It has V = {1, 2, 3, …,

2n-1} as its vertex set and ED = {(1,2), (1,2n-1), (2,1), (2,2n-1), (3,4), (3,2n-1), (4,3), (4,2n-1), (5,6), (5,2n-1), (6,5), (6,2n-1), …, (2n-3,2n-2), (2n-3,2n-1), (2n-2,2n-3), (2n-2,2n-1)} as its edge set. As seen in the figure, dark lines represent the elements of edge set. Light grey lines are also displayed in 0 that represent ordered pairs involving the last (i.e. 2n-1th) row of E11. These light grey lines are not part of the edge set ED as presence of even a single ordered pair involving the last (i.e. 2n-1th) row of E11 in the edge set would bring about a 3-cycle. Thus, the only possible solution to avoid a 3-cycle in directed graph DE11 is e11.2n-1,j = 0 (for j = 1, 2, …, 2n-2).

For condition (iv), Figure 6 shows the undirected graph GE11 that has V = {1, 2, 3, …, 2n-1} as its vertex set, EG = {(1,2), (3,4), (5,6), …, (2n-3,2n-2)} as its edge set and RE11 = {1, 3, 5, …, 2n-1}. As seen in Figure 5, the edges in GE11 correspond to the 2-cycles in GE11. Besides, the elements of RE11 are painted black. However, as no black vertex is allowed to have precisely one white neighbor, and each white vertex must have at least one white neighbor according to the definition [15], vertices {2, 4, 6, …, 2n-2} are also painted black to prevent the elements of RE11 from having a single white neighbor. As a consequence, in every RE11-coloring of the undirected graph GE11, all vertices are black. Therefore, condition (iv) is also satisfied for e11.2n-1,j = 0 (j = 1, 2, …, 2n-2). Likewise, condition (v) can be proved for non-negative control parameters as the undirected graph GE11 admits a (V ∼ RE11)-complete matching. This is true according to [15] since the determinant of E11 has some non-zero term in its expansion and condition (iii) is satisfied.

Thus, it is proved that the sufficient conditions for E11 (or A11) to be a stable matrix are e11.2n-

1,j = 0 (for j = 1, 2, …, 2n-2), which are the same as

⎭⎬⎫dk(ωbn - ωbk + bnPonp - bkPokp) = 0

dk(bnωGnPonp - bkωGkPokp) = 0 (for k = 1, 2, …, n-1) (37)

or ⎭⎬⎫ωbk + bkPokp = C1

bkωGkPokp = C2

(for k = 1, 2, …, n) (38)

where C1 and C2 are some positive constants.

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2n-1

1

e11.2n-3,2n-1

e11.2n-1,2n-3

e11.2n-1,2n-2

e11.2n-2,2n-1

e11.4,2n-1

e11.2n-1,4 e11.2n-1,3

e11.3,2n-1 e11.2,2n-1

e11.2n-1,2

e11.1,2n-1

e11.2n-1,1

2 3 4 2n-3 2n-2 e11.2n-3,2n-2

e11.2n-3,2n-2

e11.4,3

e11.3,4

e11.2,1

e11.1,2

Figure 5. A sketch of the directed graph DE11

2n-1

1 2 3 4

2n-3 2n-2

Figure 6. A sketch of the undirected graph GE11

Physical Significance of Proposition 3: The sufficient conditions developed in Proposition 3 for stability of real power flow in the

parallel microgrid are more rigid as compared to those of a chain microgrid presented in [11]. These are very conservative in ensuring stability and represent physical conditions that provide intuitive observations. The first relationship of (38) requires the gain of the frequency restoring integrator ωbk to be small when the transient droop frequency ωdtk (or bkPokp) is large and vice-versa. This implies that if there is a large transient change in frequency at a particular DG, its

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frequency should be restored to its final steady state value faster by appropriately choosing ωbk. Furthermore, the second relationship of (38) requires that the ratio of the transient droop frequency ωdtk to that of the time-constant of the frequency restoration transient τGk (or 1/ωGk) is a constant. Typically, the transient droop frequency signifies that amount of load transient power that is being taken up by a particular DG in the pool of generators. Therefore, this condition implies that the DG that takes up a bulk of the load transient to have smaller restoration time constant, which further states that it should reach its final steady state power contribution to the pool, and vice-versa. These conditions together ensure that DGs that tend to be stressed more in supplying power will return to their nominal conditions sooner than others.

Proposition 4: The sufficient conditions for ensuring stability of a microgrid in parallel con

bk k okp 1

bkωGkPokp = C2

) (39)

where C1 and C2 are someo propositions that the sufficient conditions for

sta

qk qk k_max ko

and

bk k okp 1

)

where C1 and C2 are some potability of the overall parallel microgrid are the

co

5. GRID-INTERFACED MODE OF OPERATION it convenient to operate in a grid-

inte

k

ck = 1 - cg

k

ndk = 1 - dg

(40)

where ck and dk are computed after into considerat parameters (Vg and Xg) of the grid

the grid-interfaced mode of operation voltage angle between nth DG and PCC

figuration are ωqk, Dqk > 0, ΔVk_max < Vko (for k = 1, 2, …, n) and

⎫ω + b P = C

⎭⎬ (for k = 1, 2, …, n

positive constants. Proof: It was determined in the previous twbility of reactive power control in the parallel microgrid is

ω , D > 0 and ΔV < V (k = 1, 2, …, n)

that for real power control in the parallel microgrid is

⎭⎬

bkωGkPokp = C2

(for k = 1, 2, …, n

sitive constants.

⎫ω + b P = C

Therefore, the sufficient conditions for smbination of the above two conditions.

The configuration of the control in a microgrid makesractive mode, and seamless transfer between the two modes of operation. The conditions

developed in the previous section may be readily extended to address the small signal stability of grid-interfaced mode of operation. In this case, the utility grid is considered as an infinite bus with no incremental change in voltage, angle or frequency, i.e. ΔV1 = 0, Δδg = 0 and Δωg = 0. If the parallel microgrid is connected from the PCC to the infinite bus of voltage E through a tie-line of inductance Xg, the following relations are true

n

⎭⎬⎫∑

= 1

∑ = 1

taking ioninterface tie-line.

As a result, in

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(Δ

A = [A11]2nx2n ⊕ [A22]nxn (41)

where A11 =

-ωb1 -b1(ωb1-ωG1)Po1p 0 … 0 0

- 1 1 bndn

-

bndn

and

A22 =

-ωq1⎛ ⎞1+

Dqtie.1.p

δnp) is considered as an additional state variable and thus the state vector now becomes xT = [Δωh1 Δδ1p Δωh2 Δδ2p Δωh3 … Δδn-1.p Δωhn Δδn.p ΔV1 ΔV2 … ΔVn]. The system matrix is then determined as

⎣⎢⎢⎢⎡

⎦⎥⎥⎥⎤(1-d1) -b (1-d )Po1p d2 … dn Po.n.p

0 0 ωb2 … 0 0

d1 b1d1Po1p -(1-d2) … dn Po.n.p

. . . … . .

d1 b1d1Po1p d2 … -(1-dn) -bn(1-dn)Po.n.p

, (42)

⎣⎢⎢⎢⎡

⎦⎥⎥⎥⎤

⎝ ⎠Dq1+

c1ωq1V1oDq1X1

c2ωq1V1oDq1X1

…cnωq1V1o

Dq1X1

c1ωq2VDq2X2

2o -ωq2⎛1+

D⎝ ⎠

⎞qtie.2.pDq2

+c2ωq2V2o

Dq2X2…

cnωq2V2oDq2X2

. . .

VnoDqnXn

…

c1ωqn c2ωqnVnoDqnXn

… -ωqn 1+⎝⎛

⎠⎞Dqtie.n.p

Dqn+

cnωqnVnoDqnXn

(43)

Proposition 2 is true for this case also as

ωqi⎛ ⎞Dqtie.i.p

⎝ ⎠1+ Dqi > DqiXi

ωqiVio cj = ωqiVio∑

j = 1DqiXi

n (1 - cg) (for i = 1, 2, …, n) (44)

Likewise, Proposition 3 can be pr for this case after making the similar transformation, E11

P11 =

-1 -b1Po1p 0 0 … 0

-

Thus, the conditions f of the microgrid are exte

oved = P11 A11 P11

-1 where

⎣⎢⎢⎢⎡

⎦⎥⎥⎥⎤0 1 0 0 … 0

0 0 -1 -b2Po2p … 0

. . . . … .

d1 -d1b1Po1p -d2 -d2b2Po2p … -dnbnPo.n.p

0 d1 0 d2 … dn

(45)

or guaranteed small signal stability stated in (39) nded for grid-connected mode of operation.

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6. DISCUSSION In the case of reactive power flow in microgrid, the sufficient conditions for stability of

parallel structure are identical to those of the chain architecture presented in [11]. The sufficient conditions derived for stability of real power flow in the parallel microgrid (39), on the other hand, are rigid as compared to those of a chain. It is to be noted that for a parallel microgrid, it is not necessary to satisfy the conditions

⎭⎬⎫ωbk + bkPokp = C1

bkωGkPokp = C2

(for k = 1, 2, …, n) , (46)

to prevent instability. This is despite the fact that under such an event, one or more of conditions (i) through (v) among the necessary and sufficient conditions for sign stability [15, Theorem 2] are not met. This is due to the fact that sign stability is only one among a set of non-exclusive indicators that are guarantors of the stability of a matrix.

In order to illustrate this case, a sensitivity analysis is conducted in this section for the eigenvalues of a 5-DG parallel microgrid. Consider 5 DGs to be operating in the baseline case with identical parameters as in Table 1 that are designed in accordance with the guidelines provided in [11]. The tie-lines between each unit and the PCC are also assumed to be identical for all DGs (equal to 0.1 p.u. reactance) so as to guarantee that conditions in (38) are met by this baseline case. The eigenvalues of the system under this situation are determined from MathCAD® software package as in Table 2. As seen in this table, the baseline case is stable as it has all eigenvalues in the left half of the s-plane. The most dominant eigenvalue is at -0.968. However, it is also observed from the eigenvalues that there are repeated values among them, while the eigenvalues of a chain microgrid, in contrast, are real and distinct in nature [11].

Table I. System parameters for the generation controller

V2o 1.0 p.u. Vgrid 1.0 p.u. b π rad./p.u. Kbβ ½ Dq 10 p.u. ωG 1 rad./s ωq 1 rad./s

Lf 0.97 mH

Rf 0.21 Ω

Cf 30 μF

Table II. System eigenvalues of a 5 DG parallel microgrid for the baseline case

2 32.448 0.968 32.448 0.968 32.448 0.968 32.448 0.968

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For sensitivity analysis, values of one of the parameters are varied to violate one or more of conditions (i) through (v) among the necessary and sufficient conditions for sign stability [15, Theorem 2]. Jeffries et al have also proposed Lemmas [15, Lemma 1 – Lemma 3] that suggest possibilities of migration of eigenvalues into the right half of s-plane when the conditions are violated. It is observed that conditions (ii) and (iii) can get easily violated.

The parameter chosen as the variable for this analysis is ωG5. By increasing or decreasing ωG5 from 1 rad./s, these conditions (ii) and (iii) get violated. Using MathCAD® software, the migration of eigenvalues is investigated as the parameter ωG5 is varied. It is observed that the dominant eigenvalue of -0.968 stays the same regardless of the increase in ωG5. On the other hand, if ωG5 is decreased below 1 rad./s, the migration of the dominant eigenvalue that was initially at -0.968 is towards the imaginary axis of the complex s-plane as illustrated in Figure 7. However, it is verified that this eigenvalue does not reach the imaginary axis regardless of the decrease in the value of parameter ωG5.

This case study illustrates for a parallel microgrid of 5 DGs that violation of the sufficient conditions in (38) of Proposition 3 do not cause any instability in the microgrid. However, it remains to be proven that the violation does not cause any instability for a general n DG parallel microgrid. In any case, a stronger set of necessary and sufficient conditions would certainly provide a tighter constraint for controller design.

Figure 7. Locus of the dominant eigenvalue of the real power flow in a 5-DG parallel microgrid as ωG5 is reduced below 1 rad./s.

7. EFFECT OF VARIATION IN TIE-LINE R/X RATIO ON THE DYNAMIC BEHAVIOR OF MICROGRID

The dynamic behavior of microgrid was analyzed in the previous sections assuming purely inductive tie-lines. This is in accordance with the traditional power flow studies that focus on high voltage transmission lines with R/X ratio small enough that it can be neglected [17]. On the other hand, the electric wire resistance is significant in medium voltage distribution systems and in buildings. In such installations, the R/X ratio cannot be neglected as indicated in [18]. In this

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section, the effect of variation in R/X ratio on eigenvalues of the microgrid is analyzed for the simple case of a single DG connected to the infinite bus.

Figure 8(a) displays the schematic of a single DG connected to the infinite bus through a tie-line of inductance X1,g and R/X ratio that is denoted by σ. The small-signal state-variable schematic of this system under zero input conditions is given in Figure 8(b). As seen in the figure, a finite non-zero value of σ creates coupling between the real and reactive power loops.

The system differential equations of Figure 8(b) are

ddt Δωh1 = - ωb1 Δωh1 -

b1 b1/β1 ωG1

1 + σ2 ( )Po1g Δδ1g + σDqtie1g ΔV1 , (47)

ddt Δδ1g = - Δωh1 -

b1

1 + σ2 ( )Po1g Δδ1g + σDqtie1g ΔV1 (48)

and ddt ΔV1 = - ωq1 ⎝

⎛⎠⎞

⎝⎛

⎠⎞1 +

11 + σ2 ⎝

⎛⎠⎞Dqtie1g

Dq1 ΔV1 -

σ1 + σ2 ⎝

⎛⎠⎞Po1g

Dq1 Δδ1g (49)

The system matrix in this case is constructed as

A =

⎣⎢⎢⎡

⎦⎥⎥⎤-ωb1

-b1(b1/β1)ωG1Po1g

1+σ2-b1(b1/β1)ωG1σDqtie.1.g

1+σ2

-1-b1Po1g

1+σ2-b1σDqtie.1.g

1+σ2

0ωq1σPo1g

Dq1(1+σ2) -ωq1 ⎝⎛

⎠⎞1 +

11+σ2⎝

⎛⎠⎞Dqtie1g

Dq1

(50)

As observed in Figure 8(b), the coupling between real and reactive power loops (for σ ≠ 0) makes it no longer possible to decouple matrix A into A11 and A22 corresponding to the real and reactive power controls. Therefore, the eigenvalues of the system matrix A, as a whole, need to be analyzed to determine the system dynamic behavior for σ ≠ 0.

To investigate the effect of variation in R/X ratio on the dynamic behavior of the single DG connected to infinite bus system, the dominant eigenvalues of A are plotted in Figure 9 for typical system parameters as σ varies from 0 through 5 using MathCAD® software package. As seen in Figure 9, the three eigenvalues advance towards the imaginary axis in the left half of complex s-plane as the system order is increased. Thus, the dynamic behavior of the single DG connected to infinite bus is greatly affected by variation of the R/X ratio.

This section has presented a simple case of a single DG connected to the infinite bus through a tie-line of finite R/X ratio (σ≠0). The key observations made here can be extended to the case of n DGs connected to the PCC in a parallel microgrid. It is reasonable to accept that a finite non-zero R/X ratio would greatly affect the dynamic behavior of a parallel microgrid - since the real and reactive power loops of the parallel microgrid would have coupling between them, and the system matrix A cannot be represented as a direct sum of submatrices A11 and A22. For these reasons, it is by and large desirable to have a very small R/X ratio for the interconnection of DGs in a microgrid. Accordingly, in the medium voltage distribution systems and buildings that contain electric wiring that is highly resistive in nature, it may be desirable to consider discrete inductors (reactors) along the tie-lines. This way, the effective R/X ratio of tie-lines (including the reactors) becomes very small that for all practical purposes is almost negligible. After such a

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Infinite Bus

E∠0o

DR1

V1∠δ1

PL1, QL1

X1,g σX1,g

(a) Single-line diagram

b1

Δωh1(s)

Δω1(s)

Kbβ1

1s

Δδ1(s)

1/Dq1

ωb1

s

ωq1

s

ΔV1(s)

Dqtie1,g

1 + σ2

Po1,g

1 + σ2

σPo1,g

1 + σ2

σDqtie1,g

1 + σ2

(b) Small-signal state-variable schematic under zero input conditions

Figure 8. Single DG connected to the infinite bus through a tie-line of R/X ratio σ

0

30

60

90

120

150

180

210

240

270

300

330

10

8

6

4

2

0

EIGi 0,

EIGi 1,

EIGi 2,

Figure 9. Dominant eigenvalues of the single DG connected to infinite bus as R/X ratio σ is increased from 0 through 5

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design adjustment is made, the real and reactive power flows between the DGs become decoupled and Propositions 1 through 4 in Sections 3 and 4 turn out to be valid.

7. CONCLUSIONS This paper has presented a dynamic analysis of generation control scheme consisting of active

power-frequency and reactive power-voltage controllers for the inverter based DGs. These droop-based controllers that allow decentralized operation of the microgrid without communication between the DGs. Dynamic behavior of a microgrid consisting of parallel connected DGs is commenced with a composite formulation that establishes the voltage, frequency and phase angle at the point of common coupling as a function of interconnection impedances and corresponding quantities of the individual DG units. Small-signal models were developed for microgrids consisting of DGs connected in a parallel topology. An investigation on dynamic (small-signal) behavior of an n DG parallel microgrid is carried out using eigenvalue analysis and sufficient conditions were developed to guarantee their small-signal stability by way of testing the sign stability of the system matrix.The sufficient conditions, thus developed, are very conservative in nature as they recommend a design criteria as well as provide broad guidelines, viz., (i) locate the smaller frequency droop DG of higher generation capacity electrically closer to the PCC making it the dominant DG in the parallel microgrid, and (ii) design the frequency restoration time constant (i.e. 1/ωG) of the dominant DG to be large enough for easier tracking of its frequency by the (smaller rated) DGs farther from the PCC. While this paper has presented the theoretical small-signal stability analysis for an n DG parallel microgrid, simulation and experimental results that illustrate the underlying concepts can be found in the published literature, for example with a parallel UPS microgrid as in [16].

ACKNOWLEDGMENT This work was supported in part by grant #02524744 System Integration of Distributed

Generation, from the National Science Foundation to the University of Wisconsin-Madison. The authors gratefully acknowledge support from Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC), and the shared research facilities of NSF funded Center for Power Electronics Systems (CPES).

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16. J.M. Guerrero, N. Berbel, J. Matas, J.L. Sosa, and L.G. de Vicuna, Control of Line-Interactive UPS Connected in Parallel Forming a Microgrid, Industrial Electronics, 2007. ISIE 2007. IEEE International Symposium, pp. 2667–2672, 2007.

17. A. R. Bergen, Power System Analysis, Prentice Hall, Inc., Englewood Cliffs, NJ, 1986. 18. National Electrical Code® Wire tables.

ABOUT THE AUTHORS M. Illindala received B.Tech. in 1995 from National Institute of Technology, Calicut, India, M.S. in 1999 from Indian Institute of Science, Bangalore, and Ph.D. in 2005 from University of Wisconsin-Madison, all in Electrical Engineering. Since October 2005, he has been working as a Power Electronics Engineer at the Technology & Solutions Division of Caterpillar Inc. His research interests include power quality, utility applications, distributed generation, and control systems.

G. Venkataramanan received the B.E. degree in electrical engineering from the Government College of Technology, Coimbatore, India, the M.S. degree from the California Institute of Technology, Pasadena, and the Ph.D. degree from the University of Wisconsin, Madison in the years 1986, 1987 and 1992 respectively. After teaching electrical engineering at Montana State University, Bozeman, he returned to University of Wisconsin, Madison in 1999, where he is currently a Professor and an Associate Director of the Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC).

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Small Signal Stability Of A Microgrid With Parallel Connected Distributed Generation