1

Abstract--This paper presents a Newton Raphson current

equation for the solution of power flow analysis in microgrid, comprising 2n current injection equations and m active power equations written in polar coordinates. The reactive power mismatch of PV bus is introduced as new state variable, which leads to a simple procedure for converting bus model between PV bus and PQ bus. In conversion process, the submatrix Y* is retained, and only few rows and columns are added or deleted. The proposed method is also applied to other occasions where the polar coordinates are adopted. Moreover, the models of microsources suitable for power flow analysis are summarized according to their operating characteristics.

Index Terms-- Current Injections, Power Flow, Polar Coordi-nates, Microgrid, Microsources.

I. INTRODUCTION

he Consortium for Electric Reliability Technology Solutions (CERTS) is pioneering the concept of the

CERTS Microgrid as "an aggregation of loads and microsources operating as a single system providing both power and heat. The majority of the microsources must be power electronic based to provide the required flexibility to insure operation as a single aggregated system" [1].

The microsources, the small scale distributed energy resources (DER of < 500kW), integrated into electricity distribution systems with power electronic interface, mainly include wind energy systems, photovoltaic systems, full cells, micro-turbines and battery energy storage systems.

The conventional distribution systems, whether radial type systems or network type systems, are generally designed to operate without any generation at load demand sides. The introduction of microsources on the distribution systems can bring significant influence on the flow of power and voltage conditions at customers and utility equipment [2]. Traditional generations are classified as PQ, PV or slack bus for the flow power analysis. The models of microsources, adopted in power flow analysis, are determined in accordance with the operating characteristics of various microsources. Several traditional power flow analysis methods for distribution systems are invalid by reason of neglecting the microsources

Yanping Zhang is with the School of Electrical Engineering, Southeast

University, Nanjing 210096, China. (E-mail: zhangyanping@seu.edu.cn). Yuping Lu is with the School of Electrical Engineering, Southeast

University, Nanjing 210096, China. (E-mail: luyupin@seu.edu.cn).

operating characteristics or lacking of the capability of solving systems including PV bus.

Distributed generators are treated as constant PQ or PV buses and storage systems are regarded as constant current buses but for purposes of the load flow the PQ model is adequate in [3]. The special models of distributed generators are not explicit. An improved PQ model and an RX model of wind farms are proposed under the assumption of wind energy generation by means of asynchronous machines in [4].

In recent years, some current injection methods for power flow analysis have been proposed to solve the power flow problem [5]-[6]. In [5], the use of the set of 2n current inject-ion equation is proposed, written in rectangular coordinates. A dependent variable QΔ is introduced for each PV bus

together with an additional voltage equation imposing the constraint of zero deviations in the bus voltage. Furthermore, except for PV buses, the Jacobian matrix has the elements of the (2 x 2) off-diagonal blocks equal to those of the nodal admittance matrix expanded into real and imaginary coordinates. The elements of the (2 x 2) diagonal blocks need be updated at each iteration according to the bus load model.

Another current equation power flow method is proposed in [6], in which the key idea is to solve an augmented system in which both bus voltages and current injections appear as state variables, and both power and current mismatches are zeroed. Instead of combining the nodal equations and the bus constraints into a single set of two nonlinear equations, the Newton’s method is applied to the two primitive sets of equations.

The current equations of the above-mentioned papers are expressed in rectangular coordinates or in a mix of polar and rectangular coordinates. Nevertheless, a variety of tools based on polar coordinates have been widely applied for power system analysis, such as power system dynamics analysis and state estimation. Furthermore, the operating data of power grid can been collected only in amplitude and phase format. Thus, in some occasions, power flow method based on polar coordinates is more suitable.

In this paper, the Newton’s method is applied to solve an augmented system based on polar coordinates, in which both bus voltage magnitudes and angles appear as state variables. Moreover, a new state variable, reactive mismatch QΔ , is

introduced for each PV bus. Active power mismatch equations, also expressed in polar coordinates, are imposed as the constraint for PV buses. When the state variables QΔ are

A Novel Newton Current Equation Method on Power Flow Analysis in Microgrid

Yanping Zhang and Yuping Lu, Senior Member, IEEE

T

978-1-4244-4241-6/09/$25.00 ©2009 IEEE

2

employed, this is convenient for converting bus model between PV bus and PQ bus In addition, the models of microsources suitable for power flow analysis are considered according to the their operating characteristics.

II. SUMMARY OF MODELING MICROSOURCE

The following three generation technologies are used for distributed generation systems: synchronous generator, asynchronous generator and power electronic interface [7]. Synchronous generators are typically utilized if the generator capacity exceeds a few MW. Asynchronous generators, with or without converter, are commonly used for wind energy system. Power electronic converters are used in photovoltaic systems, full cells, micro-turbines and battery energy storage systems. Owing to the small capacity of microsources, asynchronous generation and power electronic interface technologies are applied in microgrid.

On the assumption that a generated active power and a power factor are given, wind energy systems can be modeled as PQ buses, the reactive power of which can be calculated by (1) [4]:

)1(22

2 PV

X

XX

XXVQ

mc

mc +−=

Where: V is voltage, P is the active power, X is sum of the stator and rotor leakage reactances, Xc is the reactance of the capacitors bank, Xm is the excitation reactance.

Fuel cell, as a high efficient and environmentally friendly renewable energy source is made up of the fuel processing unit, the fuel cell stack and the power conditioning unit. A grid-connected model of proton-exchange-membrane (PEM) fuel cell power plant (FCPP) is proposed in Fig.1 [8]. The output voltage and the output power as a function of the modulation index and the phase angle are given as followings:

)2(δ∠= cellac mVV

)3(sin δX

VmVP scell

ac =

( )

)4(cos2

X

VmVmVQ scellcell δ−=

Fig. 1. FCPP, inverter and grid-connection diagram

Where: Vac is AC output voltage of the inverter, m is inverter modulation index, δ is phase angle of the AC voltage, Pac is AC output power from the inverter, X is reactance of the line connecting the fuel cell to the load, Vs is grid voltage and Q is reactive output power from the inverter.

The reactive power and AC output voltage can be

controlled by the modulation index m, and the active power can be controlled by phase angle of the AC voltage. The control principles are similar to those used in the reactive and active control of conventional generators. In conventional synchrono-us, the active output power is controlled by changing in steam input and rotor speed; moreover, the reactive and generator terminal voltage are controlled by controlling the excitation voltage. Thus, the fuel cell is treated as a PV bus for power flow analysis. It is worth noting that it would to be treated as a PQ bus when the reactive power of the inverter exceeds its limits. The output reactive of fuel cell is fixed at the violated limit.

A large scaled photovoltaic system acts as a voltage source by using the voltage control device. At the moment, the active power is determined by photovoltaic system model and the voltage magnitude is specified. Thus, it can be treated as a PV bus. For most small scaled photovoltaic systems, using the current control devices can be treated as PQ buses. In this condition, the current to be output by the power converter unit (PCU) is controlled to match a reference value from the grid providing a high power factor (PF). Most commercially available units specify an output power factor close to 1 under normal conditions and no lower than 0.9 [9].

Micro-turbines consist of the generator, air compressor, and turbine mounted on air bearings. Because the rotating shaft of the generator turns between 15,000 and 90,000 revolutions per minute, the generator provides a high frequency AC voltage source (angular frequencies up to 10, 000 rad/sec) [10]. For this reason, a power electronic interface is required between the micro-turbines and the AC loads. The power electronic interface can guarantee that the output power frequency AC voltage fixes at a given value. Thus, micro-turbines are treated as PV buses in power flow analysis.

For photovoltaic systems and micro-turbines, when their reactive outputs extend beyond their limits, they are needed to switch PQ buses.

III. CURRENT INJECTION POWER FLOW

A. Basic Equations

The current mismatch expressed in terms of the voltage polar coordinates for a given bus k is:

)5()(

1* ∑

=−−=Δ

n

iiki

k

spk

spk

k EYE

jQPI

Where:

{ }nk ,,1L= : n is the total number of buses;

spk

spk QP , :net scheduled active and reactive power at bus k.

)sin(cos kkkj

kk jVeVE k θθθ +== : complex voltage

phasor at bus k .

)sin(cos*kkk

jkk jVeVE k θθθ −== − : complex conju-

gate voltage phasor at bus k.

3

kkV θ, :voltage magnitude and angle at bus K.

kikiki jBGY += : thik ),( element of nodal admittance

matrix.

)6()()( kLkGsp

k PPP −=

)7()()( kLkGspk QQQ −=

The effect of voltage level on the system loads is represent-ed by polynomial equations, as follows:

)8()( 20)( kpkppkkL VcVbaPP ++=

)9()( 20)( kqkqqkkL VcVbaQQ ++=

Where:

kk QP 00 , : constant power components of load at bus k

)10(1=++ ppp cba

)11(1=++ qqq cba

Equation (5) defines the current injection mismatches corresponding to the system buss, which can be expressed in terms of its real and imaginary parts as follows:

)12()sincos(

sincos)Re(

1∑

=

−

−+=Δ

n

iiikiiki

k

kspkk

spk

k

VBG

V

QPI

θθ

θθ

)13()cossin(

cossin)Im(

1∑

=

+

−−=Δ

n

iiikiiki

k

kspkk

spk

k

VBG

V

QPI

θθ

θθ

And written as follows :

)14(sin

cos

sin

cos1)Im(

)Re(

1i

i

in

i kiki

kiki

k

k

spk

spk

spk

spk

kk

k

VGB

BG

PQ

QP

VI

I

×⎥⎦

⎤⎢⎣

⎡×⎥

⎦

⎤⎢⎣

⎡ −

−⎥⎦

⎤⎢⎣

⎡×⎥

⎦

⎤⎢⎣

⎡

−×=⎥

⎦

⎤⎢⎣

⎡

ΔΔ

∑= θ

θ

θθ

Equation (14) is expanded in Taylor series with respect to

( )spk

spk QPV ,,,θ and the nonlinear terms are neglected,

which yields

⎥⎦

⎤⎢⎣

⎡

ΔΔ

×⎥⎦

⎤⎢⎣

⎡

−×+

Δ×⎥⎦

⎤⎢⎣

⎡×⎥

⎦

⎤⎢⎣

⎡ −−

Δ×⎥⎦

⎤⎢⎣

⎡×⎥

⎦

⎤⎢⎣

⎡

−×−

Δ××⎥⎦

⎤⎢⎣

⎡−×⎥

⎦

⎤⎢⎣

⎡ −−

Δ×⎥⎦

⎤⎢⎣

⎡−×⎥

⎦

⎤⎢⎣

⎡

−×=⎥

⎦

⎤⎢⎣

⎡

ΔΔ

∑

∑

=

=

spk

spk

kk

kk

k

ii

in

i kiki

kiki

kk

k

spk

spk

spk

spk

k

iii

in

i kiki

kiki

kk

k

spk

spk

spk

spk

kk

k

Q

P

V

VGB

BG

VPQ

QP

V

VGB

BG

PQ

QP

VI

I

θθθθ

θθ

θθ

θθθ

θθθ

cossin

sincos1

sin

cos

)15(sin

cos1

cos

sin

cos

sin1)Im(

)Re(

1

2

1

Where:

spk

spk QP ΔΔ , :net scheduled active and reactive power

mismatch at bus k .

B. Equation for PQ Buses

In the case of a PQ bus k , because the net scheduled

active and reactive power spkP , sp

kQ remains unchanged, that

is

)16(0

0⎥⎦

⎤⎢⎣

⎡

=Δ=Δ

spk

spk

Q

P

Thus, to apply Newton’s solution algorithm, considering all system buses as being of the PQ type, Equation (15) can be written as follows:

)17(

)Im(

)Re(

)Im(

)Re(

)Im(

)Re(

1

1

***1

***1

*1

*1

*11

1

1

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ΔΔ

ΔΔ

ΔΔ

×

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ΔΔ

ΔΔ

ΔΔ

n

n

k

k

nnnkn

knkkk

nk

n

n

k

k

V

V

V

YYY

YYY

YYY

I

I

I

I

I

I

θ

θ

θ

M

M

LL

MMMMM

LL

MMMM

LL

M

M

The diagonal and off-diagonal elements in (17) are 22× blocks and have the following composition:

)18(*⎥⎦

⎤⎢⎣

⎡=

kiki

kikiki LJ

NHY

Where:

)sincos(1

)cossin(

ksp

kkspk

k

kkkkkkkkk

PQV

BGVH

θθ

θθ

−+

+×=

4

)()cossin( ikBGVH ikiikiiki ≠+= θθ

)sincos(1

)sincos(

2 kspkk

spk

k

kkkkkkkk

QPV

BGN

θθ

θθ

+−

−−=

)()sincos( ikBGN ikiikiki ≠−−= θθ

)sincos(1

)sincos(

kspkk

spk

k

kkkkkkkkk

QPV

BGVJ

θθ

θθ

++

−×−=

)()sincos( ikBGVJ ikiikiiki ≠−×−= θθ

)cossin(1

)sincos(

2 kspkk

spk

k

kkkkkkkk

QPV

GBL

θθ

θθ

−−

+−=

)()sincos( ikGBL ikiikiki ≠+−= θθ

Equation (17) can be written in compact form:

)19()Im(

)Re(⎥⎦

⎤⎢⎣

⎡

ΔΔ

×⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

ΔΔ

VLJ

NH

I

I θ

C. Representation of PV Buses

In the case of a PV bus k , only the net scheduled active

power spkP maintains invariable, the reactive power mismatch

spkQΔ becomes a dependent variable of (15). This is,

)20(0

0⎥⎦

⎤⎢⎣

⎡

≠Δ=Δ

spk

spk

Q

P

A single active power mismatch equation, expressed in terms of the voltage Polar coordinates, is introduced as an additional linearized equality constraint:

)21(0=⎥⎦

⎤⎢⎣

⎡

ΔΔ

×=ΔV

MP kical

k

θ

Where:

[ ]**kikiki RFM =

[ ]knkkki FFFF L21* =

[ ]knkkki RRRR L21* =

)()cossin( ikBGVVF kiijkikiikki ≠−−= θθ

∑≠=

−=n

kii

kiijkikiikkk BGVVF1

)cossin( θθ

)()sincos( ikBGVVR kiijkikiikki ≠+−= θθ

0=kkR

Equation (15) and (21) are combined to revise the Newton’s power mismatches formulation of the PV type at bus k :

)22(0

)Im(

)Re(

1

1

1

***1

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ΔΔΔ

ΔΔ

ΔΔ

×⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

spk

n

n

k

k

knkkk

kknkkk

calk

k

k

QV

V

V

MMM

KYYY

PI

I

θ

θ

θ

M

M

LL

LL

In (22), kK is a 12 × block and expressed in the

following composition:

)23(cos

sin

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

k

k

k

k

k

V

VK θ

θ

The proposed Newton’s algorithm, for both PQ buses and PV buses, is given by (24). Bus k , of the PV model, is connected to the tow PQ buses i and j :

)24(0

0

)Im(

)Re()Im(

)Re()Im(

)Re(

***

***

***

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

Δ

Δ

ΔΔΔΔΔ

×

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

Δ

Δ

ΔΔΔΔΔ

M

M

M

MMMMMMM

LMLL

LMLMMML

LLL

LLL

LLL

MMMMMMM

M

M

M

spk

j

j

k

k

i

i

kjkkki

jjjkji

kkjkkki

ijikii

calk

j

j

k

k

i

i

Q

V

V

V

MMM

YYY

KYYY

YYY

P

I

II

II

I

θ

θ

θ

Equation (24) also can be written in following form:

)25(0

)Im(

)Re( *

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

×⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

spk

cal QV

M

KY

PI

I θ

Generally, for a system of n buses including one slack bus and having m PV buses, there are ( )22 −n current mismatch

equations and m active power mismatch equations in (25). There are no equations for the slack bus. The matrix dimension of nonzero submatrices within the Jacobian matrix is )1(2)1(2 −×− nn , mn ×− )1(2 and )1(2 −× nm , respectively.

The current mismatches in (25) given by (12) and (13), and the active mismatch at PV bus k is computed by (26):

5

)26()sincos(1∑

=

+−=Δn

ikikikikiik

spk

calk BGVVPP θθ

At a genetic iteration ( )1+h , the voltage corrections are

solved and applied to the bus voltages, and the net scheduled reactive power spQ concerning PV buses are updated

following (31):

( )( ) ( )( ) ( )( ))27(

1 hspk

hspk

hspk QQQ Δ+=+

D. Bus Type Conversion

During the course of iteration, when a PV bus’s reactive power extends beyond its limits, it is fixed at the violated limit and its voltage magnitude is relaxed. Thus, the PV bus has been converted to a PQ bus, bus with specified active and reactive power. Later, during the iteration, if the bus voltage shows tendency to return and the reactive power again falls within the limits, the bus will be switched back from PQ to PV. In conversion process, only the rows and columns in terms of K and M within the Jacobian matrix are deleted (PV buses are converted to PQ buses) or added (PQ buses are

converted to PV buses), and the submatrix *Y is retained.

IV. ITERATIVE ALGORITHM

Step 1) Assemble the nodal admittance matrixY . Step 2) Initialize the iteration counter 0=h ; Initialize all

state variables ( )hV , ( )hθ for all system buses; Initialize

all state variables ( )0=Δ hsp

kQ for all PV buses.

Step 3) Calculate the current injection mismatches (Equation (12), (13)) and calculate the active power mismatches for all PV buses (Equation (27))

Step 4) If { } ε≤ΔΔΔ calPII ),Im(),Re(max goto step8; else,

goto step 5. Step 5) Assemble the Jacobian matrix. Step 6) Solve the system of (25). Step 7) Update all the state variables using (27) and following

equations;

( ) ( ) ( )hhh θθθ Δ+=+1

( ) ( ) ( )hhh VVV Δ+=+1

Increment the iteration counter 1+= hh ; Return step 3.

Step 8) Print the results. Due to high R/X ratio of some distribution systems which

causes the Jacobian Matrix to be ill conditioned, the convergence performance of the Newton method is deteriora-ted. To guarantee better convergence rate of the proposed method, the weight iterative of improvement method [11] is applied in step 6.

V. EXPERIMENTAL APPLICATIONS

The proposed method has been coded in C++ using the sparse storage techniques, and compared with the

conventional NR method and the back/forward sweep method [12] in polar coordinates. The test cases are 30 buses [13], 69 buses [14], and a practical distribution system whose size is 173 buses. There are no PV buses in those systems.

A. Convergence rate analysis TABLE I

CONVERGENCE COMPARISON

Cases NR

Method Back/Forward Sweep Method

Proposed Method

30 Buses 4 3 4 69 Buses 4 4 4 173 Buses 5 4 5

Table I compares the number of iterations required to

achieve convergence when the iteration convergence threshold is 10-5 pu. From Table I, in the case of all PQ buses, it is seen that the proposed method is as robust as the back/forward sweep method and the conventional NR method.

B. Number of PV Buses

Table II illustrates the number of iterations in three cases, in which the microsources are installed in the 30 buses system. In case 1, there is only a microsource, a fuel cell power plant treated as PV bus at bus 9. In cases 2, bus 5 is allowed to be a PV bus as well. And in case 3, we install micro-turbine at bus 22 treated as a PV bus, additionally. The iterations don’t increase with the number of PV buss.

TABLE II ITERATION NUMBER WITH PV BUS NUMBER

Cases PV Bus Number

Iteration number

Case 1 1 4 Case 2 2 4 Case 3 3 4

C. PV Bus Converted to PQ Bus

In the 30 buses system, a full cell system is installed at bus 9. Its output active power equals to 0.32 pu, and its output reactive power ranges from 0 pu to 0.16 pu. Due to grid-connected by using the power electronic interface, bus 9 is treated as a PV bus. In first iteration, the output reactive power of bus 9 QΔ reaches 4.928195 pu. If we don’t convert

bus 9 from PV bus to PQ bus, the final output reactive power equals to 5.67556 pu. The result exceeds its limit. The results of voltage and output reactive power before and after converting bus type are listed in table III. It is worthy of note that the PV bus reactive power can be expediently obtained in terms of the reactive power mismatch QΔ instead of the

reactive power calculation. TABLE III

RESULTS BEFORE AND AFTER CONVERSION

Before

conversion After

conversion Voltage magnitude 1 0.85507

Voltage angle -0.17709 -0.02051 Reactive power 5.67556 0.16

6

CONCLUSIONS

In this paper, the models of microsources within distribution system analysis are discussed according to their operating characteristics. Wind energy systems and small scaled photo-voltaic systems can be modeled as PQ buses. Moreover, photovoltaic systems and micro-turbines with power electronic interface can be modeled as PV buses.

In addition, this paper presents an alternative method to solve the load flow for microgrid. The reactive power mismatch of each PV bus is introducing as new state variable, which is convenient for converting bus model between PV bus

and PQ bus. In conversion process, the submatrix *Y is retained. Only the rows and columns in terms of K and M within the Jacobian matrix are deleted (PV buses are converted to PQ buses) or added (PQ buses are converted to PV buses). As a result of introducing the dependent variable QΔ , the dimension of the Jacobian matrix increases.

Due to the high-sparseness of the elements of the augmented parts concerning PV buses, the incremental requirements for storage and computation in the iterative process are not obvious after using sparse storage technique. The proposed method is also applied to other occasions where the polar coordinates are adopted.

VI. REFERENCES [1] Lasseter, Robert, Abbas Akhil, Chris Marnay, John Stevens, Jeff Dagle,

Ross Guttromson, A.Sakis Meliopoulos, Robert Yinger, and Joe Eto (2003, October ), The CERTS Microgrid Concept. California Energy Commission, Sacramento, CA, Tech. Rep. P500-03-089F.[Online]. Available: http://certs.lbl.gov/pdf/50829.pdf

[2] P.P.Barker, R.W de Mello, "Determining the impact of Distributed Generation on Power Systems. Part 1. Radial Distribution Systems," in Proc. 2000 IEEE Power Engineering Society summer meeting, pp. 1645-1656, July 2000

[3] Y. Zhu, K. Tomsovic, "Adaptive power flow method for distribution systems with dispersed generation," IEEE Trans. Power Delivery, vol. 17, pp. 822-827, July 2002

[4] A.E. Feijoo, J. Cidras, "Modeling of wind farms in the load flow analysis," IEEE Trans. Power systems, vol. 15, pp. 110-115, Feb. 2000

[5] V.M. da. Costa., N. Martins., and J.L.R. Pereira., "Developments in the Newton Raphson power flow formulation based on current injections," IEEE Trans. Power systems, vol. 14, pp. 1320-1326, Nov. 1999

[6] A.G. Exposito, E.R. Ramos, "Augmented rectangular load flow model," IEEE Trans. Power systems, vol. 17, pp. 271-276, May 2002

[7] T. Ackermann, V. Knyazkin, "Interaction between distributed generation and the distribution network: operation aspects," in Proc. IEEE/PES Transmission and Distribution Conference and Exhibition 2002: Asia pacific , vol. 2, pp. 1357-1362, Oct. 2002

[8] M.Y El-Sharkh, A. Rahman, M.S. Alam, A.A. Sakla, P.C. Byrne, and T. Thomas, "Analysis of active and reactive power control of a stand-alone PEM fuel cell power plant," IEEE Trans. Power systems, vol. 19, pp. 2022-2028, Nov. 2004

[9] Miu Karen, Carneiro, and Sandoval, "Assessing Impacts of Distributed Generation on Distribution Systems Analysis Tools," in Proc.2007 Power Engineering Society General Meeting, pp. 1-6, June 2007

[10] R. Lasseter, "Dynamic models for micro-turbines and fuel cells," in Proc.2007 Power Engineering Society Summer Meeting, vol. 2, pp. 761-766, July 2001

[11] Zhang Chun-mei, Yao Sheng, "On the Research of Weight Iterative Improvement Method for Morbid State Linear Systems", Journal of Anqing Teachers College(Natural Science), pp. 78-79, Feb. 2004

[12] D. Shirmohammadi, H.W. Hong, A. Semlyen, G.X. Luo, "A Compen-sation-Based Power Flow Method for Weakly Meshed Distribution and

Transmission Networks," IEEE Trans. Power Systems, vol. 3, pp. 753-762, May 1988

[13] R.G. Cespedes, "New method for the analysis of distribution networks," IEEE Trans. Power Delivery, vol. 5, pp. 391-396, Jan. 1990

[14] M.E. Baran and F.F. Wu, " Optimal capacitor placement on radial distribution systems," IEEE Trans. Power Delivery, vol. 4, pp. 725-734, Jan. 1989

VII. BIOGRAPHIES

Yanping Zhang was born in Nanjing, Jiangsu province, China, in 1974. He graduated from school of Electrical Engineering , Southeast University and received B.S. degree in 2000. Now he is pursing Ph. D. degree in Southeast University. His current research interest is distributed generation of electric power and microgrid technology. His E-mail is zhangyanping@seu.edu.cn Yuping Lu was born in Danyang, China, in Oct, 1962. He was received the Ph.D. degree in electrical engineering from the City University, UK in 2003. He is working as a professor in Southeast University of China. His research interests are power system protection, especially digital relaying of generator-transformer unit, and protection and control technique in distribution system with DGs.