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Equivalencing the Collector 

System of a Large Wind Power 

Plant 

Preprint 

E. Muljadi and C.P. Butterfield 

National Renewable Energy Laboratory 

A. Ellis and J. Mechenbier 

Public Service Company of New Mexico 

J. Hochheimer and R. Young 

FPL Energy LLC 

N. Miller and R. Delmerico 

General Electric International, Inc. 

R. Zavadil and J.C. Smith 

Utility Wind Interest Group 

To be presented at the 2006 IEEE Power Engineering Society 

General Meeting 

Montreal, Quebec, Canada 

June 18–22, 2006 

Conference Paper 

NREL/CP-500-38940 

January 2006 
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EQUIVALENCING THE COLLECTOR SYSTEM OF 

A LARGE WIND POWER PLANT 

A. Ellis J. Mechenbier 

Public Service Company of New Mexico 

Alvarado Square, MS 0604 

Albuquerque NM 87158 

E. Muljadi C.P. Butterfield 

National Renewable Energy Laboratory 

1617 Cole Blvd 

Golden, CO 80401 

J. Hochheimer R. Young 

FPL Energy LLC 

700 Universe Blvd. 

Juno Beach, FL 33408

N. Miller R.Delmerico 

General Electric International, Inc. 

1 River Road, Bldg. 2-605 

Schenectady, New York 12345

R. Zavadil J.C.Smith 

Utility Wind Interest Group 

2004 Lakebreeze Way 

Reston, VA 20191

Abstract— As the size and number of wind power 

plants (also called wind farms) increases, power 

system planners will need to study their impact on 

the power system in more detail. As the level of wind 

power penetration into the grid increases, the 

transmission system integration requirements will 

become more critical [1-2]. 

A very large wind power plant may contain 

hundreds of megawatt-size wind turbines. These 

turbines are interconnected by an intricate collector 

system. While the impact of individual turbines on 

the larger power system network is minimal, 

collectively, wind turbines can have a significant 

impact on the power systems during a severe 

disturbance such as a nearby fault. Since it is not 

practical to represent all individual wind turbines to 

conduct simulations, a simplified equivalent 

representation is required. 

This paper focuses on our effort to develop an 

equivalent representation of a wind power plant 

collector system for power system planning studies. 

The layout of the wind power plant, the size and type 

of conductors used, and the method of delivery 

(overhead or buried cables) all influence the 

performance of the collector system inside the wind 

power plant. Our effort to develop an equivalent 

representation of the collector system for wind power 

plants is an attempt to simplify power system 

modeling for future developments or planned 

expansions of wind power plants. Although we use a 

specific large wind power plant as a case study, the 

concept is applicable for any type of wind power 

plant. 

Index Terms—wind turbine, wind farm, wind power 

plant, wind energy, aggregation, equivalence, 

distribution network, collector system, power 

systems, renewable energy. 

I. INTRODUCTION

Although it is very important to understand the 

dynamics of individual turbines [3-4], the collective 

behavior of the wind power plant and the accuracy in 

modeling the collector systems are also very critical in 

assessing wind power plant characteristics. Among 

other aspects, the design of collector systems for wind 

power plants seeks to minimize losses and voltage drops 

within budgetary constraints. This philosophy is 

generally applied regardless of the size of the wind 

power plant, the types of the turbines and reactive power 

compensation. 

 Within a wind power plant, wind turbines are placed 

optimally to harvest as much wind energy as possible. 

The turbine layout in a large wind power plant on a flat 

terrain is different from the layout of a wind power plant 

located on mountain ridges. The different layouts will 

have different impacts on the line impedances to the grid 

interconnection bus. 

 This paper describes an analytical approach that can 

be used to derive the equivalent representation of a wind 

power plant collector system. Many textbooks on 

distribution system modeling are available [5], but this 

paper focuses on modeling wind power plant collector 

systems in particular. To illustrate the methodology, we 

used data from the New Mexico Wind Energy Center 

(NMWEC), which is interconnected to the transmission 

grid owned and operated by Public Service Company of 

New Mexico (PNM). To independently verify the 

calculation, we used two different concepts. The first 

concept is based on circuit analysis, and the second one 

is based on load flow analysis. Although we used the 

Positive Sequence Load Flow (PSLF TM) program from 

General Electric for our process, other programs, such as 

Power Systems Simulation for Engineers (PSSETM) from 

Siemens Power Technologies Inc., can also be used. 

Figure 1 shows a simplified one-line diagram for the 

single-machine equivalent of the NMWEC. Bus 10999 

is the point of interconnection to the utility grid. Note 

1
 

that this node is represented as an infinite bus, such that 

it is not necessary to include any details about the system 

external to the wind power plant. The parameters that we 

seek to derive are highlighted 

Section II describes the steps we used to derive the 

equivalent impedance of a power system network. 

Section III presents two types of simplification, and 

Section IV presents and tabulates the result of the 

calculation. Section V presents the load-flow calculation, 

and Section VI the result and summary. 

 Figure 1. Simplified one-line diagram of a typical wind 

power plant connected to a substation. 

The parameter that the developer provided for PNM 

planning study purposes are listed on last row of table 1. 

II. STEP BY STEP DERIVATION 

A. General overview and assumptions 

In this section, we describe the background of the 

circuit simplification. A modern utility-size wind 

turbine generates electrical power at a low voltage level 

(typically 575V or 690V). Current utility-scale wind 

turbine ratings range from 1.5 MW to 5 MW per turbine. 

Each wind turbine is electrically attached to a pad 

mounted transformer that steps up the voltage to a 

medium voltage level, typically 34.5kV. 

At the 34.5 kV level, wind turbines are connected to 

each other in a string or “daisy chain” configuration 

(Fig. 2). Underground cables are most commonly used 

as trunk lines. Three or more turbines may be connected 

in this way to trunk lines, which then connect to one of 

possibly several feeder circuits that use a larger 

conductor. The feeder circuits can be underground 

cables or overhead lines. These feeders connect to a 

34.5 kV bus that is connected to the wind power plant’s 

substation transformer. The substation transformer steps 

up the voltage from a sub-transmission voltage level to a 

transmission voltage level (60 kV or above). 

In the following derivation, we based our equivalent 

circuit on apparent power losses (real power losses and 

reactive power losses). We made the following 

assumptions to derive the general equation for a circuit 

within a wind power plant: 

- The current injection from all wind turbines is 

assumed to be identical in magnitude and angle. 

- Reactive power generated by the line capacitive 

shunts is based on the assumption that the 

voltage at the buses is one per unit. 

B. Connection at the trunk line level

The first step is to derive the equivalent circuit for 

two or more turbines connected in a daisy-chain 

configuration (Figure 2). 

The equivalent circuit of the daisy-chain network 

shown in Figure 2 is represented in Figure 3. Note that 

the pad-mounted transformer is considered to be part of 

the generator itself. At this stage, we are only interested 

in the equivalent impedance of the collector system, 

excluding the pad-mounted transformers. Each of the 

currents shown is a phasor quantity, as follows: 

 Im = Im / θm 

 

In this paper, a boldfaced variable indicates a phasor 

quantity. For instance, I1 represents the current out of 

the wind turbine 1. The magnitude and angle of the 

phasor I1 are I1 and θ1, respectively. Since current 

injections from each turbine are assumed to be identical, 

we obtain the following: 

I1= I2 = I3 = I4 = I 

34.5 kV 345kV

10998 10995

Wind Farm 10999

(R + jX) j B 

values to be 

determined.

(0.002+j0.002) 

Taiban Mesa

Infinite Bus

10997 10996

0.570 kV 34.5kV

(0.0014+j0.0828)

(RXFMR + jXXFMR) 

values to be 

determined.

Figure 3 a) single series daisy-chain 

 b) equivalent representation of circuit (a) 

ZS

IS ZSIS

 1 2 3 4 

I1 I2 I3 I4 IS

(a) 

(b) 

Z1 Z2 Z3 Z4

 666 

 R 10 

131 

10666 

-

136

(a) (b)

Figure 2 a) single series daisy-chain physical diagram 

 b) equivalent representation of circuit (a) 

2
 

Therefore, the total current in the equivalent 

representation is given by: 

IS = n I

The voltage drop across each impedance, can be easily 

derived as follows. 

The voltage drop across Z1 = ∆VZ1 = I1 Z1 = I Z1

The voltage drop across Z2 = ∆VZ2 = (I1 + I2) Z2 = 2 I Z2

. 

The voltage drop across Z4 = 

∆VZ4 = (I1 + I2 + I3+ I4) Z4 = 4 I Z4 

The power loss at each impedance, can be computed as: 

SLoss_Z1 = ∆VZ1 I1

* 

= I1 I1

*

 Z1 = I I* Z1 = I

2 Z1 

SLoss_Z2 = ∆VZ2 (I1 + I2)

* = 2 I Z2 (2 I)*

 = 22 I

2 Z2

. 

. 

SLoss_Z6 = ∆VZ6 (I1 + I2 + I3 + I4)

*

 

 = 42 I

2 Z4

 Since IS = n I, the power loss equation can be 

simplified as follows: 

2 222

Tot_loss 1 2 3 4

2 2

Tot_loss 1 m

S = I (Z + 2 Z + 3 Z + 4 Z )

S = I m Z n

∑m=

output current of a single turbine

index

number of turbines in a string

I

m

n

=

=

=

where

The equations for the simplified circuit can be written as 

folllows: 

Zm represents the individual series impedances. 

C. Connection at the feeder level 

Consider parallel branches connected to the same 

nodes as shown in Figure 4. Each branch has a unique 

impedance and is connected to a group of wind turbines. 

Let us consider a simple three-branch configuration 

connected to the same node. For this configuration, we 

made the following assumptions: 

- Equivalent circuits for three circuits have been found 

following the methodology described in the previous 

section II.A. 

- Originally, each trunk circuit equivalent consisted of 

n1, n2 and n3 turbines respectively 

- All turbines are producing identical output in 

magnitude and phase angle. 

- The impedance from each group to the point of 

interconnection is Z1, Z2, Z3 respectively 

 

The output currents from each group are given as 

follows: 

I1 = n1 I

I2 = n2 I

I3 = n3 I 

The total output from the parallel branches are give as: 

IP = I1 + I2 + I3

IP = n1 I + n2 I + n3 I

IP = (n1 + n2 + n3) I 

Losses in individual branches: 

SZ1 = I1

2

Z1 

SZ2 = I2

2

Z2 

SZ3 = I3

2

Z3 

 

The total loss is: 

SZP = IP

2

ZP

By substitution, we derived the following equations: 

IP

2

ZP = I1

2

Z1 + I2

2

Z2 + I3

2

Z3 

 = I2

 n1

2

Z1 + I2 n2

2 Z2 + I2

 n3

2 Z3 

ZP = (n1

2

Z1 + n2

2 Z2 + n32 Z3) / (n1 + n2 + n3)

2 

2

Tot_loss S S

2

1 m

S 2

S = I Z 

m Z

Z = 

n

n

∑m=

Figure 4. Parallel connection of three groups of 

turbines through series impedances Z1, Z2 and Z3. 

ZP

IP

Z1 Z2 Z3 

I1 I2 I3 

IS

n1 n2 n3

3
 

The above equation can be further simplified: 

Zm represents the equivalent impedance of the individual 

parallel groups. 

D. Connection at the 34 kV main bus 

Consider the series of parallel branches represented 

by Figure 5. The circuit shown in Figure 5 represents 

the parallel combination of several feeders at 34.5kV. 

For this configuration, we made the following 

assumptions: 

- All turbines are producing identical output in 

magnitude and phase. 

- Between two points of inter-connection, there 

are a series of impedances, Z1S, Z2S, and Z3S. 

- Each group of wind turbines has series 

impedance Z1p, Z2p, Z3p 

- Each branch has unique impedance and is 

connected to a group of wind turbines. 

- Originally, each trunk circuit equivalent 

consisted of n1, n2 and n3 turbines respectively 

The output currents from each group are given as: 

I1 = n1 I

I2 = n2 I

I3 = n3 I 

 

The total output from the parallel branches are give as: 

IT = n1 I + n2 I + n3 I 

IT = (n1 + n2 + n3) I 

The voltage drop across the impedances can be 

computed: 2

1 m m

P 2

1 m

n Z

Z = 

n 

n

m

n

m

=

=

⎡ ⎤ ⎣ ⎦

∑

∑

∆VZ1P = I1 Z1P = n1 I Z1P 

∆VZ1S = I1 Z1S = n1 I Z1S 

∆VZ2P = I2 Z2P = n2 I Z2P 

∆VZ2S = (I1 + I2) Z2S = (n1I + n2I) Z2S 

∆VZ3P = I3 Z3P

∆VZ3S = (I1 + I2 + I3) Z3S = (n1I + n2I + n3I) Z3S 

Based on losses across each impedance, the impedance 

loss can be computed as: 

SLoss_Z1P = ∆VZ1P I1

* 

= I1 I1

*

 Z1P = n1

2 I

2 Z1P

SLoss_Z2P = ∆VZ2P I2

* 

= I2 I2

*

 Z2P = n2

2 I

2 Z2P

SLoss_Z3P = ∆VZ3P I3

* 

= I3 I3

*

 Z3P = n3

2 I2 Z3P

SLoss_Z1S = ∆VZ1S I1

*

 = n1

2

 I I* Z1S = n1

2 I

2

 Z1S

SLoss_Z2S = ∆VZ2S (I1 + I2)

*

 = (n1 + n2)

2 I

2

 Z2S

SLoss_Z3S = ∆VZ3S (I1 + I2 + I3)

*

= (n1+ n2 + n3)

2 I

2

 Z3S

The equivalent total loss can be computed as: 

Total Loss = SZTLOSS = IT

2 ZT

By equating the total loss to the sum of the individual 

impedance loss, we can find the equivalent impedance in 

a series-parallel circuit as shown in Figure 5. 

2 22 2 2

T T 1 1P 2 2P 3 3P

2 2

1 1S 1 2 2S 1 2 3 3S

222

T 2 1 1P 2 2P 3 3P

12 3

2 2

1 1S 1 2 2S 1

I Z = I [(n Z + n Z + n Z ) +

 (n Z +(n + n ) Z + (n + n + n ) Z ]

1 Z [(n Z + n Z + n Z ) + (n + n + n )

 (n Z +(n + n ) Z + (n 

=

2

2

2 3 3S

2

2

1 11

T 2

1

+ n + n ) Z ]

Z

np ns ns

i iP j iS

i ij

np

i

i

nZ n Z

n

= ==

=

⎛ ⎞

+ ⎜ ⎟ ⎝ ⎠ = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

∑ ∑∑

∑

E. Shunt representation 

Consider an equivalent circuit for the transmission 

line shown below. Because the nature of the capacitance 

generates reactive power that is proportional to the 

square of the voltage across them, and considering that 

the bus voltage is close to unity under normal conditions, 

the representation of the shunt B can be considered as 

the sum of all the shunts in the power systems network. 

This assumption is close to reality under normal 

condition. Figure 5. Parallel series connection of three 

groups of turbines. 

ZPS IPS 

Z1P Z 2P Z3P 

I1 I 2 I3 IT 

Z1S Z 2S Z3S 

 n1 n2 n3 

4
 

With the assumption presented above, we can 

compute the total shunt capacitance within the wind 

power plant as follows: 

 

1

n

tot i

i

B B == ∑

F. Pad mount transformer representation 

The pad mount transformer must be represented to 

process the entire wind power plant. The equivalent 

circuit can be scaled so that the resulting voltage drop 

across the (leakage) impedances and the reactive and 

real power losses are equal to the sum of individual 

reactive and real losses of the turbines. 

The equivalent representation for the entire wind 

power plant can be computed as the impedance of a 

single transformer divided by the number of the turbines. 

R + j X 

B/2 B/2 

ZPMXFMR_WF = ZPMXFMR_WTG /nturbine

As an example, the pad-mount transformer impedance 

for the NMWEC is: Figure 6. Representing the line capacitance within a 

wind farm. 

ZPMXFMR_WTG = (0.3572 + j 3.3370) p.u. 

The number of turbines is nturbine = 136 turbines. 

Using the equation above, the equivalent impedance for 

the pad-mount transformer represented by a single 

turbine for the entire wind power plant is: 

ZPMXFMR_WF = ZPMXFMR_WTG /nturbine

ZPMXFMR_WF = (0.0026 + j0.02454) p.u.

III. DIFFERENT EQUIVALENCE CONFIGURATIONS 

A. Equivalence of complete circuit (EOCC) 

We seek to find the equivalent collector impedance 

taking into account all conductors in the collector 

system. We will use the NMWEC collector system to 

illustrate the equivalencing methodology. The complete 

circuit representation for the NMWEC is illustrated in 

Figure 8. In this diagram, blocks D1 through D23 

represent groups of turbines connected as a daisy-chain 

network (refer to Figure 2). Several daisy-chain 

networks are connected to nodes N1 through N8. Node 

N1 through node N8 are connected to bus 10998, the 

low-voltage side of the substation transformer. To 

derive the equivalent collection system parameters (R, X 

and B in Fig. 1) for this collector system, the following 

procedure is suggested: 

- Analyze each block of daisy chain connections 

(e.g. D1, D2 ... D23). Use the method described 

in Section II B to find the equivalent impedance 

if the collector subsystem represented by blocks 

D1, D2, D3, etc. As discussed in Section B 

below, one option is to ignore the impedance of 

the daisy-chain connection. This may be 

justifiable in location where the line section is 

very short. 

- Several blocks are connected to a node (e.g. D1 

through D5 are connected to node N1 and blocks 

D6 through D10 are connected to node N2). We 

can use the method described in section II D to 

find the equivalent impedance in this 

simplification. 

 0.570 kV 34.5kV

 0.

(0.3572+j3.3370)

570 kV 34.5kV

(0.3572+j3.3370)

Turbine 

#1 

Turbine 

#136 

10997 10996 

0.570 kV 34.5kV 

(0.3572+j3.3370)/136 = 

 

 Wind Farm

 (136 turbines) (0.0026+j0.02454) 

Figure 7. Pad mount representation within the wind farm. 

5
 

- In some of the blocks, for example the blocks 

connected to Node 3 through Node 8, the block 

connections are computed by using the method 

described in section II B. 

- The next step is to connect Nodes N1 through 

N8 to bus 10998 and find the equivalent 

impedance within the wind power plant. The 

method described in section II C is used to 

complete this task. 

- The equivalent shunt capacitance, representing 

the capacitance of the cables and overhead lines, 

is then computed using the method described in 

section II E. 

- Finally, we can compute the equivalent 

impedance of the pad mount transformer using 

the method described in section II F. 

B. Equivalence of major lines (EOML) 

This representation ignores the impedance in the 

daisy-chain connections. For example, in Figure 8, only 

connections from Node 1 through Node 8 are 

considered. The daisy-chain connections inside blocks 

D1 through D23 are ignored. The method described in 

section II C is used to solve this equivalent circuit 

representation. The letters m1 through m8 indicate the 

number of turbines connected to node N1 through node 

N8. 

The equivalent shunt capacitance representing the 

capacitance of the cables and overhead lines is computed 

using the method described in section II E. The 

equivalent impedance of the pad mounted transformer is 

computed using the method described in section II F. 

C. Comparison of different types of representations 

Table 1 compares the calculated results from 

different circuit representations and the equivalent 

impedances. The EOCC and EOML collector system 

Table 1. Equivalent Circuit of Collector System 

Circuit Impedance-Shunt Admittance 

Representation R X B 

EOCC 0.01345 0.0497 0.1004 

EOML 0.008973 0.03527 0.0605 

PNM 0.0107 0.0214 0.0699 

Table 2. Other Impedances 

From - To R X 

HV Line 

345 KV 10995 - 10999 0.002 0.002 

34.5kV/345kV 

Transformer 10998 - 10995 0.0014 0.0828 

WTG 

Transformer 0.57kV/34.5kV 0.0026 0.0245 

10999 

345 kV

Taiban Mesa 

Infinite Bus

10999 

345 kV

Taiban Mesa 

Infinite Bus

10999 

345 kV

10995 

345 kV

10998 

34.5 kV

m1 m2

m5 

m4 

m7 

m3 

m6 

10995 

345 kV

10995 

345 kV

10998 

34.5 kV

10998 

N1

N4

N2

N3

N5

N6

N7

N8

m8 

Figure 9. Major lines in the wind farm.

m1 m2

m5 

m4 

m7 

m3 

m6 

N1

N4

N2

N3

N5

N6

N7

N8

m8 

Figure 9. Major lines in the wind farm.

34.5 kV

D2 

D3 

D4 

D5 

D1 

D7

D8

D9

D10

D6

D15 

D16 

D14 

D13 

D12 

D11 

D23 

D22 

D19 

D20 

D18 

D17 

D21 

N8

N3

N1 N2

N4 N6

N5 N7

Figure 8. Complete collector system network in 

the wind farm. 

6
 

impedance for the wind power plant are calculated using 

the methods described above. As expected, the “major 

lines” representation has a smaller impedance value and 

a smaller shunt representation value compared to the 

“complete circuit” equivalent impedance. The EOML 

has a lower resistance but a higher inductance than the 

equivalent impedance that Public Service of New 

Mexico has been using. From Table 1, we can conclude 

that the resulting equivalent impedance is lower if we do 

not include the trunk circuits. For example the resistive 

R is lower by 33%, the inductive part X is lower by 

30%, and the capacitance is lower by 40%. The last row 

(labeled PNM) of Table 1 is based on previously 

calculated impedance. The resistance R is calculated 

from real power losses and X is calculated by estimating 

X/R ratio of the collector system in the wind power plant 

and B values is estimated by adding the individual Bs. 

The impedance value of the pad-mount (WTG) 

transformer represents the impedance of the equivalent 

transformer for 136 turbines. The equivalent pad-mount 

transformer can be computed as shown in Table 2. 

This approach for calculating X, R and B using EOCC 

suggested by the authors has been adopted by the wind 

developer for estimating equivalent impedances of 

proposed wind power plants for interconnection study 

purposes. 

IV. LOAD FLOW ANALYSIS

A. Power flow comparison 

To validate the results of the calculation for 

equivalencing the collector systems using the method 

presented in section III, we use a load flow analysis. We 

modeled the complete circuit in the PSLF program, 

which consists of 136 turbines. All the impedances in 

the collector systems were entered in the load-flow 

model. We also modeled the equivalent representations 

derived using the EOCC and EOML methods described 

above to facilitate computation of losses with the load 

flow program for comparison purposes. To approximate 

the performance at the point of interconnection it is 

important to account for the losses in the collector 

system. The resulting apparent power MVA and the 

current flows in the branches are tabulated in Table 3. 

Table 3 shows the three types of representations. Note, 

that the current flowing in the line segment or 

transformers is based on the actual values (i.e. it is not in 

per unit values). There is a significant difference in the 

scaling because of the different voltage levels. The low 

Table 3. MVA and Amps Comparison 

Table 4. Losses Comparison (in MW and MVAR) 

 

Table 5. Collector System Losses 

Table 6. Distributed Line Impedance Z (p.u.), B, X/R and B/R Calculated from Losses 

Full Load From To Complete (136WTG) EO CC EO ML

Current Flow Bus Bus MVA AMPS MVA AMPS MVA AMPS

HV Line 345kV 10995 10999 199.5 317 210.5 335.5 212.2 338.2

HV Transformers 10995 10998 199.5 317 210.9 335.5 212.6 338.2

Dist. Lines (WF) 10998 10996 202.7 3328.7 200.5 3354.6 202.1 3382.5

WTG Transfomers 10996 10997 204 3323 203.5 3393 203.8 3405.8

Full Load From To Complete (136WTG) EO CC EO ML

Losses Comparison Bus Bus Ploss Qloss Ploss Qloss Ploss Qloss

HV Line 345kV 10995 10999 0.72 0.72 0.8 0.8 0.82 0.82

HV Transformers 10995 10998 0.57 32.75 0.56 33.27 0.57 33.83

Dist. Lines (WF) 10998 10996 4.13 5.33 5.48 10.17 3.69 8.46

WTG Transfomers 10996 10997 1.36 9.52 1.07 10.07 1.08 10.15

Losses Complete (136WTG) EOCC EOML

Ploss 4.13 MW 2.02% 2.69% 1.81%

Qloss 5.33 MVAR 2.61% 4.99% 4.15%

7
 

voltage at the wind turbine (570 volt), the medium 

voltage (34.5 kV) between the high side of the pad 

mount transformer and the low side of substation 

transformer, and high voltage (345 kV) after the high 

side of the substation transformer. 

The power loss shown in Table 4 is based on the 

load-flow calculation. The real power loss is caused by 

the resistive part of the impedance. The reactive power 

loss is the balance of reactive power absorbed by the line 

inductance and the reactive power injected by the 

susceptance of the lines. In the load-flow calculation, 

only the total reactive power loss is listed, the two 

components of VAR sources are not separated. The real 

power loss is the largest in the collector system. In the 

full circuit representation (136 turbines), the real power 

loss is computed as the sum of the collector system 

losses. The real and reactive losses in the collector 

system in the wind power plant are tabulated in Table 5. 

As shown in Table 5, both the EOCC and the EOML 

give a very good approximation of the real power loss. 

The reactive power losses in the collector system, both 

EOCC and EOML, have larger values than the complete 

representation of 136 wind turbines installed in the wind 

power plant. This is a result of the conservative 

assumption that the output currents from all turbines are 

equal in magnitude and in phase angle. 

B. Impedance calculated from losses 

From section IV A, we tabulated the results from the 

load flow analysis and then, we can compute the 

impedance from the calculated losses and calculated 

branch currents. In this section (section IV B), we are 

trying to estimate the equivalent impedance by looking 

at collector system internal losses using the complete 

system (136 turbines). From Tables 3 and 4, the real and 

reactive components of the impedance can be retraced to 

check the load-flow calculation. The resistive part of the 

impedance can be calculated from the real power loss 

and the current flow. 

2 3

Loss

pu

base

P R

I Z =

where, the Zbase is the base impedance calculated on 100 

MVA system base. 

2

LL base _

base

base

V

Z

S =

The inductance can be calculated from the reactive 

power loss. Here, the reactive power caused by the 

inductance in the lines must be extracted from the total 

reactive power calculated by load flow. The total 

reactive power minus the reactive power generated by 

the shunt capacitance (Bpu) is the reactive power 

generated by the inductance in the impedance. Note that 

the Bpu was calculated previously during the circuit 

reduction and is listed in Table I. 

 

2

_

2 3

tot Loss pu pu base

pu

base

Q VBS X

I Z

− =

The equivalent impedance was calculated and tabulated 

in Table 6 based on the equations above. 

Based on the result of this study, the authors 

recommendation is that New Mexico Energy Center be 

modeled with the following collector system 

impedances: 

a) pad mounted step-up transformer impedance 

Rxfm + j Xxfm = (0.0026+ j0.0245) p.u., 

b) collector system impedance: R+ j X = 0.0104+j 

0.0388 p.u. with B = 0.1004 p.u.

Notice that X and B values for the collector 

system impedance are appreciably higher than the 

ones initially supplied by PNM. 

In the process of designing a collector system, an 

estimation of total real power losses (excluding pad 

mounted transformers) is often available. At full load, 

losses may range from 1.5% 6o 3%. It is possible to 

estimate the equivalent impedance by looking at 

collector system from the internal real losses. 

The inductance can be estimated by assuming a 

suitable X/R and B/R ratios. Typical X/R ratio for 

collector systems may range from 6 to 10, depending on 

the relative amount of underground and overhead lines. 

For the NMWEC, for example, for the NMWC, X/R is 

approximately 3.8, from which X can then be estimated. 

As stated before, susceptance can be computed directly 

as shown in Section IIE. 

V. CONCLUSION

This paper describes methods used to develop an 

equivalence of a collector system in a large wind power 

plant. We derived equations to find the equivalent 

circuits for three different network configurations using 

simple electric circuit theory with the results shown in 

Table 1. These calculations allow rigorous computation 

of equivalent collector system impedance for virtually 

any collector system size and configuration. We also 

used load flow to verify the results of the equivalencing 

methodology. By using the losses in the collector 

system, we can compute the equivalent impedances and 

admittances as shown in Table 6. 

The ratio of X/R is very consistent for the EOCC, 

EOML and Complete representation. While the ratio 

B/R for EOCC and EOML is very consistent, the value 

of B/R for complete equivalent circuit is over predicted. 

It is also shown that the data from PNM for X/R and B/R 

are under predicted compared to the other calculations. 

8
 

Very large and diverse wind power plants can be 

represented by dividing the large wind power plant into 

smaller groups of wind turbines with similar attributes. 

The method described in this paper is intended to help 

utility planners get a quick and simple calculation for 

predicting the losses and behavior of the planned wind 

power plant with a good level of confidence. 

This methodology has a practical application. This 

approach for calculating X, R, and B using EOCC was 

tested and validated by the authors and has been adopted 

by the wind developer for estimating equivalent 

impedance of proposed wind power plants for 

interconnection study purposes. Recommendations were 

provided to PNM to adjust the equivalent circuit 

representation for this project. 

Rather than estimating the impedance by visual 

inspection as initially used, much more accurate result 

can be obtained by taking into account the number of 

wind turbine generators that are injecting currents into 

each branch. 

One thing that we did not have to time to incorporate 

in this study is assessing the Qmax and Qmin capability of 

the wind power plant. This reactive power capability is 

an important aspect of the wind power plant that will be 

continued in the future investigation. 

VI. ACKNOWLEDGMENT

We acknowledge the support of the U.S. Department 

of Energy, Public Service Company of New Mexico, 

FPL Energy, General Electric and Utility Wind Interest 

Group (UWIG). We also thank William Price from 

General Electric International Inc., for help and 

discussions during the development of this project. 

VII. REFERENCES

[1] Zavadil, R.; Miller, N.; Ellis, A.; Muljadi, E. “Making Connections,” 

Power and Energy Magazine, IEEE, Vol. 3, Issue 6, Nov.-Dec. 2005, 

pp. 26-37. 

[2] Zavadil, R.M.; Smith, J.C. “Status of Wind-Related U.S. National and 

Regional Grid Code Activities,” Power Engineering Society General 

Meeting, June 12-16, 2005, pp. 2892-2895. 

[3] Miller, N.W.; Sanchez-Gasca, J.J.; Price, W.W.; Delmerico, R.W. 

“Dynamic Modeling of GE 1.5 and 3.6 MW Wind Turbine-Generators 

for Stability Simulations,” Power Engineering Society General Meeting, 

IEEE, Vol. 3, July 13-17, 2003, pp. 1977-1983. 

[4] Miller, A.; Muljadi, E.; Zinger, D. "A Variable-Speed Wind Turbine 

Power Control," IEEE Transactions on Energy Conversion, Vol. 12., 

No. 2, June 1997 pp. 181-186. 

[5] Kersting, W.H. Distribution System Modeling and Analysis, CRC Press, 

Boca Raton, FL, 2001. 

9
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Equivalencing the Collector System of a Large Wind Power Plant: 

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This paper focuses on our effort to develop an equivalent representation of a wind power plant collector system for 

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wind turbine; wind farm; wind power plant; wind energy; aggregation; equivalence; distribution network; collector 

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