A New Pi-Equivalent Model of Wind Turbine Generating System for Load Flow Analysis

This paper proposes a method to incorporate a fixed-speed (or near fixed-speed) wind turbine generating system (WTGS) in the load flow analysis of interconnected electric power systems. The present paper also proposes a new steady-state model of the WTGS. The pi-equivalent circuit of the induction generator of the WTGS is employed in developing the new model. The model requires only one set of equations to describe the WTGS steady-state performances. It is, therefore, much simpler than the conventional pi-equivalent model.

Introduction

Recently, the utilization of wind energy as a source of electrical power generation has increased significantly. However, high penetration of renewable energy sources (i.e., WTGS) in the electrical power system can cause operational problems and affect the system's steady-state and dynamic performances [1]–[3]. It has widely been known that the steady-state performance of a power system is normally evaluated using load (or power) flow analysis. Therefore, to properly assess the system's steady-state performance, it is crucial to develop an accurate and valid WTGS model to be used in the study.

In the context of WTGS modeling, several researchers have proposed some interesting methods [4]–[13]. The method in [4]–[7] treats machine stator impedance, rotor impedance, and magnetic core circuit admittance as system branches in the load flow calculation. Also, in the proposed method [4]–[7], turbine mechanical power is treated as a negative load. In this way, a conventional load flow technique can be employed in the analysis without modifying the program source code. Various mathematical steady-state models for WTGS have been developed and proposed in [8]–[13]. In the method [8]–[13], the resulted mathematical model is used together with the existing load flow problem formulation (i.e., the formulation of the system without WTGS). The combined set of equations is then solved simultaneously.

In this paper, a new pi-equivalent model of fixed-speed WTGS is proposed. Unlike the previous pi model developed in [10], where two sets of equations (i.e., machine power and rotor voltage equations) are used to describe the WTGS, the proposed model in the present work is much simpler. Only one set of equations (machine power equations) is required to describe the WTGS steady-state performances. The proposed model has also been validated and tested using a representative multi-bus power system.

Method

Formulation of Load Flow Problem

Load flow study can be outlined as calculating the voltage magnitude and angle at each bus in a power system for a specified value of power generation and system load. After all bus voltages have been computed, the power flows and losses in all of the system transmission lines can then be determined. Formulation of the load flow problem is generally derived using a set of equations describing the power system network performance in terms of admittance. This set of nonlinear equations is then combined with the equations of bus power injection to obtain:

(1) $$\begin{array}{r}{\text{S}}_{\text{Gi}}-{\text{S}}_{\text{Li}}-{\text{V}}_{\text{i}}\sum _{\text{j}=1}^{\text{n}}{\text{Y}}_{\text{ij}}^{\ast }{\text{V}}_{\text{j}}^{\ast }=0\end{array}$$

where:

i = 1, 2, …, n: bus number

n: total number of buses

S_Gi = P_Gi + jQ_Gi: generation/supply at bus i

P_G, Q_G: active, reactive power generation

S_Li = P_Li + jQ_Li: load/demand at bus i

P_L, Q_L: active, reactive power load

V_i = |V_i|e^jδi: phasor of voltage at bus i

Y_ij = |Y_ij|e^jθij: phasor of ij-th element of bus admittance matrix

By splitting up the real and imaginary parts of (1), there will be two equations for each power system bus. It can also be observed that for each system bus, there are four unknown variables (i.e., P_G, Q_G, |V|, and δ).

Therefore, to find a unique solution to the set of (1), the values of the two variables should be specified. To do this, the usual practice in load flow analysis is to define three types of buses: (1) slack bus, (2) PQ bus, and (3) PV bus. By using this definition, the unknown (to be determined) and known (specified) variables for each bus are given in Table I.

Table I. Bus and Variable in Load Flow Formula

No	Bus	Specified variable	Calculated variable
1	Slack	|V| and δ = 0	PG and QG
2	PQ	PG = QG = 0	|V| and δ
3	PV	PG and |V|	QG and δ

New Pi-Equivalent Model of WTGS

A fixed-speed WTGS connected to an electric power system is shown in Fig. 1. In the figure, S_g is WTGS electrical power output; P_m is turbine mechanical power input; and SCIG stands for Squirrel Cage Induction Generator. It is to be noted that the value of P_m is always known, since it is readily available from the power curve provided by the turbine manufacturer.

Fig. 1. WTGS connected to power system.

Fig. 2 shows a steady-state equivalent circuit of the SCIG. In Fig. 2, R_R(1-s)/s can be thought of as the resistance through which the mechanical power is dissipated. In terms of impedances, the steady-state equivalent circuit of the SCIG is shown in Figs. 3 and 4. It is to be noted that the circuits in Figs. 3 and 4 are both equivalent. Y-Δ transformation has been used to convert the T-circuit in Fig. 3 into the pi-equivalent circuit in Fig. 4. Formulation of the WTGS mathematical model will be based on the pi-equivalent circuit since the derivation is much more straightforward.

Fig. 2. Electrical model of SCIG.

Fig. 3. Electrical model of SCIG in terms of impedances (Y circuit).

Fig. 4. Electrical model of SCIG in terms of impedances (Δ circuit).

Based on Fig. 2, the impedances of Z_S, Z_R and Z_M in Fig. 3 are written as:

(2) $$\begin{array}{l}{Z}_S=R_S+j{X}_S\\ Z_R=R_R+j{X}_R\\ Z_M=j{R}_c{X}_m/(R_c+j{X}_m)\end{array}$$

Whereas, in Fig. 4, Z_a, Z_b and Z_c impedances can be computed using:

(3) $$\begin{array}{l}{Z}_a=Z_R+Z_M+Z_R{Z}_M/Z_S\\ Z_b=Z_S+Z_M+Z_S{Z}_M/Z_R\\ Z_c=Z_S+Z_R+Z_S{Z}_R/Z_M\end{array}$$

From Fig. 4, it can be shown that the formulation for turbine mechanical power input (S_m = P_m + j0) in terms of WTGS terminal voltage (V_S) and electrical power output (S_g) is of the following form (detailed derivation can be found in the Appendix):

(4) $$\begin{array}{r}{\text{S}}_{\text{m}}={\text{V}}_{\text{S}}{\text{V}}_{\text{S}}\ast {\text{Z}}_{\text{1}}+{\text{S}}_{\text{g}}{\text{Z}}_{\text{2}}+{\text{S}}_{\text{g}}\ast {\text{Z}}_{\text{3}}+\frac{{\text{S}}_{\text{g}}{\text{S}}_{\text{g}}\ast }{{\text{V}}_{\text{S}}{\text{V}}_{\text{S}}\ast }{\text{Z}}_4\end{array}$$

where Z₁, Z₂, Z₃ and Z₄ are calculated using the formulas given in Appendix (Eq. A.12).

Equation (4) is the proposed steady-state mathematical model of WTGS. Incorporation of the model into load flow analysis of an electric power system is discussed in the next section.

WTGS Model Integration

For a power system embedded with WTGS, the load flow problem formulation (1) and (4) is then simultaneously solved. In the solution searching process, the additional variable to be determined is the WTGS electrical power output (S_g). Table II shows the unknown and known variables for each power system bus.

Table II. Bus and Variables for System with WTGS

No	Bus	Specified variable	Calculated variable
1	Slack	|V| and δ = 0	PG and QG
2	PQ	PG = QG = 0	|V| and δ
3	PV	PG and |V|	QG and δ
4	WTGS	PG = QG = 0	|V|, δ, Pg, and Qg

Results and Discussion

Model Validation

To validate the proposed WTGS model, the results from the model are compared with those obtained by the pi model in [10], [11]. The same data as in [10], [11] is used in the validation process. All of the data are in pu and have the following values.

i) WTGS terminal voltage: V_S = 1$⌞$0^o

ii) SCIG parameters: R_S = 0.00571, X_S = 0.06390, R_R = 0.00612, X_R = 0.18781, R_c = 30, X_m = 2.78

Table III shows the results of Sg (WTGS power) calculations for various values of P_m (turbine power). It can be seen that the results from the proposed model are in exact agreement with those obtained by the pi model in [10], [11]. This agreement indicates that the proposed model is accurate and valid.

Table III. Validation Results

Pm (in pu)	Sg (in pu)
	New Pi model	Pi model [10], [11]
1.0	0.9546 – j0.6265	0.9546 – j0.6265
0.5	0.4644 – j0.4173	0.4644 – j0.4173
0	−0.0326 – j0.3516	−0.0326 – j0.3516
−0.5	−0.5357 – j0.4192	−0.5357 – j0.4192
−1.0	−1.0461 – j0.6432	−1.0461 – j0.6432

Application in Multi-Bus Power System

This section investigates the application of the proposed new pi-model in the load flow study of a multi-bus power system installed with WTGS. The system used in the investigation is based on a 5-bus power system [14]. A single line diagram of the system is shown in Fig. 5. It has been assumed that the WTGS is connected to bus 4 via a power transformer. Tables IV–VI show the system data (all data are in pu on a 100 MVA base).

Fig. 5. Equivalent circuit of SCIG.

Table IV. Branch Data

Branch	Bus p – q	Series impedance (Z)	Shunt admittance (Ysh/2)
1	1 – 2	0.02 + j0.06	j0.030
2	1 – 3	0.08 + j0.24	j0.025
3	2 – 3	0.06 + j0.18	j0.020
4	2 – 4	0.06 + j0.18	j0.020
5	2 – 5	0.04 + j0.12	j0.015
6	3 – 4	0.01 + j0.03	j0.010
7	4 – 5	0.08 + j0.24	j0.025
8	4 – 6	j0.10	0

Table V. Bus Data

Bus	|V|	δ	Generation	Load	Note
1	1.06	0	–	0	Slack
2	1.05	–	0.40 + j-	0.20 + j0.10	PV
3	–	–	0	0.45 + j0.15	PQ
4	–	–	0	0.40 + j0.05	PQ
5	–	–	0	0.60 + j0.10	PQ
6	–	–	0	0	WTGS

Table VI. WTGS Data

Component	Parameter
SCIG	Stator: RS = 0.020; XS = 0.100
	Rotor: RR = 0.018; XR = 0.180
	Magnetic core: Rc = 50; Xm = 3.2
Transformer	Impedance: ZT = j0.1

Results of the load flow analysis (i.e., WTGS terminal voltages and power outputs) are presented in Table VII Load flow calculations were carried out for various turbine mechanical power (P_m) values. Mechanical power values ranging from 0.1 to 1.0 pu were used in the analysis. These values represent low to high wind conditions. It is to be noted that the same results are also obtained when the pi model [10], [11] is used in the analysis. These results also indicate that the proposed model is accurate and valid.

Table VII. WTGS Voltage and Power

Pm	Voltage magnitude	Power output
0.1	0.9782	0.0798 – j0.2937
0.2	0.9786	0.1785 – j0.3038
0.3	0.9777	0.2764 – j0.3192
0.4	0.9755	0.3734 – j0.3401
0.5	0.9719	0.4696 – j0.3668
0.6	0.9669	0.5648 – j0.3998
0.7	0.9600	0.6589 – j0.4400
0.8	0.9511	0.7518 – j0.4888
0.9	0.9394	0.8429 – j0.5485
1.0	0.9237	0.9317 – j0.6237

Conclusion

A method to model and incorporate a fixed-speed (or near fixed-speed) wind turbine generating system (WTGS) in the load flow analysis of interconnected electric power systems has been proposed in this paper. The pi-equivalent circuit of the induction generator of the WTGS has been employed in developing the new model. The model requires only one set of equations to describe the WTGS steady-state performances. It is, therefore, much simpler than the conventional pi-equivalent model. The proposed method has also been validated and tested using a representative multi-bus power system.