A new T-circuit model of wind turbine generator for power system steady state studies

Received Sep 16, 2020 Revised Nov 13, 2020 Accepted Dec 5, 2020 Modeling of wind power plant (WPP) is a crucial issue in power system studies. In this paper, a new model of WPP for steady state (i.e. load flow) studies is proposed. Similar to the previous T-circuit based models, it is also developed based on equivalent T-circuit of the WPP induction generator. However, unlike in the previous models, the mathematical formulation of the new model is shorter and less complicated. Moreover, the derivation of the model in the present work is also much simpler. Only minimal mathematical operations are required in the process. Furthermore, the rotor voltage value of the WPP induction generator is readily available as an output of the proposed new model. This rotor voltage value can be used as a basis to calculate the induction generator slip. Validity of the new method is tested on a representative 9-bus electrical power system installed with WPP. Comparative studies between the proposed method (new model) and other method (previous model) are also presented.


INTRODUCTION
Modeling of WPP is a crucial issue in steady state and dynamic analyses of modern electric power system. Several interesting models have been investigated to enable incorporating the WPP into the studies. In the context of power system steady state (i.e. load flow) studies, some recent methods in the modeling can be found in . The present paper investigates a new method for modeling WPP to be incorporated into modern power system load flow analysis.
Steady state model of WPP is developed where equivalent T-circuit of the WPP induction generator has been used as a basis in the model derivation [2, 16-18, 22, 23]. For each WPP, the developed model formulation consists of two nonlinear equations. Also, the unknown quantities (i.e. quantities to be determined or calculated) in the formulation are only the WPP active-and reactive-power output. However, in the method proposed in [2, 16-18, 22, 23], the WPP has been represented with relatively long and complex mathematical expressions.
In the present paper, a new model of WPP is proposed. Similar to the models discussed in [2, 16-18, 22, 23], it is also developed based on equivalent T-circuit of the WPP induction generator. However, the derivation process of the new model is much simpler than that proposed in [2,16,17,18,22,23]. Only minimal mathematical operations are required in the process. As a result, simpler WPP steady state model can be obtained. Moreover, the rotor voltage value of the WPP induction generator is readily available as an output of the proposed model. This rotor voltage value can be used as a basis to calculate the induction generator slip. The validity of the proposed method is then tested on a representative 9-bus electrical power system installed with WPP. To be systematic, the rest of the paper is arranged as follows: Section 2 gives the load flow problem formulation of a general power system without WPP. Section 3 addresses the proposed method of WPP modeling. Incorporation of the model into load flow analysis is also discussed in this section. Validity of the proposed method is investigated in section 4. Effects of shunt capacitor installation on WPP steady state performances (voltage profile, power output and slip) are also presented in this section. Finally, some important conclusions of the present work are given in section 5.

FORMULATION OF LOAD FLOW PROBLEM
The formulation of power flow problems can be obtained through equations that describe the performance of power system network in the forms of admittance. These equations are then combined with the formulations of bus power injection to obtain [24]. To be able to find a valid solution to (1), in the power flow analysis, the following three types ofsystem buses are usually defined: reference (slack), generator (PV) and load (PQ) buses as shown in Table 1. This definition is intended to make the number variables equal to the number of equations so that a correct and valid solution to (1) is obtainable.

PROPOSED MODEL OF WPP
In Figure 1(a), a WPP connected to bus k of an electric power system is shown. The main energy converter of the WPP is SCIG (squirrel cage induction generator). Separately, the SCIG is shown in Figure 1(b). Mechanical power input to the SCIG is Pm, and the SCIG (i.e. WPP) electrical power output is Sg=Pg+jQg. Figures 2 and 3 show the steady state equivalent circuits of SCIG. In Figure 3, ZS, ZR and ZM represent the impedances of stator, rotor and magnetic core circuits, respectively. These impedances are given by: It is also to be noted that in Figure 2, s is the machine (i.e. SCIG) slip, and R2(1-s)/s is a variable resistance where the turbin mechanical power Pm is assumed to be dissipated into. Based on Figures 2 and 3, the dissipated power can be computed using: Also, by looking at Figure 3, the formulas for WPP electrical power output (Sg) and turbine mechanical power input (Sm) can be written as:  (4) and (5), the stator current (IS) as a function of electric power output (Sg) and stator voltage (VS), and the rotor current (IR) as a function of mechanical power input (Pm) and rotor voltage (VR) are formulated as: Furthermore, by applying basic electric circuit theories (i.e. Kirchhoff's and Ohm's laws) to circuit in Figure 3, the following equations can also be obtained (derivation is given in the appendix): Substituting (6) and (7) into (8)and (9) will result in: In (10) and (11), are the proposed steady state model of WPP. It is to be noted that, in (10) and (11), VS is the stator voltage which is also the voltage at WPP terminal (i.e. V=|V| e j ), and VR=|V|e j is the rotor voltage. Moreover, the active and reactive power generations (PG and QG) at WPP bus, are also the WPP active and reactive power outputs (Pg and Qg), or: PG=Pg and QG=Qg. All of the equations to be solved and quantities to be determined in the complete load flow formulation are presented in Table 2. It is also worth mentioning here that there is an alternative expression to (11). This alternative formulation can be found in the appendix.

Test system
The new WPP model proposed in section 3 will be tested by using a power system shown in Figure 4. This power system is based on 9-bus system adopted from [25]. The test system is then modified by adding a WPP to bus 8 via a step-up transformer. Data for the test system (including the WPP) are shown in

Results and discussion
The load flow studies in the present work are carried out for various values of Pm (i.e. from 0.1 to 1.0 pu). The calculation results in terms of voltage magnitude and electric power output of the WPP are shown in Table 6. As a comparison, results of the calculation when the previous T-circuit model [23] is used are also shown in the table. It can be seen that both results are in exact agreement which indicates that the model proposed is also valid. To support the WPP reactive-power demand and improve system voltage profile, the WPP can be equipped with a shunt capacitor. Table 7 shows the calculation results when capacitor of 0.15 pu in capacity is installed. Slip (s) of the WPP induction generator is also calculated and shown in Table 8. Slip is determined based on the induction generator rotor voltage (VR), and computed using the formula given in the appendix. To have a better observation, the results in Tables 6-8 are also presented in the forms of graphs as shown in Figures 5-8. Figure 5 shows, as predicted, that the shunt capacitor is able to support the WPP reactive-power demand and improves system voltage profile. Figure 6 shows a graph where WPP active-power output (Pg) is plotted against turbine mechanical power input (Pm). It can be seen that the relationship is linear (i.e. Pg is always proportional to Pm). However, the value of Pg is a little bit smaller than Pm as there are some losses in the WPP induction generator.
Plot of WPP reactive-power demand (-Qg) against turbine mechanical power input (Pm) is shown in Figure 7. Unlike the graph in Figure 6, the relationship between -Qg and Pm is not linear (i.e. it is exponential). This relationship indicates that WPP requires much higher amount of reactive-power with the increase of turbine mechanical power (i.e. WPP active-power output).  Figure 8 shows the graphs of negative slip (-s) against turbine mechanical power input (Pm). It can be seen that for WPP without shunt capacitor, the values of slip vary in the range of -0.1080% and -1.1732%. Whereas, for WPP with shunt capacitor, the range of variation is slightly smaller (-0.1011% and -1.0807%). These results are expected since system with shunt capacitor has a better voltage profile and will lead to a higher (or less negative) value of slip.

CONCLUSION
A new steady state model of WPP has been presented in this paper. Similar to the previous T-circuit based models, it is also developed based on equivalent T-circuit of the WPP induction generator. However, unlike in the previous models, the mathematical formulation of the new model is shorter and less complicated. Moreover, the derivation process of the new model is much simpler. Only minimal mathematical operations are required in the process. Also, the rotor voltage value of the WPP induction generator is readily available as an output of the proposed model. This rotor voltage value can be used as a basis to calculate the induction generator slip. The validity of the proposed method has been tested on a representative 9-bus electrical power system installed with WPP. Comparative studies between the proposed method (new model) and other method (previous model) have also been carried out. Results of the studies confirm the validity of the new model. Then, (A.6) can be rewritten as: