Two-port network model of fixed-speed wind turbine generator for distribution system load flow analysis

Load flow analysis has always been used in determining the steady-state operation of an electric power or distribution system. For conventional power system without wind turbine generator, the method for load flow analysis has been well established. However, for modern system embedded with wind turbine generator, the investigation of analysis method is still an active research area. This paper proposed a new method to integrate fixed-speed wind turbine generator into distribution system load flow analysis. The proposed method is derived based on two-port network theory where the parameters of induction generator of the wind turbine generator are embedded in general constants of the two-port network. The proposed method has been tested and verified using a representative electric distribution system.

In (1), SG, SL, V and Y are generated power vector, load power vector, nodal voltage vector and nodal admittance matrix, respectively. For n-node power system, these vectors and matrix are of the forms: where: SGi : generated power flowing into node i SLi : load power flowing out node i Vi : voltage at node i Yij : element ij of admittance matrix After (1) has been solved and all of the quantities have been determined, the line flows and losses can also be calculated. It is to be noted that distribution system is usually fed at one bus (substation bus), and the voltage at this bus is known or specified. Therefore, only the voltages at the remaining buses (load buses) need to be computed. Table 1 shows the known and unknown quantities of the distribution system load flow formulation. For conventional system without WTGS, (1) is the only set of equations that needs to be solved to obtain the load flow solution. However, for modern system with WTGS, additional equation is needed to facilitate and incorporate the WTGS into the load flow analysis. The modeling and integration of the WTGS will be discussed in the following section.

Proposed Model of WTGS and Its Integration 2.2.1. Two-Port Network Theory
In the present work, development of fixed-speed WTGS model will be based on two-port network mathematical model where nodal power equations are used in the model derivation. The general two-port network containing passive impedances is presented in Figure 1. In two-port network theory, it can be shown that the voltage/current relationship is of the form: In (4), A, B, C and D are the general two-port network constants. The values of these constants depend on the network components, i.e. their impedances and admittances. The (3) is equivalent to the following two (5a) and (5b). The development of the proposed WTGS model based on (5) will be explained in the next section.  Figure 1. General two-port network Figure 2(a) shows fixed-speed WTGS connected to a distribution system [16][17][18][19][20]. Power converter of the WTGS is squirrel cage induction generator (SCIG). The SCIG has mechanical power input Pm and electrical power output Sg= Pg+jQg as can be seen in Figure 2 [21][22][23] where R1, X1, R2, X2, Rc and Xm denote stator resistance, stator leakage reactance, rotor resistance, rotor leakage reactance, core loss resistance and magnetic reactance, respectively. R2(1-s)/s is the dynamic resistance and its value depends on slip s. Power of the dynamic resistance represents the mechanical power Pm delivered by the wind turbine to SCIG. The value of this power depends mostly on the wind speed and it can be determined using the power curve provided by the turbine manufacturer. In the proposed method, the WTGS model is obtained by viewing the SCIG equivalent circuit of Figure 3(a) as the two-port network depicted in Figure 3 (b). In Figure 3 (b), the impedances Z1, Z2 and Z3 are given by:

Proposed Model of WTGS
it is to be noted that for the two-port network of Figure 3 Figure 3. Steady-state equivalent circuit of SCIG based on Figure 3 (b), the SCIG electrical power output is formulated as: (8) and the mechanical power delivered by wind turbine to the SCIG is calculated as: Application of two-port network equations in the above SCIG mechanical power input equation, or by substituting (5a) and (5b) into (9), gives: or: on using (8) in (11), i.e. substituting I1 in (11) by Sg*/V1*, the following equation is obtained: In (12) is the proposed model for fixed-speed WTGS. It is to be noted that V1 in (12) is the voltage at WTGS bus (or it is equal to Vk in Figure 3 (a)). Incorporation of the model into the load flow formulation (1) will be discussed in the next section.

Integration of WTGS Model
For modern system containing fixed-speed WTGS, solution to the power flow problem can be found by simultaneously solving (1) and (12). It can be seen that (12) is the additional equation to the formulation in (1). Whereas, the additional quantity that need to be calculated is the WTGS power output Sg. Table 2 shows the known and unknown quantities in the load flow formulation of the system with WTGS. It is to be noted that since sets of the equations, i.e

Results and Analysis 3.1. Test System
The proposed method for incorporating WTGS in distribution system load flow analysis is tested by using the 33-bus system [25]. This distribution network has the system voltage of 12.66 kV. One line diagram of the distribution system is shown in Figure 4, and the detail of the system data can be found in [25]. In the present work, it is assumed that the distribution system has one WTGS and it is connected to bus 33. Data for the SCIG of WTGS is shown in Table 3. All of the data are in pu on 1 MVA base. Results of the load flow analysis for the test system are presented in the following section.

Results and Analysis
In this paper, power flow calculations were carried out for various values of mechanical power Pm. The mechanical powers ranging from 0.1 to 1.0 pu were taken in the investigation. These values represent the low speed and higher speed wind conditions. It is to be noted that all of the computations were done on PC, and the proposed method were implemented as MATLAB codes (m-files).
Results of the calculations (in terms of WTGS voltage/power and substation power) are shown in Table 4. It is to be noted that the results of the proposed method are accurate and in excellent agreement with those of the method in [16][17][18]. These results confirm that the proposed two-port model is valid and can be used as a method for incorporating fixed-speed wind turbine generator into distribution system load flow analysis.
Since fixed-speed WTGS is usually equipped with shunt capacitance to support the reactive-power consumed by induction generator of the WTGS, the effects of the capacitance installation are also investigated in this paper. Table 5 shows the results of load flow analysis (in terms of WTGS voltage/power and substation power) when the capacitance with the capacity of 0.5 pu is installed. To observe the effects more clearly the results are also presented in graphical forms in Figures 5 and 6.   Table 5 clearly shows that shunt capacitance is able to support the WTGS reactive-power demand as indicated by the improvement of WTGS voltage profile. Although its value is slightly less than mechanical power input (due to losses in induction generator), active-power generation of WTGS is always proportional to the mechanical power input as shown in Figure 5 (a). However this is not the case for the variation of WTGS reactive-power. With the increase in mechanical power, the WTGS reactive-power increases almost exponentially as shown in Figure 5 (b). In other words, the increase in WTGS active-power generation requires higher increase in WTGS reactive-power demand. This result is also confirmed by Figure 6 (b). Figure 6 (a) shows that with the increase in mechanical power, the active-power supplied by distribution substation decreases linearly. This result is expected because with the increase in mechanical power, the WTGS active-power generation also increases, and therefore more active-power can be delivered by WTGS to support the system load demand. Consequently, with the increase in WTGS mechanical power, distribution substation can deliver less active-power since part of the system load is supplied by WTGS.

Conclusion
A simple method for incorporating fixed-speed WTGS into a distribution system load flow analysis has been presented in this paper. The development of the proposed method is based on two-port network theory where equations from the electric circuit theory have been utilized to derive the proposed WTGS mathematical model. By integrating the model into the load flow analysis, the steady-state operations of the system (including the WTGS) can then be The proposed method has been tested and verified using a representative test system, i.e. 33-bus distribution network.