Power Flow Analysis and Reactive Power Compensation of Grid Connected Wind Energy Conversion Systems

Abstract

The power flow analysis is the basic tool for analyzing the steady state operation of any power system. It also provides the necessary initial conditions to investigate the dynamic performance of the system. This chapter discusses the power flow analysis of grid connected wind energy conversion systems (WECS). The power flow analysis with WECS is quite complicated, unlike the analysis with conventional sources, because: 1. The power injected into the grid by WECS depends on the instantaneous wind speed, which varies unpredictably. 2. Most WECS use induction generators. Therefore, the operating slip of the machine has to be determined. The machine operates at a slip for which the mechanical power developed by the turbine is equal to the electrical power developed by the induction generator. This chapter outlines two methods (1) sequential method of power flow and (2) simultaneous method of power flow analysis for grid connected WECS. Both the methods of power flow are tested on a sample system and the results are presented. This chapter also looks into the effect of various types of reactive power compensation, namely, shunt, series and series-shunt compensation, on the steady state performance of WECS equipped with squirrel-cage induction generators. A cost effective method to strengthen the given network between the point of common coupling (PCC) and rest of the grid is proposed. The effect of compensation in improving the penetration level of wind energy into the grid is also analyzed.

Appendices

Appendix I

Data for 9-Bus Radial System

1. Transmission Line Data (All Lines)

Resistance	0.24 Ω/km
Reactance	0.48 Ω/km
Susceptance	2.80 μS/km
Length	20.0 km

Table A.1 Transformer data

Parameter	Load transformer data (all)	Step up transformer data (at the wind bus)	Feeding transformer data
Rated apparent power	0.63 MVA	1.0 MVA	25 MVA
Rated voltage of MV side	15 kV	15 kV	110 kV
Rated voltage of LV side	0.4 kV	0.69 kV	15 kV
Nominal short-circuit voltage	6 %	6 %	11 %
Copper loss at rated power	6 kW	13.58 kW	110 kW

2. Load Data (All Loads)

0.150 + j0.147 MVA

3. Coefficients of C _P —λ Curve

C1	0.5
C2	67.56
C3	0.4
C4	0
C5	1.517
C6	16.286
Gear box ratio	67.5

Fig. A.1

C_P—λ curve for the considered wind turbine

Asynchronous Generator Data (Δ-Connection)

Stator resistance	0.0034 Ω
Rotor resistance	0.003 Ω
Stator leakage reactance	0.055 Ω
Rotor leakage reactance	0.042 Ω
Magnetizing reactance	1.6 Ω

Appendix II

Jacobian Matrix for the 9-bus System

For the system considered the Jacobian is a (17 × 17) matrix as below. The numbers in the matrix represent the row and column of the elements. The analytical expressions for these elements are given in the subsequent sections.

$$ \left[ {\begin{array}{*{20}c} {(1,1)} & \cdots & {(1,8)} & {} & {(1,9)} & \cdots & {(1,16)} & {} & {(1,17)} \\ \vdots & { \ldots J_{1} \ldots } & \vdots & {} & \vdots & { \ldots J_{2} \ldots } & \vdots & {} & { \ldots J_{5} \ldots } \\ {(8,1)} & \cdots & {(8,8)} & {} & {(8,9)} & \cdots & {(8,16)} & {} & {(8,17)} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {(9,1)} & {} & {(9,8)} & {} & {(9,9)} & {} & {(9,16)} & {} & {(9,17)} \\ \vdots & { \ldots J_{3} \ldots } & \vdots & {} & \vdots & { \ldots J_{4} \ldots } & \vdots & {} & { \ldots J_{6} \ldots } \\ {(16,1)} & {} & {(16,8)} & {} & {(16,9)} & {} & {(16,16)} & {} & {(16,17)} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {(17,1)} & { \ldots J_{7} \ldots } & {(17,8)} & {} & {(17,9)} & { \ldots J_{8} \ldots } & {(17,16)} & {} & {(17,17) J_{9} } \\ \end{array} } \right] $$

1 Elements of J ₁

$$ \frac{{\partial P_{i} }}{{\partial \theta_{i} }} = imag\left[ {\mathop \sum \limits_{j \ne i} V_{i}^{*} V_{j} Y_{ij} } \right] $$

$$ \frac{{\partial P_{i} }}{{\partial \theta_{j} }} = - imag\left[ {\mathop \sum \nolimits V_{i}^{*} V_{j} Y_{ij} } \right] $$

2 Elements of J ₂

$$ \frac{{\partial P_{i} }}{{\partial |V_{i} |}}|V_{i} | = real\left[ {\mathop \sum \limits_{j \ne i} V_{i}^{*} V_{j} Y_{ij} } \right] + 2|V_{i} |^{2} real\left( {Y_{ij} } \right) $$

$$ \frac{{\partial P_{i} }}{{\partial |V_{j} |}}|V_{j} | = real\left[ {\mathop \sum \nolimits V_{i}^{*} V_{j} Y_{ij} } \right] $$

Diagonal element for WECS bus (bus 9),

$$ J\left( {8,16} \right) = J\left( {8,16} \right) - \frac{{2|V_{9} |^{2} r_{e} }}{{(r_{e}^{2} + x_{e}^{2} )}} $$

3 Elements of J ₃

$$ \frac{{\partial Q_{i} }}{{\partial \theta_{i} }} = real\left[ {\mathop \sum \limits_{j \ne i} V_{i}^{*} V_{j} Y_{ij} } \right] $$

$$ \frac{{\partial Q_{i} }}{{\partial \theta_{j} }} = - real\left[ {\mathop \sum \nolimits V_{i}^{*} V_{j} Y_{ij} } \right] $$

4 Elements of J ₄

$$ \frac{{\partial Q_{i} }}{{\partial |V_{i} |}}|V_{i} | = - imag\left[ {\mathop \sum \limits_{j \ne i} V_{i}^{*} V_{j} Y_{ij} } \right] - 2|V_{i} |^{2} real\left( {Y_{ij} } \right) $$

$$ \frac{{\partial Q_{i} }}{{\partial |V_{j} |}}|V_{j} | = - imag\left[ {\mathop \sum \nolimits V_{i}^{*} V_{j} Y_{ij} } \right] $$

Diagonal element for WECS bus (bus 9),

$$ J\left( {16,16} \right) = J\left( {16,16} \right) - \frac{{2|V_{9} |^{2} x_{e} }}{{(r_{e}^{2} + x_{e}^{2} )}} $$

$$ {\text{where}}, Z_{e} = Z_{S} + Z_{m} ||Z_{r} $$

5 Elements of J ₅

$$ J\left( {8,17} \right) = \frac{\partial }{\partial s}\left[ {\frac{{V^{2} r_{e} }}{{(r_{e}^{2} + x_{e}^{2} )}}} \right] = \frac{{|V_{9} |^{2} }}{{(r_{e}^{2} + x_{e}^{2} )^{2} }}\left[ {\left( {x_{e}^{2} - r_{e}^{2} } \right)\frac{{\partial r_{e} }}{\partial s} - 2r_{e} x_{e} \frac{{\partial x_{e} }}{\partial s}} \right] $$

6 Elements of J ₆

$$ J\left( {16,17} \right) = \frac{\partial }{\partial s}\left[ {\frac{{ - V^{2} r_{e} }}{{(r_{e}^{2} + x_{e}^{2} )}}} \right] = \frac{{ - |V_{9} |^{2} }}{{(r_{e}^{2} + x_{e}^{2} )^{2} }}\left[ {\left( {r_{e}^{2} - x_{e}^{2} } \right)\frac{{\partial x_{e} }}{\partial s} - 2r_{e} x_{e} \frac{{\partial r_{e} }}{\partial s}} \right] $$

7 Elements of J ₇ : 0

8 Elements of J ₈

$$ J\left( {17,16} \right) = \frac{{2|V_{9} |x_{m}^{2} R_{r} /s}}{{\left[ {\left( {R_{1} + \frac{{R_{r} }}{s}} \right)^{2} + \left( {x_{1} + x_{r} } \right)^{2} } \right]\left[ {R_{s}^{2} + \left( {x_{s} + x_{m} } \right)^{2} } \right]}} $$

$$ {\text{where}}, Z_{1} = Z_{s} ||Z_{m} $$

9 Elements of J ₉

$$ J\left( {17,17} \right) = \frac{{\partial \varDelta P_{m} }}{\partial s} $$

These are given by Eqs. (7.13) to (7.18).