Investigation of Wind Farm on Power System Voltage Stability Based on Bifurcation Theory

Zhiyuan Zeng, Xianqi Li, Jianzhong Zhou, Yongchuan Zhang

School of Hydropower and Information Engineering, Huazhong University of Science and Technology, China

Abstract—Voltage stability has been well investigated for the traditional power system using bifurcation theory since two decades ago. This paper studies the impact of wind farm on voltage stability of power system with and without reactive power compensation devices. The static reactive compensation devices including static capacitor banks and static var compensators (SVC) are used to improve the maximum loadability. The continuation method for power flow is used to obtain the system PV curves and determine the maximum loadability. The type of instability could possibly be of either Hopf. bifurcation or saddle node bifurcation. The equivalent wind farm model is established to replace the whole wind farm with a high number of wind turbines. The IEEE 14 bus benchmark system is used to demonstrate the reactive power compensation devices to support the voltage stability after wind farm integration into a power grid.

Keywords-Voltage Stability Hopf Bifurcation Wind FarmStatic Reactive Compensation

I. INTRODUCTION

The ability to maintain voltage stability has become a growing concern for the planning and operating today’s stresses power systems. It has been lengthy studied using bifurcation theory for the traditional power systems in the past two decades [1][2]. Wind energy has become one of the most important and promising sources of renewable energy all over the world, mainly because it is considered to be nonpolluting and economically viable with the rapid development of related wind turbine technology. With wind farms capacities continuously increasing, the impact of wind farm integration on voltage stability has attracted more concerns. The dynamic of a large class of power systems can be represented by parameter dependent differential-algebraic models of the form

),,,(0),,,(

pyxgpyxfx

λλ

==

•

(1)

When the parameters and/or p (such as load of the system) vary, the stable equilibrium points may lose its dynamic stability at local bifurcation points. These equilibrium points are asymptotically stable if all the eigenvalues of the system state matrix have negative real parts. As the parameters change, the eigenvalues associated with the corresponding equilibrium point change as well. Upon parameters variation, local bifurcation analysis of equilibria of the DAE model often

results in three major bifurcations, saddle node (SN), Hopf and singularity induced (SI) bifurcations. The point where a complex conjugate pair of eigenvalues reach the imaginary axis with respect to the changes in ( , p), say (x0, y0, 0, p0), is known as a Hopf bifurcation point. More detailed discussions about the stability of power systems DAE model are in [3]. The fast growth of wind generation has led to concern about the effect of wind power on the voltage stability of the power grid. Compared to the conventional power plants, wind parks exhibit certain singular characteristics, especially for variable speed wind turbines, which not only produce reactive power but also absorb it. The reactive power supporting capability directly changes the voltage stability of power grid. In [4], an aggregate model of a grid-connected wind farm for power stability has been studied. It concentrated on the shaft systems of the wind turbines when a simplified aggregate model of the wind farm is used in voltage stability investigations. In [5], it analyzed the voltage stability in a weak connection wind farm. The possibility of network voltage drop and instability are investigated by the detailed electromagnetic transient simulation program.This paper is structured as follows: the foundation of power

system stability and bifurcation analysis is described in section II. In section III, wind farm model is established. Simulation results are presented which illustrate various effects on the power system voltage stability bifurcations theory in Section IV. In section V, a summary is given and the conclusion of this paper is presented.

II. BASIC BACKGROUND

Static and dynamic approaches were presented in [6] to analyze voltage stability. Static approaches like sensitivity analysis, modal analysis and P-V and Q-V methods for voltage stability assessment use a system condition or snapshot for voltage stability evaluation. They usually solve power flow equations of the network with specific load increments until the point of voltage collapse is reached. These techniques allow examination of a wide range of system conditions and can provide much insight into the nature of this phenomenon by computation of the contributing factors. For small disturbance analysis, the dynamic approach is based on the eigenvalue computation of the linearized system, while for large disturbance analysis a complete time domain simulation is required.

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Typically, PV curve analysis is widely used in the industry for analyzing voltage stability problems [7]. Continuation Power Flow (CPF) presents a way to plot complete PV curves by automatically changing the value of the loading parameter ( ). The principle of continuation is a mathematical technique in which the path of an established solution of a system of equations is followed around parameter space when a control parameter is varied. Implementing a predictor---corrector scheme, a continuation algorithm can trace the path of an already established solution as the parameters are varied [8]. Bifurcation analysis is appropriate for voltage stability since it is observed that the instability that may occur usually coincides with the singularity of system state Jacobian, which is the necessary condition of saddle-node bifurcation. If a constant power load is used as the bifurcation parameter, the SNB point corresponds to the nose point of the P−V or Q−Vcurve. The physical explanation is that the load power reaches its maximum which can be transmitted through the network. Mathematically, the Jacobian matrix has a zero eigenvalue when evaluated at the SNB point. When Hopf bifurcation occurs, the system suddenly starts oscillating, and then it may keep on periodic oscillating or lose stability at last by increasing the amplitude continuously [9].

III. WIND FARM MODEL 3.1 wind turbine model A complete wind turbine model includes wind speed model, wind generator model, and corresponding controller model. In this paper, the composite wind model presented in [10] is used, which has been demonstrated accuracy and efficiency. Wind generator and controller are used the simplified models, which is given in detail in [11]. 3.2 Aggregated model A complete model of a wind farm with a high number of wind turbines, may lead to compute an excessive and impractical number of equations. The size of the wind farm model may be reduced by aggregating several wind turbines with similar incoming wind into a bigger turbine called aggregated model. The mechanical and electric parameters per unit are preserved, and the nominal power is increased up to the sum of the nominal power of the whole set of turbines to obtain the parameters of the aggregated turbine [12].

Due to the wide range of distribution and geographically variations, sometimes a large wind farm will cover several square kilometers. Therefore, the wind farm equivalent model includes an aggregated model of the generation systems and a dynamic simplified model of each individual wind turbine approximating the operation points of each wind turbine according to the corresponding incoming wind speed[13] Meanwhile the equivalent wind turbine presents n-times the size of individual wind turbines, and therefore a rated power equal to n-times the rated power of individual wind turbines, where n is the number of aggregated wind turbines. For example, thirty wind turbines are equivalent into three wind turbines under the three typical wind speeds conditions and three aggregated rated powers, as shown in Fig.1.

Fig. 1. Wind farm detailed and aggregated model

Fig. 2. IEEE 14 bus benchmark system

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1

Total System Load (MW)

Bus

14 v

olta

ge (p

.u.)

30 DFIGs pre-contingency PV curve3 DFIGs pre-contingency PV curve30 DFIGs post-contingency PV curve3 DFIGs post-contingency PV curve

operating pointHBHB

Fig. 3. PV curve for the system with detailed and aggregated wind farm model under the normal and contingency conditions

IV. RESULTS

All results presented in this section are obtained from a slight modified of the IEEE 14 bus benchmark system, which was described in detail [14]. The system consists of five synchronous machines, three of which are synchronous compensators used only for reactive power support, represented with subtransient models and with IEEE type-1 exciters. There are 11 loads in the system, totaling 259 MW and 81.3 Mvar, represented as constant power loads (PQ loads). To examine the wind farm effect, the transmission lines in the IEEE 14 bus test system were modified appropriately to achieve a realistic wind penetration level. The one-line diagram of the system can be seen in Fig. 2. The original system of this reference was modified to include a wind farm composed of 30 turbines of 2 MW each. The dynamic model of the wind farm includes an equivalent model of its internal electric network; thus, an equivalent wind generator at 2 MW is considered. The wind farm is connected to the grid through 25kV/69kV transformers. The parameters of modified transmission lines and equivalent wind turbine are given in Appendix. The comparative results of PV curves between detailed and aggregated wind model are shown in Fig.3. The simulation results are very similar under the normal and contingency operating conditions. Therefore, the aggregated wind farm model can be approximately used to replace the whole wind farm to save time computation and improve efficiency. The dynamic behavior of the power system has been changed after the wind farm integration, and it is depicted in Fig. 4. The maximum loadability and Hopf. bifurcation (HB) will decrease due to wind farm replacing the synchronous generator at bus 2. The cause is the wind farm internal network decreasing the voltage margin. The internal network parameters for the step-up transformer and transmission lines are changed to verify it. With the parameters increasing, the loadability will decrease. On the contrary, the loadability will increase, and the result is shown in Fig.5. With the wind speed various, the output power of wind farm will change correspondingly. The dynamic behavior of three wind speed conditions is shown in Fig.6. With wind speed increasing, under the limit of wind turbines, the loadability will be improved, either the Hopf. bifurcations or saddle node bifurcations. The wind farm reactive power includes wind turbine and wind electric field of reactive power compensation device. First of all, should make full use of wind power generating units of reactive power capacity and its ability to regulate, if only the wind turbine reactive power capacity can not meet the needs of system voltage regulator, we need to consider the installation of wind power reactive power compensation device. Wind power reactive power compensation devices division can be switched capacitors or reactor group and, if necessary, can be used for regulating the SVC or other more advanced reactive power compensation device. A static capacitor bank to compensate for the reactive power drawn by the induction generator is connected at the bus.

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V14

(p.u

.)

Original System PV CurveDFIG System PV Curve HB

HB

SNB SNB

Fig. 4. PV curves for the system with synchronous generator and wind farm

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V14

(p.u

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Internal NetworkParameters IncreasedSystem PV Curve(*5)

Internal NetworkParameters DecreasedSystem PV Curve(*0.01)

Origianl System PV Curve

Fig. 5. PV curves for the system with wind farm after changing parameters

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High W ind Speed PV Curve

Medium W ind SpeedPV Curve

Low W ind Speed PV Curve

SNB

HB

Normal Operation----- Line 2-4 removing Contingency

Fig. 6. PV curves for the system with wind farm under the various wind speeds

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HB after DFIG withCapacitor Bank

HB after DFIG with SVC

Original System HB

Fig. 7. PV curves for the system with reactive compensation devices

The internal parameters are not changed, but the reactive compensation devices are added to improve the system voltage margin. But the phenomenon can be found that static capacitor will decrease the HB, although the loadability is increased. However, the SVC will improve the HB at the same time to increase the loadability. Moreover, the SVC is installed on the common bus of wind farm. Finally, a power stability stabilizer (PSS) is used to remove the HB and improve the stability.

V. CONCLUSIONS This paper investigates that using bifurcation theory to study the impact of wind power on the system voltage stability. The aggregated model is used to replace the whole wind farm. The result shows that the discrepancy is very small under the condition of either normal or contingency operation. Therefore, the aggregated model can be used to study the voltage stability with reducing calculation complexity. Meanwhile, traditional switched capacitor banks cannot easily follow voltage swings caused by a wind farm, since these devices are only designed to correct slowly-changing voltages that naturally occur as load cycles over 24 hours. A static var compensator (SVC) is a much better solution. This device is similar to a switched capacitor bank, but instead of mechanical switches, electronic semiconductors are used thus achieving fast and continuously-variable reactive power output.

ACKNOWLEDGMENT

The authors gratefully acknowledge Professor Claudio Cañizares in the University of Waterloo, Waterloo, Ontario, Canada and Dr. Federico Milano, for his excellent simulation software PSAT and many valuable discussions.

APPENDIX

The parameters of those transmission lines near to wind farm will be modified as follows.

REFERENCES

[1] C. A. Cañizares, Ed., “Voltage Stability Assessment: Concepts, Practicesand Tools,’’ Special Publication of IEEE Power System Stability Subcommittee, Tech. Rep. SP101PSS, Aug. 2002.

[2] T.Van Cutsem and C.Vournas, Voltage Stability of Electric Power Systems, Kluwer Academic Publishers, 1998

[3] D.J. Hill and I.M.Y. Mareels, “Stability theory for Differential/Algebraic systems with application to power systems,” IEEE Trans. Circuits and Systems, vol. 37, no. 11, pp. 1416-1423. Nov. 1990.

[4] V. Akhmatov and H. Knudsen, “An aggregate model of a grid-connected, large scale, offshore wind farm for power stability investigations- Importance of windmill mechanical system,’’ Int. J. Elect. Power Energy Syst., vol. 24, pp. 709---717, Nov. 2002.

[5] F. Zhou, G. Joos, and C. Abbey, “Voltage stability in weak connection wind farms,’’ in Power Engineering Society General Meeting, San Francisco, vol. 2, pp.1483---1488, Jun. 2005.

[6] G.K.Morison, B.Gao, and P.Kundur, "Voltage Stability analysis using static and dynamic approaches," IEEE Trans. on Power Systems, Vol. PWRS8, No.3, pp.1159-1171, Aug.1993

[7] C.W. Taylor. Power System Voltage Stability. McGraw-Hill, New York, 1994.

[8] V. Ajjarapu and C. Christy, “The continuation power flow: a tool for steady state voltage stability analysis,” IEEE Trans. on Power Systems,Vol. 7, pp. 416 - 423, Feb. 1992.

[9] Ajjarapu, V, Lee, B, “Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system,” IEEE Trans. on Power Systems, vol.7.no.1, pp.424-431, Feb. 1992.

[10] P. M. Anderson and A. Bose, “Stability simulation of wind turbine systems,” IEEE Trans. Power App. Syst., vol. 102, pp. 3791–3795, Dec.1983.

[11] F. Milano, Power System Analysis Toolbox: Documentation for PSAT version 2.0.0, Mar. 2006 [Online]. Available: http://thunderbox.uwaterloo.ca/-fmilano

[12] J G Slootweg, “Wind power, modelling and impact on power system dynamics,” PhD Thesis, Power Systems Laboratory, Delft University of Technology, 2003.

[13] L.M. Fernandez, F. Jurado, and J.R. Saenz, “Aggregated dynamic model for wind farms with doubly fed induction generator wind turbines,” Renewable Energy, vol.33, no.1, pp.129–140, 2008.

[14] S. K. M. Kodsi and C. A. Cañizares, “Modeling and simulation of IEEE 14 bus system with Facts controllers, Tech. Rep.,” Univ. Waterloo, ON, Canada, [Online] Available: http://www.power.uwaterloo.ca., Mar.2003.

Static Capacitor Bank parameters Power

rating (MW)

Voltage rating (kV)

Frequency rating (HZ)

Susceptance (p.u.)

SC 1 22 25 60 0.8 SC 2 20 25 60 0.8 SC 3 18 25 60 0.8

SVC parameters Parameters values

Type 1 Power (MVA) 59.3

Regulator time constant--Tr (s) 5 Regulator gain--Kr (p.u./p.u.) 1000

Reference voltage (p.u.) 1.00 Bmax (p.u.) 1.00 Bmin (p.u.) -1.00

Constant wind speed parameters. Resistance

(p.u.) Reactance

(p.u.) Susceptance

(p.u.) Line 1 2*0.05695 2*0.17388 2*0.034 Line 3 2*0.01938 2*0.05917 2*0.0528 Line 7 2*0.05403 2*0.22304 2*0.0492

(*The parameters about wind speed without being listed will be zero)

Wind turbine parameters Rated

power (MW)

Active power (p.u.)

Normal wind speed (m/s)

Air density

rho (kg/m3

)

Filter time constant

tau(s)

WT 1 22 0.667 14.88 1.225 4.0 WT 2 20 0.667 14.50 1.225 4.0 WT 3 18 0.667 14.10 1.225 4.0