Studies on VIPI based control methods for improving voltage stability

PII:

Electrical Power & Energy Systems, Vol. 20, No. 2, pp. 141–146, 1998 q 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0142-0615/98 $19.00+0.00 S0142-0615(97)00035-5

Studies on VIPI based control methods for improving voltage stability M Nanba, Y Huang, T Kai and S Iwamoto Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169, Japan

method can be applied to preventive control. The second one determines the controls needed to maintain the specified threshold value, based on the sensitivities of VIPI with respect to control variables. This method improves the voltage stability with minimum operational actions by regulating the controllers with the maximum sensitivity and can be applied to corrective control. The obtained sensitivities can also be utilized for operator judgement. There are many controllers used for voltage and reactive power control. In this paper, the proposed methods employ transformer tap changers, static capacitors, SVCs and generator terminal voltages. Simulations are carried out for a 6-bus power system to confirm the effectiveness of the proposed methods.

Based on the concept of Voltage Instability Proximity Index (VIPI) proposed by Y. Tamura et al., this paper develops two control methods for improving voltage stability. The first method maximizes the value of VIPI by using a Successive Quadratic Programming (SQP) method to find the optimal controls under various system conditions. The second determines the controls needed to maintain the specified threshold value, based on the sensitivities of VIPI with respect to control variables. The controllers used in these methods are transformer tap changers, static capacitors, SVCs and generator terminal voltages. The proposed methods have been tested for a 6-bus power system, and successful results have been obtained. q 1997 Elsevier Science Ltd. Keywords: voltage stability, voltage control, VIPI

II. Theory I. Introduction

II.1 VIPI The load flow equation can be written as below in the rectangular coordinates.

There are many publications concerning voltage stability issues (e.g. [1–14]). The approaches can be divided into two categories: steady-state approaches and dynamical approaches. In spite of the complexity of the dynamic characteristics, steady-state approaches are still considered important for voltage stability problems. Several voltage stability indices have been proposed from the viewpoint of the steady-state approaches. For on-line monitoring and control of voltage stability, a voltage stability index should provide a margin to the critical point and is ideal if it will not give singular values near the critical point. There have been many investigations so far for the mechanism and indices of voltage stability, but there are still few publications about voltage stability control (e.g. [9– 11]). Based on these considerations, this paper focuses on the Voltage Instability Proximity Index (VIPI) [1,2] proposed by Y. Tamura et al. and develops two control methods for improving voltage stability based on the concept of VIPI. The first method maximizes the value of VIPI by using a Successive Quadratic Programming (SQP) method [15] to find optimal controls in various system conditions. This

ys ¼ y(x) ¼ y(xe þ Dx) ¼ y(xe ) þ J(xe )·Dx þ y(Dx)

(1)

where y s is the specified value vector, y load flow vector and x the bus voltage vector. The elements of y and x at the ith bus are represented below as, respectively: # 8 " P (x) > i > > (Generator Bus) > > < V2i (x) yi ¼ " # > > Pi (x) > > > (Load Bus) :

(2)

Qi (x)

"

xi ¼

ei fi

#

(3)

where P is active power, Q reactive power and V bus voltage. Assuming that there are two voltage vectors x 1 (stable) and x 2 (unstable) satisfying the same specified value vector

141

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142

Figure 1. Concept of VIPI y s, and using a singular vector a and a margin vector b, x 1 and x 2 can be expressed as follows: (4) x1 ¼ a þ b, x2 ¼ a ¹ b where x þ x2 x ¹ x2 , b¼ 1 a¼ 1 2 2 Substituting x 1 and x 2 into equation (1), we have equations (5) and (6) respectively. ys ¼ y(a þ b) ¼ y(a) þ J(a)·b þ y(b)

(5)

ys ¼ y(a ¹ b) ¼ y(a) ¹ J(a)·b þ y(b)

(6)

Adding equations (5) and (6), we obtain ys ¼ y(a) þ y(b), J(a)·b ¼ 0

(7)

Since J(a)·b ¼ 0, det J(a) ¼ 0, Thus a is a critical voltage vector, VIPI is defined as a scalar index representing the solid angle between the operating state vector y s and the critical point vector y(a) [1]. VIPI ; v ¼ cos ¹ 1

yTs y(a) kys k·ky(a)k

(8)

The relationship between the vectors in the voltage space and the node-specification space can be depicted as in Figure 1. As the power system becomes more heavily loaded, VIPI decreases and becomes zero at the critical point. The decrease is observed to be almost linear. Therefore, VIPI is considered very useful for monitoring the degree of voltage stability. II.2 SQP based control method First, we consider a SQP based control method for preventive control. As VIPI represents the angle between an operating state vector and a critical point vector, the controls should be operated as to make a larger angle for VIPI. Therefore, for the first VIPI based control method, a nonlinear programming approach that maximizes VIPI( ¼ F) under the system constraints is formulated as follows: Maximize F ¼ cos ¹ 1 subject to 2

Pmin

3

2

P(x, u)

yTs ya kys k·kya k 3

2

Pmax

(9) 3

6 7 6 7 6 7 6 Qmin 7 6 Q(x, u) 7 6 Qmax 7 6 7 6 7 6 7 6 7#6 7#6 7 6 Vmin 7 6 V(x, u) 7 6 Vmax 7 4 5 4 5 4 5

umin

2

u

3

2

PL (x, u) # PL max

3

umax

where y s, is a function of voltage vector x and control variable vector u, and y a is a function of critical voltage vectors a and u. Also, P is the vector of bus active powers, Q bus reactive powers, V bus voltage magnitudes, u controllers and P L line active power flows. By solving this problem using a SQP method, the angle between y s and y a can be widened using the obtained optimal controls and voltage stability can be improved. The SQP method is one of the most effective methods for nonlinear programming problems, which can deal with equality and inequality constraints and solve the approximated second order programming problem successively. This method can be considered as an expansion of the quasi Newton method for the optimizing problem without constraints. A general nonlinear programming problem is defined as follows: Minimize f (x)

(10)

subject to ci (x) ¼ 0

(i ¼ 1, …, me )

ci (x) $ 0

(i ¼ me þ 1, …, m)

For SQP, the formula can be written as follows and x is updated by x (kþ1) ¼ x (k) þ d. 1 Minimize =f (x(k) )T d þ dT =2 f (x(k) )d (11) 2 subject to 1 ci (xk ) þ =ci (x(k) )T þ dT =2 ci (x(k) )d ¼ 0 (i ¼ 1, …, me ) 2 1 ci (x(k) ) þ =ci (x(k) )T þ dT =2 ci (x(k) )d $ 0 2 (i ¼ me þ 1, …, m) From equation (11), it is obvious that partial derivatives of the objective function and constraints are required. In order to apply the SQP method, the problem (9) is transformed to equation (11). Partial derivatives of equation (9) with respect to the element of x are expressed as follows by using the chain rule. X ]F9 ]y X ]F9 ]y ]F 1 si ai ¼¹ þ ]xk sin F i ]ysi ]xk i ]yai ]xk

!

(12)

where F9 ¼ cos F. Partial derivatives of F9 with respect to the elements of y s and y a in equation (12) are given by X  X  2 y · y ¹ y y y ak sk si ai si ]F9 ¼ (13)  X 1=2  X 3=2 ]ysk · y2 y2 ai

si

X  X  y2ai ¹ yak ysi yai ]F9 ysk · ¼  X 1=2 ÿ 13=2 ]yak · Sy2ai y2si

(14)

Other derivatives in equation (12) can be obtained from the Jacobian of the load flow calculations. In this method, state variables are bus voltages and control variables are ratios of transformer tap changers, banks of static capacitors and admittances of SVCs. The discrete type controllers such as tap changers and static capacitors are treated as continuous type controllers at first, and then changed to discrete values.

VIPI based control methods: M. Nanba et al.

143

II.3 Sensitivity based control method The sensitivity based control method is developed for corrective control. This method improves voltage stability by regulating controllers (such as static capacitors, SVCs, transformer tap changers and AVRs) according to sensitivities of VIPI with respect to the control variables. The minimum operational actions can be obtained by regulating the controller with the maximum sensitivity one by one. The variation of VIPI ( ¼ F) can be expressed as below. 

DF ¼

]F ]x

T



Dx þ

T

]F ]xp



Dx p þ

]F ]u

T

Du

(15)

where x is a voltage vector of an operating point and x* is a voltage vector of an unstable or fictitious solution. Load flow equations (except the slack bus) can be represented as below: G(x, u) ¼ 0

(16)

The element of G is as below.

# 8" Pi (x) ¹ Psi > > > (Generator Bus) > > < Vi2 (x) ¹ Vsi2 Gi ¼ " # > > Pi (x) ¹ Psi > > > (Load Bus) :

¹1

·



]G Dx p ¼ ¹ ]xp

]G ·Du ¼ HDu ]u

¹1

·

where a is the margin constant for target determination. Figure 2 shows the concept of the control target determination and Figure 3 shows the flow chart of this control method.

III.1 Model system In order to see the effectiveness of the proposed methods, simulations have been carried out for a 6-bus power system shown in Figure 4. This system is based on Ward and Hale’s 6-bus system with static capacitors added at bus 4 and bus 6,

(18)

]G ·Du ¼ H p Du ]u

(19)

Substituting equations (18) and (19) into equation (15), we can obtain 

]F DF ¼ ]x

T



T

]F HDu þ ]xp



]F H p Du þ ]u

T

Du

(20)

Thus the sensitivity vector S of VIPI with respect to control variables can be obtained by 

DF S¼ Du

T



]F ¼ ]x

T



]F Hþ ]xp

T



]F Hp þ ]u

T

(21)

Derivations of partial derivatives can be carried out in the same way as in the first method. While in the SQP based method only continuous type controllers are considered, in this method both continuous and discrete type controllers can be taken into account. A discrete type controller can be operated step by step according to the sign of the sensitivity. For a continuous type controller, controls can be determined by the sensitivity and the specified control target value. That is, Du is determined by the following equation. Dui ¼

(F(x) ¹ Ft ) Si

(23)

III. Simulation

The variation of x and x* with respect to the change of u can be written as ]G Dx ¼ ¹ ]x

Ft ¼ F p ¹ aDF

(17)

Qi (x) ¹ Qsi



Figure 2. Concept of control target determination

(22)

where F t is the control target value and S i the ith element of S. The variation of F for load increase is represented by DF, and the control target F t is determined so that it does not become smaller than the threshold value F* in each control step as below.

Figure 3. Flow chart control

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144

Table 2. Variation of bus state [Case (1), K ¼ 1.0] Bus Before control V 1 2 3 4 5 6

P

After control Q

V

1.050 0.956 0.457 1.050 0.500 0.178 0.914 ¹0.550 ¹0.130 0.936 0.000 0.000 0.884 ¹0.300 ¹0.180 0.917 ¹0.500 ¹0.050

P

Q

1.100 0.767 0.274 1.100 0.700 ¹0.175 1.036 ¹0.550 ¹0.130 1.016 0.000 0.000 0.988 ¹0.300 ¹0.180 1.030 ¹0.500 ¹0.050

Figure 4. Model system and SVCs added at bus 3 and bus 5. The loads are the same as those of Ward and Hale’s original system. The controllers treated in this paper are static capacitors (SC), SVCs, transformer tap changers (Tap) and generator terminal voltages (VG). The parameters of the controllers are shown in Table 1. The admittance of the static capacitor bank is 0.05 p.u. The ratio r t of the transformer tap changer is calculated by rt ¼ 1 þ 0:02·pt

(24)

where p t is a tap position. In order to change loading for simulations, specified active powers and reactive powers of each bus are multiplied by the loading factor K. In addition, a fault is assumed in which transmission line L2 between bus 1 and bus 6 is tripped at K ¼ 1.1. III.2 SQP based control method In the model system shown in Figure 4, simulations are carried out for the following cases. Simulation cases Case (1): without fault Case (2): with fault

(K ¼ 0.8, 1.0, 1.2, 1.4, 1.6) (K ¼ 1.1)

Constraints for all cases are shown in the following. Constraints (p.u.) Generator bus :DP 6 0.2, DQ 6 0.2, DV 6 0.05 Load bus :P, Q fixed, 0.9 # V # 1.1 Line flow :L1, L2, L3 # 1.5; L5, L6 # 0.5; L4, L7: no constraints where D means changes from the base values. The base values are shown as the values of V, P, Q for ‘‘Before control’’ in Table 2. Figure 5 shows the variations of VIPI before and after control for each case without fault. It is obvious that the value of VIPI increases and voltage stability is improved after control. Tables 2 and 3 show the variations of each bus state before and after the control at K ¼ 1.0 in Case (1). In Table 1. Parameters of controllers Controller

Control variable

Tap changer Static capacitor SVC Generator

Tap position Bank Admittance Bus voltage

Initial value

Upper Lower limit limit

0 1 0.0 1.05

5 ¹5 4 0 1.0 ¹1.0 1.10 1.00

Figure 5. Result of SQP based method [Case (1)] Table 3. Variation of controllers [Case (1), K ¼ 1.0] Controller Tap1

Tap2

SC1

SC2

SVC1 SVC2

Before After

0 0

1 3

1 4

0.00 0.10

0 2

0.00 0.10

this example, there is a voltage violation (V 5 ¼ 0.884) and it is corrected after the control (V 5 ¼ 0.988). Tables 4 and 5 show the variations of each bus state before and after control in Case (2). In this example, there are 4 voltage violations (V 3,V 6), but they are corrected after the control (VIPI changed from 11.41 deg to 22.23 deg).

Table 4. Variation of bus state [Case (2), K ¼ 1.1] Bus

Before control V

1 2 3 4 5 6

P

After control Q

V

1.050 1.114 0.623 1.050 0.550 0.343 0.859 ¹0.605 ¹0.143 0.877 0.000 0.000 0.804 ¹0.330 ¹0.198 0.824 ¹0.550 ¹0.055

P

Q

1.100 0.913 0.274 1.100 0.723 ¹0.175 0.906 ¹0.605 ¹0.143 0.968 0.000 0.000 0.956 ¹0.330 ¹0.198 0.956 ¹0.550 ¹0.055

Table 5. Variation of controllers [Case (2), K ¼ 1.1] Controller Tap1

Tap2

SC1

SC2

SVC1 SVC2

Before After

0 ¹2

1 0

1 4

0.00 0.01

0 2

0.00 0.10

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145

Figure 6. Result of sensitivity based method [Case (3)]

Figure 7. Result of sensitivity based method [Case (4)]

III.3 Sensitivity based control method, two cases have been tested

proposed controls maintain the threshold value until K reaches the limit where all the controllers are regulated to the upper or lower limits. Note that there have been cases where VIPI detracted from the action of the largest sensitivity control unit. In that case we locked the control unit and searched for the second best controller.

Simulation cases Case (3): without fault Case (4): with fault

(K ¼ 1.0,1.5) (K ¼ 1.0,1.3)

The threshold value F* of VIPI is taken as 20 deg from experience and the step of K is taken as 0.01. Constraints of controllers are the same as in Table 1. The simulation results are shown in Figures 6 and 7. The state of the operated controllers are shown in Tables 6 and 7. For example for Case (3), the simulation starts with K ¼ 1.00 where VIPI ¼ 29.6 deg (no violation). When K is increased to 1.10, no control is activated because VIPI ¼ 24.4 deg. But when K ¼ 1.20, the control action of VG1 is activated as shown in Table 6. When K ¼ 1.28, VG1 reaches the upper limit of 1.100 and then VG2 starts to join the control. The

IV. Conclusions There have been many investigations so far into the mechanism and indices of voltage stability, but there are still few publications about voltage stability control. In this paper, two voltage stability control methods based on the concept of VIPI have been proposed. One is a SQP based control method developed for preventive control, and the other is a sensitivity based control method developed for corrective control.

Table 6. States of operated controller [Case (3)] K

Controller

Before

After

Controller

Before

After

1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34 1.36 1.38 1.40 1.42

VG1 VG1 VG1 VG1 VG1 VG2 VG2 SVC1 SVC2 SC1 SC2 Tap1

1.050 1.061 1.072 1.084 1.096 1.068 1.096 0.073 0.082 3 2 0

1.061 1.072 1.084 1.096 1.100 1.096 1.100 1.000 1.000 4 4 3

— — — — VG2 — SVC1 SVC2 SC1 SC2 — Tap2

— — — — 1.050 — 0.000 0.000 1 1 — 0

— — — — 1.068 — 0.073 0.082 3 2 — ¹5

Table 7. States of operated controllers [Case (4)] K

Controller

Before

After

Controller

Before

After

1.10

VG1 SVC1 SC2 SC1 SC1 Tap2

1.050 0.000 0 1 3 0

1.100 1.000 4 3 4 ¹5

VG2 SVC2 — — — Tap1

1.050 0.000 — — — 0

1.100 1.000 — — — 1

1.12 1.14 1.16

146

VIPI based control methods: M. Nanba et al.

In the SQP based control method, simulations are carried out for a model system based on Ward and Hale’s 6-bus system, and increase of VIPI after operation is confirmed. The convergence that is difficult in conventional methods from infeasible regions is enabled by this method. In the sensitivity based control method, simulations are carried out for the same model system. The reliability of the calculated sensitivity of VIPI with respect to controllers is confirmed from the results, and the obtained sensitivities can also be utilized for operator judgment. From the simulation results, the effectiveness of the proposed methods has been confirmed.

6.

7. 8. 9. 10.

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power transmission system by sensitivity methods. IEEE Transactions on Power Systems, 1990, PS-5(4), 1286–1293. Dobson, I. and Lu, L., Voltage collapse precipitated by the immediate change in stability when generator reactive power limits are encountered. IEEE Transactions on Circuit and Systems, 1992, CAS39(9), 762–766. Galiana, F. D. and Zeng, Z. C., Analysis of the load flow behavior near a Jacobian singularity. IEEE Transactions on Power Systems, 1992, PS7(3), 1362–1369. Lof, P. A., Anderson, G. and Hill, D. J., Voltage stability indices for stressed power systems. IEEE Transactions on Power Systems, 1993, PS-8(1), 326–335. Stankovic, A., Ilic, M. and Maratukulam, D., Recent results in secondary voltage control of power systems. IEEE/PES 1990 Winter Meeting. Atlanta, GA, February 1990. Arcidiacono, V. and Corsi, S., A real-time voltage stability index for bulk power system with secondary voltage regulation. Bulk Power System Voltage Phenomena III. Davos, August 1994, pp. 197–204. Liu, X., Ilic, M., Athans, M., Vialas, C. and Heilbronn, B., A new concept for tertiary coordination of secondary voltage control on a large power network. Conference on Decision and Control. Tucson, December 1992. Liu, X., Vialas, C., Ilic, M., Athans, M. and Heilbronn, B., A new concept of an aggregate model for tertiary control coordination of regional voltage controllers. Proceedings of the 11th PSCC. Avignon, France, August/September 1993, pp. 995–1002. Mansour, Y., Xu, W., Alvarado, F. and Rinzin, C., SVC placement using critical modes of voltage instability. IEEE Transactions on Power Systems, 1994, PS-9(2). Taylor, C. W., Power System Voltage Stability. McGraw-Hill, New York, 1993. Fukushima, M., A successive quadratic programming algorithm with global and superlinear convergence properties. Math. Programming, 1986, 35, 253–264.