Corrective rescheduling for static voltage stability control

Electrical Power and Energy Systems 27 (2005) 3–12 www.elsevier.com/locate/ijepes

Corrective rescheduling for static voltage stability control L.D. Aryaa,*, D.K. Sakravdiab, D.P. Kotharic a

Electrical Engineering Department, Shri. G.S. Institute of Technology and Science, Indore, MP 452003, India b Department of Electrical Engineering, Government Engineering College, Ujjain, MP 456010, India c Centre for Energy Studies, I.I.T. Delhi, New Delhi 110016, India Received 19 March 2003; revised 6 April 2004; accepted 8 July 2004

Abstract This paper describes a technique for improving static voltage stability by rescheduling reactive power control variables. The algorithm is based on sensitivities of minimum eigen value with respect to reactive power control variables. Objective function selected is minimization of deviation of squares of reactive power control variables subject to desired increase in minimum eigen value of load flow Jacobian. Since minimum eigen value is system wise voltage stability index increase in load bus voltages as desired is also simultaneously observed. Using Lagrangian optimization technique closed form relations for corrections in reactive power control variables have been obtained. Developed methodology has been implemented on a 6-bus and 25 bus test systems. q 2004 Elsevier Ltd. All rights reserved. Keywords: Load flow Jacobian; Eigen sensitivities; Voltage stability; Optimization

1. Introduction Voltage stability has been defined in terms of ability to maintain voltage so that when load admittance is increased load power will increase and so that both the power and voltage are controllable [1]. Voltage and power are controllable in upper region of the PV-curve. Voltage stability assessment/evaluation techniques fall into following categories (a) direct method using Lyapunov based energy function [3–6] (b) modal analysis based methods [7,8] (c) direct computation of point of voltage collapse [9,10]. It has been further observed that voltage magnitude alone is not a good indicator of voltage security. Load bus voltages may be high but the loadability limit may be very close to the present operating point [2,11]. It is demand of the day that load buses should not only have high voltage magnitude but the operating point should have sufficient distance in terms of MVA from voltage collapse point. This amounts that enhancement procedure must give due consideration to voltage stability margin. This can be incorporated via a proximity indicator. Minimum eigen * Corresponding author. Fax: C91 731 432 540. E-mail address: [email protected] (L.D. Arya). 0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2004.07.005

value of load flow Jacobian is such a proximity indicator. All eigen values of load flow Jacobian are positive in upper segment of PV-curve. At least one eigen value becomes negative in lower segment of this nose curve. At voltage collapse point one of the eigen value becomes zero. Hence, the magnitude of minimum eigen value is an indicator of relative voltage stability margin [12]. It is also important for maintaining the voltage stability to keep sufficient reactive power reserve and thus reactive power transfer must be minimized. This is important since large reactive power transfer along the transmission line is not desirable as it increases losses, equipment sizes, and rise in voltage under sudden load rejection. Moreover, if power angle across the transmission line is large, it is difficult to transfer reactive power without substantial voltage gradient [13]. The rescheduling of reactive power control variables must be based on (a) reactive power transfer minimization, (b) safe distance of the present operating point must be maintained from voltage collapse point, (c) voltage should be kept as high as possible (within permissible limits). This suggests that voltage security must be enhanced from the voltage stability viewpoint. Most of the security constrained OPF related work has assessed the voltage security based on the indices which

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depends on load bus voltage magnitudes. However, voltage instability problems have been shown to occur in systems where voltage magnitudes never decline below acceptable limits. Tiranuchit et al. [14] developed a methodology of control against voltage instabilities based on singular value decomposition (SVD). One of the disadvantages of the methodology is that large amount of CPU time is required in SVD. Overbye [15] utilized energy based sensitivities to develop a controller for voltage security enhancement. Control of static voltage stability has been attempted by Begovic and Phadke [16] using sensitivity analysis. Effect of allocation and amount of reactive power support on voltage stability margin has been discussed. An optimum reactive power planning strategy against voltage instability using continuation power flow has been developed by Ajjarapu et al. [17]. Zalpa and Cory [18] developed a technique to obtain settings of reactive power control variables so that the allocation of reactive reserves guarantees that the system does not move toward voltage collapse as load changes or line outages takes place. Bansilal et al. [19] presented a reactive power dispatch algorithm for voltage stability margin improvement using L-index and non-linear least-square optimization algorithm. A steady state voltage monitoring and control algorithm using localized least square minimization of load bud voltage deviations has been developed by Zobian and Allic [20]. The approach attempts to maintain a given voltage profile as the load demand, generation availability and network topology vary. Arya et al. [21] developed a reactive power optimization algorithm using static voltage stability index using incremental model. The algorithm has the potential to be implemented in real time and gives close form relations for reactive power control variables in terms of base case sensitivities. A voltage security assessment and control algorithm using particle swarm optimization (PSO) algorithm is discussed in Ref. [22]. Zecevic and Miljkovic [29] developed an algorithm for enhancement of voltage stability by an optimal selection of load following units. A voltage stability constrained VAR planning algorithm along with statistical approximation has been developed by Chattopadhyay and Chakrabarti [30]. Gubina et al. [31] developed an intelligent reactive power control technique using a novel local voltage control approach. In Ref. [32] a reactive power planning incorporating voltage stability methodology has been developed addressing economic issues. Milsovic et al. [33] developed a methodology for local monitoring of onset voltage collapse and emergency control in the presence of voltage sensitive loads. Song et al. [34] presented a new concept of reactive reserve based contingency constrained optimal power flows (RCCOPF) for voltage stability enhancement in view of the above discussion an algorithm has been developed by increasing the minimum eigen value of load flow Jacobian so as to maintain desired voltage profile and yet minimum shift in control variables is required and reactive power limit violation does not takes place. For these purpose

sensitivities of minimum eigen value with respect to control variables have been derived and used as incremental model to get desired voltage profile.

2. Eigen sensitivities An estimate of absolute sensitivity of eigen value of lk in relation to any element aij of square matrix [A] can be written as follows [23–25] vl=vaij Z hk ðiÞxk ðjÞ

(1)

where ijth element of square matrix [A] aij lk kth eigen value of square matrix [A] hk(i)xk(j) ith and jth element of left and right eigen vector corresponding to lk hTk , xk unit left and right eigen vectors, respectively, corresponding to lk . Further each element aij is a function of system control variables ‘a’, i.e. aij Z f ða1 ; .ai ; .ap Þ Hence, the sensitivity of lk with respect to system parameter or control variable can be written using chain rule of differentiation as follows X vlk vaij vlk Z (2) vai vaij vai i;j or X vaij vlk Z hk ðiÞxk ðjÞ vai vap i;j

(3)

Sensitivities as expressed in relation (3) were first used by Van Ness et al. [25] in analyzing large multiple feed back control systems. Such sensitivities further have been used by Lima [24] for studying transient stability of single synchronous machine system connected to infinite bus along with excitation system. Lima et al. [23] further used to assess the eigen value sensitivities with respect to parameter uncertainties.

3. Sensitivity of minimum eigen value of load flow Jacobian with respect to reactive power control variables It has already been emphasized that minimum eigen value of load flow Jacobain signifies proximity of the present operating point to the voltage collapse point. Sensitivities of minimum eigen value with respect to reactive power control variables are derived in this section.

L.D. Arya et al. / Electrical Power and Energy Systems 27 (2005) 3–12

The load flow Jacobian at solution point can be written as follows: " # H N JZ (4) M L The diagonal and off diagonal elements of sub Jacobians [H], [M], [N], [L] are given as follows [26]. Hij Z Vi Vj Yij sinðdi K dj K qij Þ Hii ZKQi K Vi2 Bii Lii Z ½Qi =Vi K Bii Vi Lij Z Vi Yij sinðdi K dj K qij Þ Nii Z Vi Gii C Pi =Vi Nij Z Vi Yij cosðdi K dj K qij Þ Mii Z Pi K Gii Vi2 Mij ZKVi Vj Yij cosðdi K dj K qij Þ

vHij Z Vi Vj Vij cosðdij K qij ÞðD2ik K D2jk Þ vUk C Yij sinðdij K qij Þ½Vj D1ik C Vi D1jk  i sj vHij Z K2Bii Vi D1ik vUk vNij Z D1k Yij cosðdij K qij Þ vUk K Vi Yij sinðdij K qij ÞðD2ik K D2jk Þ

i sj

vNij Z ½Gii K Pi =Vi2 ÞD1ik vUk vMii Z Vi Vj Vij sinðdij K qij ÞðD2ik K D2jk Þ vUk

where

C Yij cosðdij K qij Þ½Vj D1ik C Vi D1jk  i sj;

Yij magnitude of ijth element of bus admittance matrix qij phase angle of ijth element of bus admittance matrix Let us assume that x and hT are right and left eigen vectors corresponding to minimum eigen value, lmin of load flow Jacobian (x is a column vector and hT is a row vector). The procedure for obtaining x, h and lmin is given in Appendix B. Sensitivity of lmin with respect to reactive power control variable, Uk is given as follows Sl Z

5

vlmin X vlmin vHij X vlmin vNij C Z vUk vHij vU vNij vUk X vlmin vMij X vlmin vLij C C vMij vUk vLij vUk

vMij Z K2Gii Vi D1ik vUk vLij Z Yij sinðdij K qij ÞD1ik vUk C Vi Yij cosðdij K qij ÞðD2jk K D2jk Þ i sj vLij Z K½Bii C Qi =Vi2 ÞD1jk vU where D1ik Z

(5)

Using relation (1) the sensitivities vlmin/vHij, vlmin/vNij, vlmin/vMij and vlmin/vLij are as follows vlmin Z hðiÞxðjÞ vHij vlmin Z hðiÞxðj C NB K 1Þ vNij vlmin Z hði C NB K 1ÞxðjÞ vMij vlmin Z hði C NB K 1Þxðj C NB K 1Þ vLij Expressions for other partial derivatives in Eq. (5) are evaluated using the elements of load flow Jacobian as given in Eq. (4) and base case sensitivities (Appendix A), as follows:

vVj vdj vVi vd ; D1jk Z ; D2ik Z i ; D2jk Z ; vUk vUk vUk vUk

are the sensitivity coefficients picked up from Appendix A Vector of reactive power control variables, UZ[U1, U2,.Uk,.,UNC]Z[VG: Qsh: t] This means reactive power control variable consists of PV-bus voltages (VG), shunt compensations (Qsh) and settings of tap changing transformers (t).

4. Validity of the incremental linearised model The method proposed in this paper utilizes linearized model for optimization. The control corrections at each step or iteration can be limited to an extent so that solutions remain within the region of validity of linearized model. Simple rule to ensure validity of linearized model is to limit bus VAR injection changes ‘DQn’ with equation [27]. jDQn =SYj% a

(6)

Typical factor a is selected as 0.01. Sy denotes sum of all susceptances connected at that bus. It is sufficient to mention here that aZ0.01 result in

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rather wide region of validity of the linearized model for example if jSyjZ20 and aZ0.01, then for validity of linearized model DQn is limited to 0.2 pu. This is a substantial amount to be rescheduled.

5. Problem formulation and solution methodology It is proposed to enhance voltage stability including improvement in voltage profile by increasing minimum eigen value of load flow Jacobian at the operating point. In its first place rescheduling the reactive power control variable around the base point settings can do it. The problem is formulated as least square minimization problem as follows: J Z 1=2 DU T W DU

The above optimality conditions can be put in matrix form as follows: " # " # W Sl  DU  0 Z (11) m Dlmin SlT 0 Solution to matrix Eq. (11) can be written as follows: #" #   " 0 B11 B12 DU Z (12) m B21 B22 Dlmin Using the inverse by partitioning B11 Z W K1 K

W K1 SlSlT W K1 SlT W K1 Sl

B12 Z

W K1 Sl SlT W K1 Sl

SlT DU Z Dlmin

B21 Z

SlT W K1 SlT W K1 Sl

[DU] vector of reactive power control variables changes [Sl] sensitivity vector of minimum eigen value with respect to reactive power control variable [W] diagonal matrix known as weighting matrix

B2 Z K

[W] matrix limit the excursion on control variables. Further since minimum eigen value is a system wise proximity indicator, when its value is enhanced via corrective rescheduling, the bus voltage also increased [7]. At each stage reactive power limited using the Eq. (6). Following Lagrangian function is formed

½DU Z

(7)

Above performance index is minimized subject to following constraint.

L Z 1=2 DU T W DU K mðDlmin K SlT DUÞ

(8)

where m is a Lagrangian multiplier. Specified change required in minimum eigen value, DlminZltKlmin,o lt threshold value of minimum eigen value lmin,o minimum eigen value in base case condition Dlmin should be specified in such a way that after achieving the optimum control correction Eq. (6) is satisfied at each generator bus and buses where shunt compensation are present. The optimality conditions are obtained as follows vL Z W DU C Slm Z 0 vU

(9)

vL Z Dlmin K SlT U Z 0 vm

(10)

1 SlT W K1 Sl

Solution for optimum control corrections can be written from Eq. (12) as follows:

m ZK

W K1 Sl Dlmin SlT W K1 Sl

Dlmin Sl W K1 Sl T

(13)

(14)

It is observed from Eq. (12) only computation of [B12] is required to get optimum control corrections. Specific expression for mth control correction is written from Eq. (13) as follows: Sl W K1 DUm Z PNC m m2 K1 Dlmin PZ1 Slp W

NC Total number of control variables. If violation of any control correction occurs that can be brought within limit by increasing respective weighting factor Wi. The details of computational algorithm are described with the help of flow chart as given in Fig. 1. It has been found that as minimum eigen value of load flow Jacobian is raised due to control action, load bus voltages also improve. If some voltage limit violation exists than as is normally done a quadratic penalty term is added to the performance index Eq. (7) and thus voltage can be brought within limits.

L.D. Arya et al. / Electrical Power and Energy Systems 27 (2005) 3–12

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Fig. 1. Computational flow chart for rescheduling reactive power control variables.

6. Results and discussions The methodology developed in this paper for voltage profile as well as static voltage stability enhancement has been implemented on a 6-bus and 25 bus test systems. The systems datas have been taken from Ref. [28].

The 6 bus system in fact has 6-reactive power control variables. Bus no. 1 and 2 are generator buses. On load tap changers (OLTC) are connected in line no. 4 and 7 near buses 6 and 4, respectively. Also capacitive shunt compensation (Qsh) has been provided at bus no. 4 and 6. Limits on shunt compensation have been assumed

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Table 1 6-Bus base case load flow results

Table 3 Optimized load flow solution for 6-bus test system

Bus no.

PG(pu)

QG (pu)

Voltage (pu)

Angle rad.

Shunt compensation

1 2 3 4 5 6

1.96 0.50 0.00 0.00 0.00 0.00

1.52 0.57 0.00 0.00 0.00 0.00

1.00 1.00 0.68 0.6885 0.6675 0.6553

0.0000 K0.6211 K0.6783 K0.4787 K0.7772 K0.6452

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.055. Base case load flow solution very close to collapse point is given in Table 1. Minimum eigen value of load flow Jacobian obtained at this point is 0.0467. Static voltage stability limit for base case settings of control variable is PdZ2.0297 pu and QdZ0.3200 pu. Both shunt compensation and tap settings are at 0.0 and 1.0 pu, respectively. At this base case condition sensitivities with respect to the control variables have been evaluated and given in Table 2. Dlmin initially selected in step of 0.2 and after three iteration the optimized solution is given in Table 3. The threshold value of minimum eigen value was selected ltZ0.600 and all load bus voltages were in the range 0.9%Vn%1.05. Optimized tap settings were obtained as 1.04 and 0.95 for line number four and seven, respectively. It is observed from Table 3 that not only the voltage profile is improved also reactive power losses are significantly reduced. Comparing Table 1 and Table 3 it is found that change in reactive power losses are 1.08 pu. Such a large reduction in reactive power losses itself signifies enhancement in voltage stability. Since, it is one of the most important objectives of power system control engineer. With optimized control variables as shown in Table 3 the real and reactive power loadability limits are 2.6739 and 0.407 pu, respectively. It is 25% increase in stability margin. Figs. 1 and 2 shows the effect of corrective rescheduling on static voltage stability limit with the help of PV-curve of bus no. 5. Similar results have been obtained for 25 bus test system [28]. This system contains five generator buses which are numbered from one to five and voltages of

Load bus no.

3

4

5

6

Load bus voltag Optimized control variables Reactive power at PV-buses

1.040

0. 9973 T4 1. 0400

0.9374

0.9884

V1 1.155

V2 1.141

Qg1 0.7137

Qg2 0.2456

T7 0.95

Qsh4 0.05

Qsh6 0.055

these buses are control variable. Table 4 shows the solution under stressed condition where lminZ0.0535. Static voltage stability limit (loadability) with base case settings of PV-buses is PdZ21.535 and QdZ 6.156 pu. In obtaining these limit load was increased at each bus in proportion to base case condition with maintaining constant power factor. A threshold value of minimum eigen value selected is ltZ0.55. Limits on load bus voltages are assumed as 0.9 and 1.05. Star marked buses in Table 4 are buses where voltage limit violation has taken place in base case condition. Again three iterations were performed to get threshold value of minimum eigen value. DlminZ0.2 was assumed for each step or iteration. Table 5 shows optimized PV-bus voltage settings and load flow solution corresponding to base case loading conditions. This also shows reactive power generation required at PV-buses. Stability limit with optimized settings is obtained using continuation power flow as PdZ28.86 pu and QdZ7.75 pu. thus an

Table 2 Sensitivity coefficient of minimum eigen value with respect to control variables for 6-bus test system Control variables

Sensitivity

V1 V2 Tap T4 Tap T7 Qsh4 Qsh6

8.3035 2.5867 0.2113 1.0177 1.2650 1.2001

Fig. 2. P–V curve for bus no. 5 for 6-bus IEEE test system with and without reactive power control.

L.D. Arya et al. / Electrical Power and Energy Systems 27 (2005) 3–12

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Table 4 25 Bus no base case load flow results Bus no.

PG

QG

Voltage

PL

QL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

7.3486 3.5748 5.2988 1.4076 6.9480 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

7.2316 1.3332 1.9049 4.0802 2.2339 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1.0000 1.0000 1.0000 1.0000 1.0000 0.9154 0.8534* 0.8446* 0.8174* 0.8672* 0.8521* 0.8442* 0.9160 0.6329* 0.6527* 0.7358* 0.8578* 0.8374* 0.9266 0.8660* 0.7569* 0.6523* 0.7943* 0.5339* 0.5551*

5.9000 0.2950 1.4750 0.8850 0.7375 0.4425 0.4425 0.7375 0.4425 0.4425 0.1475 0.2950 0.7375 0.5900 0.8850 0.8850 1.7700 0.4425 0.4425 0.7375 0.5900 0.5900 0.4425 0.4425 0.7375

1.7550 0.0810 0.4590 0.2700 0.2160 0.1350 0.1350 0.0000 0.1350 0.1350 0.0000 0.0000 0.2160 0.1890 0.2700 0.2700 0.5400 0.1350 0.1350 0.2160 0.1890 0.1890 0.1350 0.1350 0.2160

increase of 34% is found in static voltage stability limit with optimized settings. Further all load bus voltages have been brought within limits. Fig. 3 show PV-curve for bus no. 25 with and without optimized PV-bus Table 5 25 Bus load flow results with optimized settings of PV-buses Bus no.

PG

QG

Voltage

Angle (rad.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

6.4929 3.5748 5.2988 1.4076 6.9480 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

7.6534 1.1314 0.8102 2.9944 1.3222 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1.1503 1.0899 1.0567 1.1052 1.0737 0.9949 0.9764 0.9685 0.9397 0.9754 0.9704 0.9712 0.9903 0.9002 0.9010 0.9674 0.9849 0.9757 1.0500 1.0161 0.9611 0.9175 1.0295 0.9001 0.9145

0.0000 0.7065 0.5012 K0.0218 0.6350 0.4906 0.4423 0.4068 0.3655 0.4255 0.3500 0.3511 0.4633 K0.0343 K0.1131 K0.1154 0.3044 0.1462 0.0739 K0.1133 K0.1733 K0.2318 K0.1255 K0.3136 K0.2878

Fig. 3. P–V curve for bus no. 25 for 25-bus IEEE test system with and without reactive power control.

settings. Substantial reduction in real and reactive power loss is observed with optimized settings of PV-buses. The developed methodology compares well against those involves exact non-linear optimization. Specifically PSO technique [22]. PSO based technique is an intelligently designed search technique which do not require sensitivity coefficient and is suited well for global optimization for base point setting determination under off line condition. Where as methodology of this present paper based on sensitivity gives results quickly. Since the derived relations for control corrections are in closed form. The CPU time required are 1.79 and 2.68 s. for 6bus and 25-bus test systems. Hence this can be applied in real time. It is the opinion of the authors that this methodology is suitable for large practical system. Since all sensitivity coefficients have been derived based on load flow Jacobian, which is readily available at the end of each load flow solution or even using a state estimation for on line purpose. Moreover, even the expressions for sensitivities have been presented in closed form. Further the applicability of this technique in real time is justified in two ways: (i) It is corrective rescheduling around a base point setting which may not require in real time much of the iteration (ii) All derived relations are in closed form. Moreover, voltage instability is a slow phenomenon and one always gets time to implement in real time. Further such technique can be applied whenever minimum eigen value falls below a threshold value which is system dependent.

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7. Conclusions

½DQ Z ½DQPV : DQsh : DQT : O; .OT

An algorithm has been developed to optimize the reactive power control variables for reactive power management including static voltage stability enhancement It has been assured via performance index that rescheduling of control variables remains to a minimum level. But this minimum rescheduling is not at the risk of voltage stability. Rescheduling of control variables has been done to an extent so as to have sufficient level of stability margin by adjusting the minimum eigen value of load flow Jacobian. Further it is observed that by enhancing the minimum eigen value of load flow Jacobian not only improves the voltage profile to a desired level but also reduces reactive power losses strongly which serves as reactive power reserve for maintaining voltage stability under contingent condition. Thus reducing transmission losses reduces power transfer and keeping voltage high helps in maintaining the stability.

where

In view of above [Dd] and [DV] can be written as follows ½DV Z ½S2P½DQPV  C ½S2H½DQsh  C ½S2T½DQT  where [S2P], [S2H] and [S2T] are the submatrices picked up from [S2] ½DV Z ½S4P½DQPV  C ½S4H½DQsh  C ½S4T½DQT  where [S4P], [S4H] and [S4T] are the submatrices picked up from [S4] From above equations PV-bus voltage corrections are written as follows½DVPV  Z ½SP½DQPV  C ½SH½DQsh  C ½ST½DQT 

Appendix A

The sensitivity relationship at solution point can be obtained from fundamental power flow equationFðd; V; P; QÞ Z 0

½DQPV  Z ½SPK1 ½DVPV  C ½SPK1 ½SH½DQSH  K ½SPK1 ½ST½DQT 

(A1)

Above equations do not contain real power equations corresponding to slack bus. Total number of equations are (2NBK1);. NB denotes number of buses. Expanding (A1) around solution point following sensitivity relations can be written # " # " Dd DP Z (A2) ½J DV DQ

Dd

DV

" Z

S1

S2

S3

S4

#"

DP

#

DQ

(A3)

Since the interest is in reactive power rescheduling as load being assumed constant DP Z 0

(A7)

Substituting Eq. (A7) in Eqs. (A4) and (A5) following expressions for [Dl] and [DV] are obtained. ½Dd Z ½S2P½SPK1 ½DVPV C½S2H K S2PSPK1 SH !½DQSH  C ½S2T K S2PSPK1 ST½DQT 

(A8)

½DV Z ½S4P½SPK1 ½DVPV C½S4H K S4PSPK1 SH !½DQSH  C ½S4T K S4PSPK1 ST½DQT 

#

(A6)

where[SP],[SH], and [ST] are submatrices picked up from [S4P],[S4H] and [S4T], respectively.Now Eq. (A6) can be solved for [DQPV] as follows

Appendix A1. Sensitivity relationship [21]

or "

[DQPV] is reactive power injection change vector at PVbuses [DQsh] is Var injection change vector at compensated buses [DQT] effective Var injection changes vector due to OLTC position changes

(A9)

Linearizing the reactive power flow expression for the lines having OLTC, the equivalent reactive power injection change vector can be expressed in terms of tap position change vector [DT] as follows½DQT  Z ½AT½DT

(A10)

substituting [DQT] from Eq. (A10) in Eqs. (A8) and (A9) the following sensitivity matrices are obtained.

From Eq. (A3) it follows that½Dd Z ½S2 ½DQ

(A4)

½DV Z ½D1½DU

(A11)

½DV Z ½S4 ½DQ

(A5)

½Dd Z ½D2½DU

(A12)

and DQ is given as follow:

where

L.D. Arya et al. / Electrical Power and Energy Systems 27 (2005) 3–12

References

½D1 Z ½S4PSPK1 : S4H K S4PSPK1 SH : S4T K S4PSPK1 STAT ½D2 Z ½S2PSPK1 : S2H K S2PSPK1 SH : S2T K S2PSPK1 STAT Also substituting Eq. (A10) in Eq. (A7) following incremental relationship between reactive power changes at generator buses and control variables is obtained. ½DQPV  Z ½C½DU

(A13)

where ½C Z ½SPK1 : SPK1 SH : KSPK1 STAT ½DU Z ½DVPV : Dsh : DT

Appendix B Appendix B1. Determination of minimum eigen value and eigen vectors Define matrix A as follows " AZ

H

N

M

L

11

#K1

Step-1 Select initial vector X(0) Such that jjX(0)jjZ1 Step-2 Set iteration count kZ1 Step-3 X(K)ZAX(kK1) Step-4 S(k)ZjjXKjj Step-5 Check if jS(K)-S(KK1)j!3(Tolerance) If yes then go to Step-8. Step-6 X(K)ZX(k)/Sk Step-7 KZKC1 and repeat from Step-3. Step-8 Minimum eigen value lminZ1/S(K) and right eigen vector xZX(K)/S(K) and stop.

To obtain hT, left eigen vector of load flow Jacobian, above process is repeated on the inverse matrix of transpose of load flow Jacobian.

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