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Optimal static voltage stability improvement using a numerically stable SLP algorithm, for real time applications
Electrical Power and Energy Systems 21 (1999) 289–297
Optimal static voltage stability improvement using a numerically stable SLP algorithm, for real time applications R. Raghunatha a, R. Ramanujam a, K. Parthasarathy b,*, D. Thukaram b a
International Development and Engineering Associates Ltd., Bangalore, India b Indian Institute of Science, Bangalore, India
Received 8 October 1997; received in revised form 12 February 1998; accepted 12 February 1998
Abstract A numerically stable sequential Primal–Dual LP algorithm for the reactive power optimisation (RPO) is presented in this article. The algorithm minimises the voltage stability index C 2 [1] of all the load buses to improve the system static voltage stability. Real time requirements such as numerical stability, identification of the most effective subset of controllers for curtailing the number of controllers and their movement can be handled effectively by the proposed algorithm. The algorithm has a natural characteristic of selecting the most effective subset of controllers (and hence curtailing insignificant controllers) for improving the objective. Comparison with transmission loss minimisation objective indicates that the most effective subset of controllers and their solution identified by the static voltage stability improvement objective is not the same as that of the transmission loss minimisation objective. The proposed algorithm is suitable for real time application for the improvement of the system static voltage stability. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: Static voltage stability; Reactive power optimisation; S L P algorithm
1. Introduction The incidents of voltage collapse [2–3], have drawn the attention of the power system industry to the problem of 1. Identification of the critical buses or areas of the system, most susceptible to voltage collapse. 2. Initiation of corrective measures to enhance the static voltage stability margin
the state estimator output and are most effective when used in conjunction with the knowledge of the relative position of the bus voltage phasors. The proposed RPO algorithm in this article minimises the index C2 of all the load buses to improve the system static voltage stability. This is equivalent to minimising the L-1 norm of the VCPI C2. Some of the requirement of optimal power flow for the real time applications are as follows. 1. The algorithm should have smooth convergence characteristic and should be numerically stable. 2. Power systems normally operate in a quasi-steady state. Hence, it is essential to identify the most effective subset of controllers and reschedule the same [17]. 3. It is essential to maintain adequate reserve margin in controllers for use in emergency states.
For static analysis, the task of identification of critical buses or areas of system, from view point of voltage stability, is usually done using voltage collapse proximity indicators (VCPIs). Although several voltage collapse proximity indicators are proposed in the literature, we have used the indices, proposed in the references [4–5], for the purpose of evaluating the proposed primal–dual SLP algorithm [6]. References [7–12] addresses some of the measures and algorithms aimed towards the corrective measures for the improvement of the system voltage stability. In reference [1], the authors have introduced two new VCPIs C1 and C2. These indices are briefly described in Section 2. These indices use the information available from
In the proposed primal–dual SLP algorithm the above features are easily implemented. The proposed algorithm was initially developed with the transmission loss minimisation object [13–16], which is reported to have several short comings. The following are some of the features of the transmission loss minimisation algorithm, reported in the literature.
* Corresponding author. Tel.: ⫹91080-334441; fax: ⫹91080-3341683. E-mail address: [email protected] (K. Parthasarathy)
1. The SLP algorithm for transmission loss minimisation do not exhibit smooth convergence owing to non-linear
0142-0615/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S0142-061 5(98)00048-9
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Fig. 1. Simple flow chart of EMS functions.
variation of the transmission loss as a function of control variables. 2. To achieve controlled convergence, it is necessary to use approximations or special techniques in the basic LP algorithm. Some of the modifications suggested in the basic LP algorithm are (i) Use of restricted control step sizes and limits [13,14] and (i) Use of intricate control logic [16]. 3. Further, heuristic measures may be required to identify the most effective subset of controllers for curtailing the number of controllers [16]. In the proposed primal–dual LP algorithm none of the above difficulties are present either with the transmission loss minimisation objective or with the voltage stability improvement objective. The implementation of the algorithm is straight forward and does not require any intricate control logic or approximations. Accurate minimisation of the objective is usually achieved in two to three cycles, without exhibiting any oscillations in the convergence characteristic. In the presence of over voltages, the feasibility of the solution is usually improved with reduced optimality. 2. The static voltage stability improvement objective formulation In reference [1], the authors have introduced two new VCPIs C1 and C2. These VCPIs are most effective when used in conjunction with the knowledge of the relative position of the bus voltage phasors. Further, the indices C1 and C2 may be directly computed from the output of the state estimator and hence are suitable for real time applications.
The performance of these indices compare well with those of the other VCPIs presented in the literature [4,5].The index C2 has a value of about 0.5 at no load condition and a value of about 1.0 at voltage collapse point. For greater voltage stability margin, the index C2 should be as small as possible. If the voltage stability indicator C2 of all load buses are minimised, then the system static voltage stability margin must increase. Minimising voltage stability indicator of all the load buses is equivalent to minimising the L-1 norm of the index C2 of all the buses. The development of such an objective function is given in this section. The basic philosophy used in developing the indices C1 and C2 is that the bus voltage phasors contain all the necessary information about the system static state. Hence, if the relative position of the bus voltages are examined with reference to a suitable voltage phasor, it should be possible to identify the most critical buses from the view point of static voltage stability. Two reference voltages Vcs and Vcg, referred to as centroid of the system voltage space and centroid of the generator voltage space are suggested in [1] to identify critical bus voltages and a measure of their proximity to voltage collapse. The centroid voltages Vcs and Vcg are computed by comparing the equilibrium conditions of a power system with that of the equilibrium conditions of rigid bodies. It is shown in [1] that the voltage Vcg is the phasor average of all the generator bus voltages. The voltage stability index C1 and C2 for the ith bus is given by Vcg ⫺Vi ; 1a C1i jVi j Vcg ; 1b C2i 2jVi jcos di ⫺dcg Where, Vi is the bus voltage under consideration, d i is the phase angle of the ith bus voltage and d cg is the phase angle of Vcg. The changes in the index C2i are then computed as 2C2 2C2i 2C2i DjVi j ⫹ DC2i i D Vcg ⫹ Ddcg 2j V i j 2dcg 2 Vcg ⫹
2C2i D di ; 2 di
2
The dependent variables of Eq. (2) are then expressed as a function of control variables u, as explained in Appendix B. Using Eq. (2), the voltage stability improvement objective may be stated as X 3 minimise DC2i C T u: subject to the usual constraints on the control and dependent variables (Appendix A). Fig. 1 shows a simple functional flow chart of EMS functions. The voltage insecurities are detected by blocks 3 (base
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case) and 6 (contingency case). If the system voltage stability margins are inadequate, optimal corrective (block 4) or preventive (block 7) control scheduling using voltage stability improvement objective is required to improve the system static voltage stability.
3. The primal–dual LP algorithm [6] In this section some of the features and implementation details of the primal–dual LP algorithm used are presented. 1. The initial condensed simplex tableau for the primal– dual algorithm need not be optimal or feasible. 2. The optimisation problem is always solved as a maximisation problem rather than as a minimisation problem. 3. All possible pivots by the primal (maximisation) or the dual (minimisation) algorithm are examined. The pivot that influences the objective most is selected. 4. A pivot selected by the primal algorithm will improve the optimality while usually reducing the upper bound feasibility margin. A pivot selected by the dual algorithm will improve the feasibility while attempting to maintain optimality. In general, as applied to power systems, improvement in lower bound feasibility is accompanied by improvement in the optimality and correction of upper bound violations normally results in reduced optimality. 5. The non basis variables associated with the cost coefficients of the objective row will always have zero value. Only the basis variables may have non zero solution. 6. If the initial tableau is control feasible, i.e., all control variables are within their limits and only the primal algorithm is used, the final tableau will always be control feasible. This is true even if unrestricted or restricted control ranges are used. Consequently, the solution from the primal algorithm may be directly implemented without using any approximations. This will usually result in improved objective and improved lower bound feasibility of dependent variables. 7. A pure dual pivot, does not necessarily result in a control variable feasible tableau. This appears to be the cause for the problems encountered with the transmission loss minimisation objective. In the actual implementation the primal–dual algorithm was used subject to the following conditions. 1. The pivot should result in a new tableau, which is always control feasible. 2. The new tableau should always be more feasible than the previous one. The new tableau after satisfying the conditions (1) and (2) will be usually always more optimal than the previous tableau and will be usually less optimal in the presence of predominantly upper bound violations. These conditions ensure that the final controller solutions can always be implemented with improved system performance objective.
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Approximations such as restricted step size for controllers as used in former reported algorithms will never be required. It is possible that the conventional algorithm without the conditions (1) and (2) may result in more optimal or feasible solution which does not necessarily guarantee feasible controller solution. The conditions (1) and (2) require that, the effect of the pivot is evaluated prior to actual pivoting algorithm. The added computation because of this is quite small, as the evaluation is carried out on a single column of the condensed simplex tableau. The condition (1) implies that irrespective of the range specified for the controller variables, the final solution for the controllers will always be within the specified limits. Hence, the need for restricted step sizes or ranges of control variables or intricate control logic are not necessary. The final solution obtained may be directly implemented without any approximation. Appendix C provides an illustration that clarifies the utility of the conditions (1) and (2).
4. Curtailing number of controllers and their movement From the description given in Section 3 and the conditions (1) and (2), it is evident that each LP pivoting iteration will progressively result in more feasible and more optimal tableau. Further, the primal algorithm always uses the most effective controller to improve the objective and the dual usually corrects the largest violation. In the implemented algorithm, the primal pivot is always given a priority over the dual pivot. Each pivoting iteration results in an exchange of variables from basis to non basis. Suppose we are interested, in obtaining solution for a specified curtailed number of control variables, then we only need to check the number of control variables appearing as basis variables at the end of each pivoting iteration. When the solution for the required specified number of control variables is available the pivoting iteration may be terminated and the solution may be implemented. This will result in improved optimal and feasible solution using curtailed number of controllers. As the primal pivot is always given a higher priority over the dual pivot, the resulting solution normally represent the most effective subset of the total number of controllers specified, from the view point of the specified objective. During the system operation, it may be relatively convenient to implement some controls in comparison to other controllers. Such controls maybe preferred by the power system operators for the improvement in the system performance. While identifying the effective subset of the controllers available for the improvement of the system performance, it is preferable to take into account the operators control priorities. This is very easily accomplished by the proposed primal–dual algorithm, by forcing the pivoting algorithm to give a higher priority to the controllers of the operators choice. These priorities may be specified in any
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Table 1 Improvement in the VCPI L-max VCPIL [5], C2-max VCPI C2 [1] MSV J minimum singular value of J [4] MSV Gs minimum singular value of Gs [4] Case l: VCPIs for the base case power flow and the optimised power flow (Final) Bus
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
VCPI-L Base case
Final
VCPI-C2 Base case
Final
0.09343 0.22553 0.35957 0.15025 0.12175 0.20064 0.26712 0.43733 0.54036 0.31590 0.32520 0.41233 0.30232 0.37114 0.34775 0.45307 0.19767 0.34996 0.46433 0.54169
0.08098 0.18563 0.27442 0.12457 0.10062 0.16248 0.21336 0.34263 0.40707 0.25332 0.25836 0.31600 0.24211 0.29297 0.27368 0.33883 0.15977 0.27869 0.35920 0.40691
0.51525 0.54273 0.58182 0.51827 0.50637 0.51887 0.53166 0.60449 0.64398 0.55719 0.58555 0.61924 0.55274 0.57463 0.59069 0.63263 0.54521 0.56762 0.60305 0.64048
0.50845 0.52369 0.50503 0.50534 0.49810 0.50224 0.50764 0.55556 0.53258 0.52667 0.54806 0.53069 0.52268 0.51709 0.54942 0.52715 0.52279 0.51369 0.55195 0.52933
order among any type of controllers (Generator excitations, OLTC taps, Switchable Var compensation). Similarly the operators may specify some controls selectively for curtailment, which can be easily implemented by the proposed algorithm, by avoiding pivots related to the specified controls for curtailment. As a consequence of the condition (1), the algorithm inherently respects any specified control ranges. Hence the controller movement may be restricted by specifying any desired range allowing for suitable reserve margin. Alternatively, suitable secondary objective may be used in conjunction with the primary objective, for curtailing the controller movement. As an example, we can minimise the L-1 norm of the controller movement along with the primary voltage stability improvement objective. Pivots which are consistent with both the primary and the secondary objectives are given a higher priority.
Table 3 Solution for generator excitations in pu Generator
Initial
Case 1
Case 2
Gen-1 Gen-2 Gen-3 Gen-4
1.00000 1.00000 1.00000 1.00000
1.05000 1.03800 1.05000 1.05000
1.05000 1.02700 1.05000 1.05000
5. Test cases and results The primal dual LP algorithm for the improvement of the static voltage stability was tested on several practical Indian systems. Sample results of the studies are presented in this section. 5.1. Case 1 A 24 bus Indian 400 kV EHV system was considered for the study. The base case data for the system is given in Appendix 4. The system has 4 generator excitation controls, 7 transformer tap controls and 4 shunt reactive power compensation controls. It was required to find the solution for all the 15 specified controllers for the improvement of the system static voltage stability. Optimum results were obtained in 3 load flow-optimisation cycles. The optimised solution was completely feasible without any violations. For the base case the minimum singular values of the load flow jacobian J and The minimum singular values for the matrix Gs [4] were 0.784 and 2.083 respectively. For the optimised solution, these values were 0.998 and 3.092. Table 1 gives the VCPIs C2 and L for the base case and the optimised power flow case. It is clearly seen that the proposed static voltage stability improvement objective results in improvement of the system voltage stability margin. 5.2. Case 2 This case is similar to case 1 with the difference that solution for 8 most effective controllers was required. The remaining 7 controllers were required to be curtailed. A check introduced in the primal–dual algorithm terminates the pivoting iteration as soon as solution for 8 controllers were obtained. This case is used to illustrate the ability of the algorithm for curtailing the number of controllers while
Table 2 Improvement in the VCPI:L VCPIL [5] C2 VCPIC2 [1] MSV J minimum singular value of J [4] MSV Gs minimum singular value of Gs [4]
Table 4 Solution for reactive power compensation (Qc) in MVAR. Initial Qc at all the buses 0MVAR Qc-max Maximum limit of Qc in MVAR
VCPI
Initial
Case 1
Case 2
Bus
Qc-max
Qc-case 1
Qc-case 2
C2-max L-max MSV(J) MSV(Gs)
0.64400 0.54200 0.78400 2.08300
0.55600 0.40700 0.99800 3.09200
0.58500 0.42800 0.97000 2.87700
Bus-7 Bus-16 Bus-20 Bus-24
25.00000 20.00000 30.00000 20.00000
25.00000 20.00000 30.00000 20.00000
0.00000 0.00000 0.00000 0.00000
R. Raghunatha et al. / Electrical Power and Energy Systems 21 (1999) 289–297 Table 5 Solution for transformer taps. Ranges for all transformers 1:1–0:9 Buses
Tap_initial
Tap_case 1
Tap_case 2
6–7 12–13 14–22 15–16 17–18 19–20 23–24
1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
0.92500 0.92500 0.96250 0.93750 0.96250 0.92500 0.92500
0.91250 0.90000 1.00000 1.00000 1.00000 0.90000 0.95000
improving the objective, using the most effective subset of controllers. The results for this case was obtained in a single load flow-optimisation cycle. The final solution has a single, very marginal lower bound voltage violation of 0.943 pu at bus 16. Table 2 gives the comparative results of the cases 1 and 2. It is seen that the proposed algorithm obtains near optimal solution using curtailed number of controllers. Tables 3–5, gives the final solution for the control variables for both the cases 1 and 2. 5.3. Case 3 This case is similar to case 2, with the following differences. 1. The Active and reactive power loads and specified active power generation of the system were increased by 10%. The increase in load reduces the static voltage stability margin. 2. The algorithm was used to identify 7 most effective controllers to improve the objective. For the sake of comparison the loss minimisation objective (Minimisation of slack generation) and the static voltage stability improvement objective were used. Both the objectives identified the 4 generators as the most effective controllers and the final solution for the generator excitations were identical (1.05 pu). The remaining 3 most effective controllers identified by the chosen objectives are listed in Table 6. It is seen from Table 6, that the set of most effective transformer controllers, identified by the objectives are not same. This is as a result of the difference in the relative ranking of the controllers based on the cost coefficients of the chosen objective functions. These experiments were carried out for several other increased loading Table 6 Case 2 VS: Voltage stability improvement objective. LR: Loss minimisation objective Control
Initial position
Final position VS
Final position LR
Tap Tap Tap Tap
1.00000 1.00000 1.00000 1.00000
0.90000 0.90000 0.90000 1.00000
0.90000 0.90000 1.00000 0.92500
6–7 12–13 19–20 23–24
293
conditions and for different practical Indian systems. It was observed that, the set of the most effective controllers and their solution given by the chosen objectives were not necessarily same. These experiments imply that, when the system is stressed and voltage stability margin is inadequate, the choice of the objective for reactive power scheduling is important. Table 7 gives the relative performance of the chosen objectives. It is clearly seen that the performance of the voltage stability improvement objective is better than that of the loss minimisation objective, for the chosen system operating condition. Traditionally, loss minimisation objective or voltage improvement objectives have been used for reactive power despatch. However, from the system operation point of view, system voltage stability may be more important. As the voltage stability improvement also results in loss minimisation and voltage improvement, this objective may be more preferable for reactive power despatch. 5.4. Case 4 During the system operation, it may be convenient to implement some controls in comparison to other controllers. System operators may have a higher priority for this type of controllers, although, the resulting operating condition is sub optimal. The proposed primal–dual algorithm can easily accommodate operators priorities, while determining the solution for the controllers. The pivoting algorithm simply pivots on columns of condensed simplex tableau, which provides solution for the control variables of operators choice. The algorithm explores other controls of lower priority, only if a feasible and optimal solution can not be obtained using the controls of operators priority. To illustrate, this flexibility of the algorithm, the system operating condition as in Case 3, was considered. The generator excitations were given lowest priority. It was required to determine, most effective 7 controllers among the remaining controls. The solution was obtained in a single load flow, optimisation cycle. All the transformer controllers other than the one connected between the buses 23–24 and the capacitor compensation at bus 24 were identified as the most effective 7 controllers by the algorithm. The solution for all the transformer were set at 0.9 pu (lower bound) and the compensation at bus 24 was set at 20 MVAR (upper bound). The final values of the VCPIs for this case is listed in Table 8. It is clearly seen that the solution is less optimal than the Case 3, as a result of the curtailment of generator excitation. 5.5. Case 5 For this case a 319 bus practical Indian system was considered for the analysis. The system has different voltage levels from 66 kV–400 kV. A total of 88 controllers (22 generators, 42 MVAR compensation, 24 transformer taps) were specified for the optimisation problem. Near accurate voltage stability improvement was achieved in one load
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Table 7 Case 3 Relative performance of the voltage stability improvement objective (VS) and Loss minimisation (LR) objective Performance Indicator
Initial value
Final value VS
Final value LR
C2-max L-max MSV(J) MSV(Gs) MW-Loss
0.70900 0.66200 0.67100 1.59700 85.58700
0.61900 0.50700 0.88900 2.47000 68.59500
0.62900 0.50900 0.88500 2.44200 68.80100
flow-optimisation cycle. Further optimisation did not result in much improvement in the voltage stability margins. At the end of the 1 load low-optimisation cycle, the algorithm rescheduled only 44 of the total 88 number specified controllers. It should be noted that the curtailment of the 44 controllers was obtained as natural solution of the algorithm, rather than specified controller solution as in the cases 2–4. In other words, the algorithm has a natural characteristic of identifying the most effective subset of the controllers to achieve the minimum objective. This is as a result of the higher priority given to the primal pivot, which activates the most effective controller from view point of the improvement in the objective. The computational time required for the optimisation module for this example was 15 s for one optimisation cycle on a 200 MHz Pentium PC. The results of the VCPIs for this case is given in Table 9.
6. Conclusion A numerically robust primal–dual SLP algorithm for improving the static voltage stability margin is presented. The algorithm effectively identifies the most effective subset of the specified set of controllers to achieve the minimum objective. Convergence to accurate minimum objective is usually obtained in 2–3 load flow–optimisation cycles. The convergence characteristic of the algorithm do not exhibit any oscillations. Approximations to the SLP algorithm or intricate logic suggested in the former literature to obtain controlled convergence are not necessary with the proposed algorithm. Further, modification to the basic algorithm to obtain curtailed number and curtailed
Table 9 Case 5-319 bus system Sv Absolute sum of voltage infeasibilities in pu Qc Capacitive compensation Performance Indicator
Initial
Final
C2-max L-max MSV(J) MSV(Gs) MW-Loss Sv (pu) Qc(MVAR)
0.66400 0.69400 0.07100 0.20800 252.91000 7.60500 5.00000
0.58100 0.51900 0.08200 0.29300 230.50000 0.35800 105.00000
movement of the controllers is simple and straight forward. The algorithm very easily handles operators control priorities. These features render the algorithm suitable for real time application. Appendix A. LP problem statement The voltage stability improvement objective may be stated as Minimise f x; u C T u;
A:1
subject to g x; u 0;
A:2
umin ⱕ u ⱕ umax ;
A:3
xmin ⱕ x ⱕ xmax ;
A:4
hmin ⱕ h ⱕ hmax :
A:5
Constraints defined by Eq. (A.2) are the load flow equations to be satisfied at any operating point. Constraints defined by Eq. (A.3) and Eq. (A.4) are the minimum and maximum permissible control and dependent variable limits. Constraints defined by Eq. (A.5) are security constraints; they include the minimum and maximum permissible MVAR loading limits for generators and power carrying limits on the transmission lines in the system. Appendix B. Reduced model formulation
Table 8 Case 4 Performance Indicator
Initial value
Final value
C2-max L-max MSV(J) MSV(Gs) MW-Loss
0.70900 0.66200 0.67300 1.59700 85.58700
0.63700 0.58800 0.74100 1.95700 78.91400
On linearisation of the power flow equations around the power flow solution, we obtain 2g 2g Dx ⫹ Du 0; 2x 2u Dx ⫺
2g 2x
⫺1
2g Du; 2u
R. Raghunatha et al. / Electrical Power and Energy Systems 21 (1999) 289–297 Table 10 Single generator excitation control of the first generator at bus 1 of the 24 bus system x
S
LBS
UBS
LBBV
UBBV
QG1 QG2 QG3 QG4 V5 V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22 V23 V24 Vg1
15.40900 ⫺6.27920 ⫺23.47300 ⫺8.73290 0.91760 0.71593 0.76736 0.46618 0.24894 0.30265 0.34241 0.40818 0.42801 0.46362 0.25839 0.26772 0.43653 0.44876 0.30857 0.32298 0.17320 0.46941 0.41602 0.42883 1.00000
⫺0.43359 0.13487 0.12496 0.44730 ⫺0.01745 0.04415 0.10468 ⫺0.02265 ⫺0.14096 ⫺0.06246 ⫺0.01375 0.18323 0.24196 0.07530 0.21495 0.30446 0.06179 0.10848 0.20664 0.29737 0.05450 0.09664 0.16802 0.23219 ⫺0.05000
0.28028 ⫺0.45437 ⫺0.08805 ⫺0.11380 0.09153 0.18383 0.23500 0.19186 0.26074 0.26796 0.27830 0.42822 0.47560 0.29099 0.60196 0.67798 0.29086 0.33131 0.53071 0.60700 0.63187 0.30967 0.40839 0.46538 0.05000
6.68120 0.84688 2.93320 3.90620 0.01601 ⫺0.03161 ⫺0.08033 0.01056 0.03509 0.01890 0.00471 ⫺0.07479 ⫺0.10356 ⫺0.03491 ⫺0.05554 ⫺0.08151 ⫺0.02697 ⫺0.04868 ⫺0.06376 ⫺0.09605 ⫺0.00944 ⫺0.04536 ⫺0.06990 ⫺0.09957 0.05000
4.31880 2.85310 2.06680 0.99382 0.08399 0.13161 0.18033 0.08944 0.06491 0.08110 0.09529 0.17479 0.20356 0.13491 0.15554 0.18151 0.12697 0.14868 0.16376 0.19605 0.10944 0.14536 0.16990 0.19957 0.05000
Dx Sx Du:
B:1
Eq. (B.1) gives the sensitivity relation between the dependent and the control variables. DC2
2C2 2C2 Dx ⫹ Du 2x 2u
2C2 2C2 Sx ⫹ Du: 2x 2u
B:2
Eq. (B.2) gives the sensitivity relation between the voltage collapse proximity indicator C2 and the control variables u. This equation is used in formulating the objective function given by Eq. (3). 2h 2h Dx ⫹ Du Dh 2x 2u
2h 2h Sx ⫹ Du: 2x 2u
B:3
The loss sensitivities (Minimisation of slack generation) is obtained as follows 2Psl T 2Psl T Dx ⫹ Du DPsl 2x 2u
2Psl 2x
T
2Psl T Du: Sx ⫹ 2u
295
B:4
Appendix C. Explanation for conditions 1 and 2 An illustration is given in this section which explains the utility of the conditions (1) and (2). For the purpose of illustration a single generator excitation control of the first generator at bus 1 of the 24 bus system is considered. The first column of Table 10 (x) gives the variables under consideration (reactive power limits QG, dependent bus voltage magnitudes V and controller Vg1). The column S gives the sensitivities of these variables with respect to the generator excitation control Vg1. The columns UBBV and LBBV refer to basis variables of the initial simplex tableau related to upper bound and lower bound constraints respectively. UBS and LBS columns give the solution for the control variable Vg1 based on the pivot related to basis variables of the upper and the lower bound constraints. The range of the control Vg1 is 0.95 pu to 1.05 pu, with the initial value of Vg1 at 1.0 pu. Thus the feasible solution for change in Vg1 is in the range of ⫺0.05 to ⫹0.05 pu. Examination of the columns UBS and LBS indicate that this solution range is not satisfied by all the constrains. The solutions from only 6 pivots will result in feasible solution for the controller Vg1. These feasible solutions are indicated in bold font in Table 10. The solution from all other pivots are infeasible and cannot be implemented. While, solutions from some of the constraints overlap the feasible solution range for the controller, solutions from most other pivots never intersect the feasible solution range for the controller. It may be noted that all possible dual pivots based on infeasibilities, with the exception of the bus voltage V6, will result in infeasible solution for the controller Vg1. In contrast A primal pivot, by virtue of selection of the smallest ratio solution for the controller Vg1, will always result in a feasible solution. In the implemented version of the algorithm, whenever a dual pivot results in infeasible solution an alternative pivot was explored, that improves the feasibility while providing the feasible solution for the controller. A known problem with the primal pivot which selects the smallest ratio solution for the new basis variable is the problem of cycling. The condition (2) implies that each new tableau after pivoting should be more feasible. The feasibility of the tableau was estimated in terms of the L-1 norm of the infeasibility of the dependent variables. It may be verified that successive improvement in feasibility of the tableau will avoid the cycling problem associated with the primal pivots. Thus the proposed algorithm is numerically stable.
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Appendix D. Data for the 24 bus system
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References [1] Raghunatha R, Ramanujam R, Parthasarathy K, Thukaram D. A new and fast technique for voltage stability analysis of a grid network using system voltage space. Accepted for publication in International Journal of Electrical Power and Energy systems, Amsterdam: Elsevier. [2] North American Electric Reliability Council, Survey of voltage collapse phenomenon, August 1991. [3] IEEE Special publication 90th 358-2-PWR. Voltage stability of power systems: concepts, analytical tools, and industry experience, Prepared by the IEEE Working Group on Voltage Stability, 1990. [4] Lof PA, Smed T, Anderson G, Hill DJ. Fast calculation of a voltage stability index, IEEE Transactions on power systems, 7(1) 1992. [5] Kessel P, Glavitsch H. Estimating the voltage stability of a power system, IEEE Transactions on power delivery, PWRD-1(3) 1986. [6] Wolfe CS. Linear programming with fortran, Scott, Foresman and Company. [7] Chebbo AM, Irving MR, Sterling MJH. Reactive power dispatch incorporating voltage stability, IEE proceedings, 139(3) 1992. [8] Begovic MM, Phadke AG. Control of voltage stability using sensitivity analysis, IEEE Transactions on power systems, 7(1) 1992. [9] Bourgin F, Testud G, Heilbronn B, Verseille J. Present practices and trends on the French power system to prevent voltage collapse, IEEE Transactions on power systems, 8(3) 1993.
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[10] Kumano T, Yokoyama A, Sekine Y. Fast monitoring and optimal preventive control of voltage instability, International journal of electrical power and energy systems, 16(2) 1994. [11] Rios Zalapa R, Cory BJ. Reactive reserve management, Proceedings of the IEE, Generation, transmission and distribution, 142(1) 1995. [12] Van Cutsem T. An approach to corrective control of voltage instability using simulation and sensitivity, IEEE Transactions on power systems, 10(2) 1995. [13] Mamandur KRC, Chenoweth RD. Optimal control of reactive power flow for improvements in voltage profiles and for real power loss minimisation, IEEE Transactions on power apparatus and systems, PAS-100(7) 1981. [14] Qiu J, Shahidehpour SM. A new approach for minimising losses and improving voltage profile, IEEE transactions on power systems, PWRS-2(2) 1987. [15] Stott B, Alsac O, Monticelli AJ. Security analysis and optimisation, Proceedings of the IEEE, 75(12) 1987. [16] Alsac O, Bright J, Prais M, Stott B. Further developments in LP based optimal power flow, IEEE transactions on power systems, 5(3) 1990. [17] Soman SA, Parthasarathy K, Thukaram D. Curtailed number and reduced controller movement optimisation algorithms for real time voltage/reactive power control, Paper 94 WM 254-3 PWRS, IEEE/PES 1994 winter meeting, New York, January 30-February 3, 1994.
Optimal static voltage stability improvement using a numerically stable SLP algorithm, for real time applications R. Raghunatha a, R. Ramanujam a, K. Parthasarathy b,*, D. Thukaram b a
International Development and Engineering Associates Ltd., Bangalore, India b Indian Institute of Science, Bangalore, India
Received 8 October 1997; received in revised form 12 February 1998; accepted 12 February 1998
Abstract A numerically stable sequential Primal–Dual LP algorithm for the reactive power optimisation (RPO) is presented in this article. The algorithm minimises the voltage stability index C 2 [1] of all the load buses to improve the system static voltage stability. Real time requirements such as numerical stability, identification of the most effective subset of controllers for curtailing the number of controllers and their movement can be handled effectively by the proposed algorithm. The algorithm has a natural characteristic of selecting the most effective subset of controllers (and hence curtailing insignificant controllers) for improving the objective. Comparison with transmission loss minimisation objective indicates that the most effective subset of controllers and their solution identified by the static voltage stability improvement objective is not the same as that of the transmission loss minimisation objective. The proposed algorithm is suitable for real time application for the improvement of the system static voltage stability. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: Static voltage stability; Reactive power optimisation; S L P algorithm
1. Introduction The incidents of voltage collapse [2–3], have drawn the attention of the power system industry to the problem of 1. Identification of the critical buses or areas of the system, most susceptible to voltage collapse. 2. Initiation of corrective measures to enhance the static voltage stability margin
the state estimator output and are most effective when used in conjunction with the knowledge of the relative position of the bus voltage phasors. The proposed RPO algorithm in this article minimises the index C2 of all the load buses to improve the system static voltage stability. This is equivalent to minimising the L-1 norm of the VCPI C2. Some of the requirement of optimal power flow for the real time applications are as follows. 1. The algorithm should have smooth convergence characteristic and should be numerically stable. 2. Power systems normally operate in a quasi-steady state. Hence, it is essential to identify the most effective subset of controllers and reschedule the same [17]. 3. It is essential to maintain adequate reserve margin in controllers for use in emergency states.
For static analysis, the task of identification of critical buses or areas of system, from view point of voltage stability, is usually done using voltage collapse proximity indicators (VCPIs). Although several voltage collapse proximity indicators are proposed in the literature, we have used the indices, proposed in the references [4–5], for the purpose of evaluating the proposed primal–dual SLP algorithm [6]. References [7–12] addresses some of the measures and algorithms aimed towards the corrective measures for the improvement of the system voltage stability. In reference [1], the authors have introduced two new VCPIs C1 and C2. These indices are briefly described in Section 2. These indices use the information available from
In the proposed primal–dual SLP algorithm the above features are easily implemented. The proposed algorithm was initially developed with the transmission loss minimisation object [13–16], which is reported to have several short comings. The following are some of the features of the transmission loss minimisation algorithm, reported in the literature.
* Corresponding author. Tel.: ⫹91080-334441; fax: ⫹91080-3341683. E-mail address: [email protected] (K. Parthasarathy)
1. The SLP algorithm for transmission loss minimisation do not exhibit smooth convergence owing to non-linear
0142-0615/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S0142-061 5(98)00048-9
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Fig. 1. Simple flow chart of EMS functions.
variation of the transmission loss as a function of control variables. 2. To achieve controlled convergence, it is necessary to use approximations or special techniques in the basic LP algorithm. Some of the modifications suggested in the basic LP algorithm are (i) Use of restricted control step sizes and limits [13,14] and (i) Use of intricate control logic [16]. 3. Further, heuristic measures may be required to identify the most effective subset of controllers for curtailing the number of controllers [16]. In the proposed primal–dual LP algorithm none of the above difficulties are present either with the transmission loss minimisation objective or with the voltage stability improvement objective. The implementation of the algorithm is straight forward and does not require any intricate control logic or approximations. Accurate minimisation of the objective is usually achieved in two to three cycles, without exhibiting any oscillations in the convergence characteristic. In the presence of over voltages, the feasibility of the solution is usually improved with reduced optimality. 2. The static voltage stability improvement objective formulation In reference [1], the authors have introduced two new VCPIs C1 and C2. These VCPIs are most effective when used in conjunction with the knowledge of the relative position of the bus voltage phasors. Further, the indices C1 and C2 may be directly computed from the output of the state estimator and hence are suitable for real time applications.
The performance of these indices compare well with those of the other VCPIs presented in the literature [4,5].The index C2 has a value of about 0.5 at no load condition and a value of about 1.0 at voltage collapse point. For greater voltage stability margin, the index C2 should be as small as possible. If the voltage stability indicator C2 of all load buses are minimised, then the system static voltage stability margin must increase. Minimising voltage stability indicator of all the load buses is equivalent to minimising the L-1 norm of the index C2 of all the buses. The development of such an objective function is given in this section. The basic philosophy used in developing the indices C1 and C2 is that the bus voltage phasors contain all the necessary information about the system static state. Hence, if the relative position of the bus voltages are examined with reference to a suitable voltage phasor, it should be possible to identify the most critical buses from the view point of static voltage stability. Two reference voltages Vcs and Vcg, referred to as centroid of the system voltage space and centroid of the generator voltage space are suggested in [1] to identify critical bus voltages and a measure of their proximity to voltage collapse. The centroid voltages Vcs and Vcg are computed by comparing the equilibrium conditions of a power system with that of the equilibrium conditions of rigid bodies. It is shown in [1] that the voltage Vcg is the phasor average of all the generator bus voltages. The voltage stability index C1 and C2 for the ith bus is given by Vcg ⫺Vi ; 1a C1i jVi j Vcg ; 1b C2i 2jVi jcos di ⫺dcg Where, Vi is the bus voltage under consideration, d i is the phase angle of the ith bus voltage and d cg is the phase angle of Vcg. The changes in the index C2i are then computed as 2C2 2C2i 2C2i DjVi j ⫹ DC2i i D Vcg ⫹ Ddcg 2j V i j 2dcg 2 Vcg ⫹
2C2i D di ; 2 di
2
The dependent variables of Eq. (2) are then expressed as a function of control variables u, as explained in Appendix B. Using Eq. (2), the voltage stability improvement objective may be stated as X 3 minimise DC2i C T u: subject to the usual constraints on the control and dependent variables (Appendix A). Fig. 1 shows a simple functional flow chart of EMS functions. The voltage insecurities are detected by blocks 3 (base
R. Raghunatha et al. / Electrical Power and Energy Systems 21 (1999) 289–297
case) and 6 (contingency case). If the system voltage stability margins are inadequate, optimal corrective (block 4) or preventive (block 7) control scheduling using voltage stability improvement objective is required to improve the system static voltage stability.
3. The primal–dual LP algorithm [6] In this section some of the features and implementation details of the primal–dual LP algorithm used are presented. 1. The initial condensed simplex tableau for the primal– dual algorithm need not be optimal or feasible. 2. The optimisation problem is always solved as a maximisation problem rather than as a minimisation problem. 3. All possible pivots by the primal (maximisation) or the dual (minimisation) algorithm are examined. The pivot that influences the objective most is selected. 4. A pivot selected by the primal algorithm will improve the optimality while usually reducing the upper bound feasibility margin. A pivot selected by the dual algorithm will improve the feasibility while attempting to maintain optimality. In general, as applied to power systems, improvement in lower bound feasibility is accompanied by improvement in the optimality and correction of upper bound violations normally results in reduced optimality. 5. The non basis variables associated with the cost coefficients of the objective row will always have zero value. Only the basis variables may have non zero solution. 6. If the initial tableau is control feasible, i.e., all control variables are within their limits and only the primal algorithm is used, the final tableau will always be control feasible. This is true even if unrestricted or restricted control ranges are used. Consequently, the solution from the primal algorithm may be directly implemented without using any approximations. This will usually result in improved objective and improved lower bound feasibility of dependent variables. 7. A pure dual pivot, does not necessarily result in a control variable feasible tableau. This appears to be the cause for the problems encountered with the transmission loss minimisation objective. In the actual implementation the primal–dual algorithm was used subject to the following conditions. 1. The pivot should result in a new tableau, which is always control feasible. 2. The new tableau should always be more feasible than the previous one. The new tableau after satisfying the conditions (1) and (2) will be usually always more optimal than the previous tableau and will be usually less optimal in the presence of predominantly upper bound violations. These conditions ensure that the final controller solutions can always be implemented with improved system performance objective.
291
Approximations such as restricted step size for controllers as used in former reported algorithms will never be required. It is possible that the conventional algorithm without the conditions (1) and (2) may result in more optimal or feasible solution which does not necessarily guarantee feasible controller solution. The conditions (1) and (2) require that, the effect of the pivot is evaluated prior to actual pivoting algorithm. The added computation because of this is quite small, as the evaluation is carried out on a single column of the condensed simplex tableau. The condition (1) implies that irrespective of the range specified for the controller variables, the final solution for the controllers will always be within the specified limits. Hence, the need for restricted step sizes or ranges of control variables or intricate control logic are not necessary. The final solution obtained may be directly implemented without any approximation. Appendix C provides an illustration that clarifies the utility of the conditions (1) and (2).
4. Curtailing number of controllers and their movement From the description given in Section 3 and the conditions (1) and (2), it is evident that each LP pivoting iteration will progressively result in more feasible and more optimal tableau. Further, the primal algorithm always uses the most effective controller to improve the objective and the dual usually corrects the largest violation. In the implemented algorithm, the primal pivot is always given a priority over the dual pivot. Each pivoting iteration results in an exchange of variables from basis to non basis. Suppose we are interested, in obtaining solution for a specified curtailed number of control variables, then we only need to check the number of control variables appearing as basis variables at the end of each pivoting iteration. When the solution for the required specified number of control variables is available the pivoting iteration may be terminated and the solution may be implemented. This will result in improved optimal and feasible solution using curtailed number of controllers. As the primal pivot is always given a higher priority over the dual pivot, the resulting solution normally represent the most effective subset of the total number of controllers specified, from the view point of the specified objective. During the system operation, it may be relatively convenient to implement some controls in comparison to other controllers. Such controls maybe preferred by the power system operators for the improvement in the system performance. While identifying the effective subset of the controllers available for the improvement of the system performance, it is preferable to take into account the operators control priorities. This is very easily accomplished by the proposed primal–dual algorithm, by forcing the pivoting algorithm to give a higher priority to the controllers of the operators choice. These priorities may be specified in any
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Table 1 Improvement in the VCPI L-max VCPIL [5], C2-max VCPI C2 [1] MSV J minimum singular value of J [4] MSV Gs minimum singular value of Gs [4] Case l: VCPIs for the base case power flow and the optimised power flow (Final) Bus
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
VCPI-L Base case
Final
VCPI-C2 Base case
Final
0.09343 0.22553 0.35957 0.15025 0.12175 0.20064 0.26712 0.43733 0.54036 0.31590 0.32520 0.41233 0.30232 0.37114 0.34775 0.45307 0.19767 0.34996 0.46433 0.54169
0.08098 0.18563 0.27442 0.12457 0.10062 0.16248 0.21336 0.34263 0.40707 0.25332 0.25836 0.31600 0.24211 0.29297 0.27368 0.33883 0.15977 0.27869 0.35920 0.40691
0.51525 0.54273 0.58182 0.51827 0.50637 0.51887 0.53166 0.60449 0.64398 0.55719 0.58555 0.61924 0.55274 0.57463 0.59069 0.63263 0.54521 0.56762 0.60305 0.64048
0.50845 0.52369 0.50503 0.50534 0.49810 0.50224 0.50764 0.55556 0.53258 0.52667 0.54806 0.53069 0.52268 0.51709 0.54942 0.52715 0.52279 0.51369 0.55195 0.52933
order among any type of controllers (Generator excitations, OLTC taps, Switchable Var compensation). Similarly the operators may specify some controls selectively for curtailment, which can be easily implemented by the proposed algorithm, by avoiding pivots related to the specified controls for curtailment. As a consequence of the condition (1), the algorithm inherently respects any specified control ranges. Hence the controller movement may be restricted by specifying any desired range allowing for suitable reserve margin. Alternatively, suitable secondary objective may be used in conjunction with the primary objective, for curtailing the controller movement. As an example, we can minimise the L-1 norm of the controller movement along with the primary voltage stability improvement objective. Pivots which are consistent with both the primary and the secondary objectives are given a higher priority.
Table 3 Solution for generator excitations in pu Generator
Initial
Case 1
Case 2
Gen-1 Gen-2 Gen-3 Gen-4
1.00000 1.00000 1.00000 1.00000
1.05000 1.03800 1.05000 1.05000
1.05000 1.02700 1.05000 1.05000
5. Test cases and results The primal dual LP algorithm for the improvement of the static voltage stability was tested on several practical Indian systems. Sample results of the studies are presented in this section. 5.1. Case 1 A 24 bus Indian 400 kV EHV system was considered for the study. The base case data for the system is given in Appendix 4. The system has 4 generator excitation controls, 7 transformer tap controls and 4 shunt reactive power compensation controls. It was required to find the solution for all the 15 specified controllers for the improvement of the system static voltage stability. Optimum results were obtained in 3 load flow-optimisation cycles. The optimised solution was completely feasible without any violations. For the base case the minimum singular values of the load flow jacobian J and The minimum singular values for the matrix Gs [4] were 0.784 and 2.083 respectively. For the optimised solution, these values were 0.998 and 3.092. Table 1 gives the VCPIs C2 and L for the base case and the optimised power flow case. It is clearly seen that the proposed static voltage stability improvement objective results in improvement of the system voltage stability margin. 5.2. Case 2 This case is similar to case 1 with the difference that solution for 8 most effective controllers was required. The remaining 7 controllers were required to be curtailed. A check introduced in the primal–dual algorithm terminates the pivoting iteration as soon as solution for 8 controllers were obtained. This case is used to illustrate the ability of the algorithm for curtailing the number of controllers while
Table 2 Improvement in the VCPI:L VCPIL [5] C2 VCPIC2 [1] MSV J minimum singular value of J [4] MSV Gs minimum singular value of Gs [4]
Table 4 Solution for reactive power compensation (Qc) in MVAR. Initial Qc at all the buses 0MVAR Qc-max Maximum limit of Qc in MVAR
VCPI
Initial
Case 1
Case 2
Bus
Qc-max
Qc-case 1
Qc-case 2
C2-max L-max MSV(J) MSV(Gs)
0.64400 0.54200 0.78400 2.08300
0.55600 0.40700 0.99800 3.09200
0.58500 0.42800 0.97000 2.87700
Bus-7 Bus-16 Bus-20 Bus-24
25.00000 20.00000 30.00000 20.00000
25.00000 20.00000 30.00000 20.00000
0.00000 0.00000 0.00000 0.00000
R. Raghunatha et al. / Electrical Power and Energy Systems 21 (1999) 289–297 Table 5 Solution for transformer taps. Ranges for all transformers 1:1–0:9 Buses
Tap_initial
Tap_case 1
Tap_case 2
6–7 12–13 14–22 15–16 17–18 19–20 23–24
1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
0.92500 0.92500 0.96250 0.93750 0.96250 0.92500 0.92500
0.91250 0.90000 1.00000 1.00000 1.00000 0.90000 0.95000
improving the objective, using the most effective subset of controllers. The results for this case was obtained in a single load flow-optimisation cycle. The final solution has a single, very marginal lower bound voltage violation of 0.943 pu at bus 16. Table 2 gives the comparative results of the cases 1 and 2. It is seen that the proposed algorithm obtains near optimal solution using curtailed number of controllers. Tables 3–5, gives the final solution for the control variables for both the cases 1 and 2. 5.3. Case 3 This case is similar to case 2, with the following differences. 1. The Active and reactive power loads and specified active power generation of the system were increased by 10%. The increase in load reduces the static voltage stability margin. 2. The algorithm was used to identify 7 most effective controllers to improve the objective. For the sake of comparison the loss minimisation objective (Minimisation of slack generation) and the static voltage stability improvement objective were used. Both the objectives identified the 4 generators as the most effective controllers and the final solution for the generator excitations were identical (1.05 pu). The remaining 3 most effective controllers identified by the chosen objectives are listed in Table 6. It is seen from Table 6, that the set of most effective transformer controllers, identified by the objectives are not same. This is as a result of the difference in the relative ranking of the controllers based on the cost coefficients of the chosen objective functions. These experiments were carried out for several other increased loading Table 6 Case 2 VS: Voltage stability improvement objective. LR: Loss minimisation objective Control
Initial position
Final position VS
Final position LR
Tap Tap Tap Tap
1.00000 1.00000 1.00000 1.00000
0.90000 0.90000 0.90000 1.00000
0.90000 0.90000 1.00000 0.92500
6–7 12–13 19–20 23–24
293
conditions and for different practical Indian systems. It was observed that, the set of the most effective controllers and their solution given by the chosen objectives were not necessarily same. These experiments imply that, when the system is stressed and voltage stability margin is inadequate, the choice of the objective for reactive power scheduling is important. Table 7 gives the relative performance of the chosen objectives. It is clearly seen that the performance of the voltage stability improvement objective is better than that of the loss minimisation objective, for the chosen system operating condition. Traditionally, loss minimisation objective or voltage improvement objectives have been used for reactive power despatch. However, from the system operation point of view, system voltage stability may be more important. As the voltage stability improvement also results in loss minimisation and voltage improvement, this objective may be more preferable for reactive power despatch. 5.4. Case 4 During the system operation, it may be convenient to implement some controls in comparison to other controllers. System operators may have a higher priority for this type of controllers, although, the resulting operating condition is sub optimal. The proposed primal–dual algorithm can easily accommodate operators priorities, while determining the solution for the controllers. The pivoting algorithm simply pivots on columns of condensed simplex tableau, which provides solution for the control variables of operators choice. The algorithm explores other controls of lower priority, only if a feasible and optimal solution can not be obtained using the controls of operators priority. To illustrate, this flexibility of the algorithm, the system operating condition as in Case 3, was considered. The generator excitations were given lowest priority. It was required to determine, most effective 7 controllers among the remaining controls. The solution was obtained in a single load flow, optimisation cycle. All the transformer controllers other than the one connected between the buses 23–24 and the capacitor compensation at bus 24 were identified as the most effective 7 controllers by the algorithm. The solution for all the transformer were set at 0.9 pu (lower bound) and the compensation at bus 24 was set at 20 MVAR (upper bound). The final values of the VCPIs for this case is listed in Table 8. It is clearly seen that the solution is less optimal than the Case 3, as a result of the curtailment of generator excitation. 5.5. Case 5 For this case a 319 bus practical Indian system was considered for the analysis. The system has different voltage levels from 66 kV–400 kV. A total of 88 controllers (22 generators, 42 MVAR compensation, 24 transformer taps) were specified for the optimisation problem. Near accurate voltage stability improvement was achieved in one load
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Table 7 Case 3 Relative performance of the voltage stability improvement objective (VS) and Loss minimisation (LR) objective Performance Indicator
Initial value
Final value VS
Final value LR
C2-max L-max MSV(J) MSV(Gs) MW-Loss
0.70900 0.66200 0.67100 1.59700 85.58700
0.61900 0.50700 0.88900 2.47000 68.59500
0.62900 0.50900 0.88500 2.44200 68.80100
flow-optimisation cycle. Further optimisation did not result in much improvement in the voltage stability margins. At the end of the 1 load low-optimisation cycle, the algorithm rescheduled only 44 of the total 88 number specified controllers. It should be noted that the curtailment of the 44 controllers was obtained as natural solution of the algorithm, rather than specified controller solution as in the cases 2–4. In other words, the algorithm has a natural characteristic of identifying the most effective subset of the controllers to achieve the minimum objective. This is as a result of the higher priority given to the primal pivot, which activates the most effective controller from view point of the improvement in the objective. The computational time required for the optimisation module for this example was 15 s for one optimisation cycle on a 200 MHz Pentium PC. The results of the VCPIs for this case is given in Table 9.
6. Conclusion A numerically robust primal–dual SLP algorithm for improving the static voltage stability margin is presented. The algorithm effectively identifies the most effective subset of the specified set of controllers to achieve the minimum objective. Convergence to accurate minimum objective is usually obtained in 2–3 load flow–optimisation cycles. The convergence characteristic of the algorithm do not exhibit any oscillations. Approximations to the SLP algorithm or intricate logic suggested in the former literature to obtain controlled convergence are not necessary with the proposed algorithm. Further, modification to the basic algorithm to obtain curtailed number and curtailed
Table 9 Case 5-319 bus system Sv Absolute sum of voltage infeasibilities in pu Qc Capacitive compensation Performance Indicator
Initial
Final
C2-max L-max MSV(J) MSV(Gs) MW-Loss Sv (pu) Qc(MVAR)
0.66400 0.69400 0.07100 0.20800 252.91000 7.60500 5.00000
0.58100 0.51900 0.08200 0.29300 230.50000 0.35800 105.00000
movement of the controllers is simple and straight forward. The algorithm very easily handles operators control priorities. These features render the algorithm suitable for real time application. Appendix A. LP problem statement The voltage stability improvement objective may be stated as Minimise f x; u C T u;
A:1
subject to g x; u 0;
A:2
umin ⱕ u ⱕ umax ;
A:3
xmin ⱕ x ⱕ xmax ;
A:4
hmin ⱕ h ⱕ hmax :
A:5
Constraints defined by Eq. (A.2) are the load flow equations to be satisfied at any operating point. Constraints defined by Eq. (A.3) and Eq. (A.4) are the minimum and maximum permissible control and dependent variable limits. Constraints defined by Eq. (A.5) are security constraints; they include the minimum and maximum permissible MVAR loading limits for generators and power carrying limits on the transmission lines in the system. Appendix B. Reduced model formulation
Table 8 Case 4 Performance Indicator
Initial value
Final value
C2-max L-max MSV(J) MSV(Gs) MW-Loss
0.70900 0.66200 0.67300 1.59700 85.58700
0.63700 0.58800 0.74100 1.95700 78.91400
On linearisation of the power flow equations around the power flow solution, we obtain 2g 2g Dx ⫹ Du 0; 2x 2u Dx ⫺
2g 2x
⫺1
2g Du; 2u
R. Raghunatha et al. / Electrical Power and Energy Systems 21 (1999) 289–297 Table 10 Single generator excitation control of the first generator at bus 1 of the 24 bus system x
S
LBS
UBS
LBBV
UBBV
QG1 QG2 QG3 QG4 V5 V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22 V23 V24 Vg1
15.40900 ⫺6.27920 ⫺23.47300 ⫺8.73290 0.91760 0.71593 0.76736 0.46618 0.24894 0.30265 0.34241 0.40818 0.42801 0.46362 0.25839 0.26772 0.43653 0.44876 0.30857 0.32298 0.17320 0.46941 0.41602 0.42883 1.00000
⫺0.43359 0.13487 0.12496 0.44730 ⫺0.01745 0.04415 0.10468 ⫺0.02265 ⫺0.14096 ⫺0.06246 ⫺0.01375 0.18323 0.24196 0.07530 0.21495 0.30446 0.06179 0.10848 0.20664 0.29737 0.05450 0.09664 0.16802 0.23219 ⫺0.05000
0.28028 ⫺0.45437 ⫺0.08805 ⫺0.11380 0.09153 0.18383 0.23500 0.19186 0.26074 0.26796 0.27830 0.42822 0.47560 0.29099 0.60196 0.67798 0.29086 0.33131 0.53071 0.60700 0.63187 0.30967 0.40839 0.46538 0.05000
6.68120 0.84688 2.93320 3.90620 0.01601 ⫺0.03161 ⫺0.08033 0.01056 0.03509 0.01890 0.00471 ⫺0.07479 ⫺0.10356 ⫺0.03491 ⫺0.05554 ⫺0.08151 ⫺0.02697 ⫺0.04868 ⫺0.06376 ⫺0.09605 ⫺0.00944 ⫺0.04536 ⫺0.06990 ⫺0.09957 0.05000
4.31880 2.85310 2.06680 0.99382 0.08399 0.13161 0.18033 0.08944 0.06491 0.08110 0.09529 0.17479 0.20356 0.13491 0.15554 0.18151 0.12697 0.14868 0.16376 0.19605 0.10944 0.14536 0.16990 0.19957 0.05000
Dx Sx Du:
B:1
Eq. (B.1) gives the sensitivity relation between the dependent and the control variables. DC2
2C2 2C2 Dx ⫹ Du 2x 2u
2C2 2C2 Sx ⫹ Du: 2x 2u
B:2
Eq. (B.2) gives the sensitivity relation between the voltage collapse proximity indicator C2 and the control variables u. This equation is used in formulating the objective function given by Eq. (3). 2h 2h Dx ⫹ Du Dh 2x 2u
2h 2h Sx ⫹ Du: 2x 2u
B:3
The loss sensitivities (Minimisation of slack generation) is obtained as follows 2Psl T 2Psl T Dx ⫹ Du DPsl 2x 2u
2Psl 2x
T
2Psl T Du: Sx ⫹ 2u
295
B:4
Appendix C. Explanation for conditions 1 and 2 An illustration is given in this section which explains the utility of the conditions (1) and (2). For the purpose of illustration a single generator excitation control of the first generator at bus 1 of the 24 bus system is considered. The first column of Table 10 (x) gives the variables under consideration (reactive power limits QG, dependent bus voltage magnitudes V and controller Vg1). The column S gives the sensitivities of these variables with respect to the generator excitation control Vg1. The columns UBBV and LBBV refer to basis variables of the initial simplex tableau related to upper bound and lower bound constraints respectively. UBS and LBS columns give the solution for the control variable Vg1 based on the pivot related to basis variables of the upper and the lower bound constraints. The range of the control Vg1 is 0.95 pu to 1.05 pu, with the initial value of Vg1 at 1.0 pu. Thus the feasible solution for change in Vg1 is in the range of ⫺0.05 to ⫹0.05 pu. Examination of the columns UBS and LBS indicate that this solution range is not satisfied by all the constrains. The solutions from only 6 pivots will result in feasible solution for the controller Vg1. These feasible solutions are indicated in bold font in Table 10. The solution from all other pivots are infeasible and cannot be implemented. While, solutions from some of the constraints overlap the feasible solution range for the controller, solutions from most other pivots never intersect the feasible solution range for the controller. It may be noted that all possible dual pivots based on infeasibilities, with the exception of the bus voltage V6, will result in infeasible solution for the controller Vg1. In contrast A primal pivot, by virtue of selection of the smallest ratio solution for the controller Vg1, will always result in a feasible solution. In the implemented version of the algorithm, whenever a dual pivot results in infeasible solution an alternative pivot was explored, that improves the feasibility while providing the feasible solution for the controller. A known problem with the primal pivot which selects the smallest ratio solution for the new basis variable is the problem of cycling. The condition (2) implies that each new tableau after pivoting should be more feasible. The feasibility of the tableau was estimated in terms of the L-1 norm of the infeasibility of the dependent variables. It may be verified that successive improvement in feasibility of the tableau will avoid the cycling problem associated with the primal pivots. Thus the proposed algorithm is numerically stable.
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Appendix D. Data for the 24 bus system
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