Relaxed approach for the parallel solution of secu rity-constrai ned resc h ed u I i n g

dispatch with post-contingency

OR. Saavedra

Abstract: Some approaches for the solution of the security-constrained dispatch with post- contingency corrective rescheduling problem in a multiprocessing environment are presented. These approaches can be interpreted as relaxed versions of the classical Benders decomposition. The idea is to improve the computational efficiency when the problem is solved by using parallel processing. Results from one of these approaches implemented into an asynchronous algorithm are reported here. Some strategics for improving performance are discussed and test results with two real-life Brazilian systems are reported.

1 Introduction

The security-constrained dispatch with post-contingency corrective rescheduling problem (SCDR) is a generalisation of the security-constrained optimal dispatch problem (SCD). because it takes into account the system corrective capacities after a contingency occurs. The corrective actions depend on the available time before the protection system becomes active and on the response capability of the control equipment. When corrective capabilities are included, each post-contingency scenario is modelled as an optimisation problem and the global formulation reaches high dimen- sions, demanding the use of decomposition techniques.

Fortunately, the availability of new less-expensive parallel machines has increased significantly in recent years. Several studies have been published, involving parallel implementa- tion of the problem [l-3]. In [I]_ the application of parallel processing to the optimal power flow problem is systema- tised, presewing a good degree of portability and efficiency. When SCD is parallelised, good load balancing is obtained by using an asynchronous model [I].

The parallel solution of the SCDR using Benders decomposition gives reliable and robust performance [3]; nevertheless, from the computational point of view, poor efficiency is achieved due to the load balancing between the processors. However, it is possible to take advantage of the practical behaviour of power systems. When post-con- tingency rescheduling is considered, we can point out the following observations:

(a) All the scenarios will be tested for violations (con- tingency analysis), hut only a fcw will be needed to solve an operating subproblem (post-contingency rescheduling). This means that some processors work more than others, leading to poor load balancing.

0 IEE. 2003 IEE Pmrecd;ngy online no. 20030303 doi: IO. 1MY/ip-gtd:?0030303 Paper first reccived 23rd May 2002 and in reviscd ibnn 2nd January 2003 The author is u,ith flic Grupo de Sirtcmas de Polencia. Departamenlo de Eigmha"a de Elelriadadr, Univemidilde Fedeii do Maranhjo. SZo Luis. MA. Rrilzil

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(b) Most of the post-contingency scenarios deal with dependent violations, i.e. elimination of the worst violation in many cases leads to the clearing of other violations. (c) In most cases. post-contingency violations are mitigated or eliminated by the rescheduling capacities. When these violations are not completely cleared, the remaining violations should he eliminated by a shift (in most cases, small) of base case operating point.

In this work. two decomposition approaches based on the linearised load flow model for the solution of the SCDR in a multiprocessing environment are presented. They can he seen as relaxed versions of Benders method. Results from implementation of one of these approaches are presented and discussed. Validation tests using two real-life Brazilian systems are also included.

2 Problem formulation

The problem formulation can he stated as follows (line- arised form) [4]:

Min./ = c'x, (1)

where f is the objective function, cis the cost associated with the base-case variables, x,, are the base-case control variables and xi are the control variab~es at the post- contingency scenario ;.

Expression (2) represents the operational constraints of the system at the base case, such as load flow equations, generation and flow limits etc. Expression (3) corresponds to the constraints for each one or the NC post-contingency scenarios. Constraints (4) represent the coupling constraints (ramp constraints) between post-contingency and hase-case states. The parameter A'; gives the allowed rescheduling for the control variables when the contingency i occurs.

For real power systems_ (1H4) can reach high dimen- sions. The problem stmcture is illustrated in Fig. I , where E, and Fi represent (3) and (4). respectively.

29 I

Fig. 1 considered

The problem srmcriire when correctice cupucifies ure

Owing to the high dimensions of the problem, mathema- tical decomposition is a natural way to handle it. A classical decomposition scheme is the Benders method. which is reviewed in the following Section. In this method, the problem variables are divided into two sets. Assuming an initial known point, one of these variable subsets is fixed and the other is calculated. With this solution, the original trial point is improved sumsively until convergence is reached.

3 Review of the problem solution by using Bend- ers decomposition

The use of Benders decomposition to solve the SCDR problem allows handling of subproblems of moderate sue specifically with dimensions such as those of the optimal power flow problem.

Two variable sets are identified X.

xi , f o r i = 1, ...; NC By relaxing the constraints associated with the second set, we can determine an optimal value for x,. Note that in the general formulation of the Benders method, the second variable set also has associated cost. Here, it has zero cost associated with objective function, because the variables may only satisfy feasibility conditions. With the values determined for xo, (1H4) for i = 1, relaxing all the other NC- 1 constraints, are solved. Solutions for i = 2 to NC are obtained in the same way. As the post-contingency variables have zero associated cost, the objective is to minimise the infeasibility of the post-contingency scenario. Then, (3) and (4) can be rewritten using decomposition, as follows [4,5]:

Min f =c'x, ( 5 )

A A 2 bo (6)

wi(xo) 5 0 i= I , . . . ,NC (7)

iv j (xo) = Min d'r + dss (8)

A x x + r ? b (9)

lxo - X I S 5 A (10)

where r and s are penalty variables associated with (9) and (IO), respectively.

The problem described by (5H7) is named to the master problem. Constraints (7) represent the information supplied by subproblems (SHlO) to improve the global problem solution.

On the other hand, (SHIO) are named operating subproblems and have the same stmcture for all the NC post-contingency scenarios. The objective wi (.yo) involves the minimisation of the penalty variables r and s with associated costs d' and d", respectively. These variables can be interpreted as the amount of the constraint violation associated with the post-contingency operating point. Note that the mathematical formulation of (9) and (10) is always feasible. If wi (xo) returns a value greater than zero, the post- contingency scenario is infeasible. If >vi (+,) returns zero, this means that the rescheduling capacities have been sufficient to eliminate the post-contingency violations.

When the operating subproblem is infeasible, a linear constraint iv(x0) is built. This constraint, called Benders cut. is formed by the Lagrange multiplier at the optimal solution of (SHIO). These multipliers represent the change of the subproblem infeasibility caused by a marginal change of point x,> [SI. This constraint is added to the master problem in order to improve the current operating point x,.

3. I Benders algorithm Let S he the set of constraints or Benders cuts. At the beginning, this set is empty; this means that the problem to he solved is only given by (5) and (6) @roblem with contingency constraints relaxed). As the process progresses, S receives the indices of the Benders cuts supplied by the subproblems. The process is illustrated in Fig. 2. -

] s=o 1

I for xo, solve the subproblem @)(lo). for each contingency

I y feasible ::re$;;:;derS include index in S 1 end

Fig. 2 Benders process.

3.2 Solving the operating subproblem The Benders decomposition scheme is a technique for building constraints w(.xo) with a required precision, based on the interactive solution of a base-case and NC separate operating subproblems. It is important that the subproblem

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is feasible, even though the real one was not. The problem feasibility is guaranteed by adding slack variables to the problem. Depending on how they are included, they have a different meaning.

3.2.1 Slack variables associated only with the couplement constraints: By using variable slack associated with couplement constraints (XHlO) can be rewritten as follows:

w(x,, j = Min d's

jxo - X I - s 5 A

(11)

A x > b (12)

(13) In this formulation, constraints (12) are forced to be viable while the infeasibility is projected on the control variables X. In other words, it is assumed that post-contingency scenarios are feasible if there is sufficient corrective capacity; however this is not always true.

3.2.2 Slack variables associated only with the operational constraints: For this case, the formulation is stated as follows:

w(xoj = Mind'r (14)

A x + r > b (15)

1x0 - 4 s A (16) In this formulation, x remains feasible and subproblem infeasibility (when it exists) is projected on (IS).

In this work this last formulation is considered. In order to guarantee subproblem feasibility, fictitious generators have been associated with all the buses, as presented in the following Sections.

4 Relaxed versions

The Benders decomposition method involves the iterative generation of constraints to be included in the master problem, reducing the feasible space of the global problem. Approaches to Benders cuts can be obtained through relaxed versions of the Benders method. Although the relaxed versions provide low quality cuts, they can he calculated with less computational effort.

4.1 Approach 1: considering only the worSt violation This approach is obtained by considering only the worst post-contingency violation, ignoring the rest. The relaxed subproblem can be stated as follows:

w(x0) = Mind'r (17)

Axj 2 b,i# j (18)

I*, - X I 5 A (20)

Axi,+ r 2 bi for the worst ith violation (19)

where (19) is the constraint representing the worst violation branch. If, after solving (17H20), r retums a positive value, then the post-contingency scenario is infeasible. Then (19) is introduced into the master problem, with the limit altered from b; to b';, where bti includes the necessary correction to eliminate the post-contingency violation.

This formulation is a relaxation of the Benders method. The relaxed subproblem (17H20) is easier to solve than the classical Benders method. When only one post-contingency violation is detected, (19) is fully equivalent to the classical Benders cut.

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location of cases from subdecomposition is performed in a careful and organised way.

5 Implementation of approach 1

This work concentrates on the implementation of the first approach presented in the preceding Section. The idea behind this approach is that, in unstressed systems, and specifically in the case of active power, many violations are interdependent. This means that removing the biggest violation often leads to elimination or mitigation of other violations. Then it is reasonable to expect that. by removing the largest violation, a large part of the problem infeasibility will be eliminated. This proposal is based on the efficient Stott's dual algorithm [6], extended to include contingency constraints. The largest violation case is represented by a contingency constraint with an associated slack variablc r, following (1 7H20). The dual algorithm will try to remove this violation by using the available rcschednling options. that is by using active power generation rescheduling. This is represented by new limits for the control variables (Fig. 3). If the violation is not completely removed (r>O), a contingency constraint will be included in the master program. This constraint is given by:

2 hj (29)

t '

p r " x-A xo x i A

Fig. 3

where

Asrocimerl m ~ t s of t/zc szrhprohletn generators

h: = hi + A'A (30) Vector A comprises the distribution factors (see Appendix) and d is the vector of allowed rescheduling margin associated with the generator vextor .x, at the ith scenario. The constraint limit has been modified from hi to bIj, reflecting the contribution of the rescheduling available margin to the reduction of the ith subproblem infeasibility. The master program will include this constraint in its base and a new base-case operating point is calculated.

5.1 Formulation by using fictitious generators Fictitious generators are associated with each bus and are responsible for supplying, at higher cost, the additional generation, or load shedding, required to remove the post- contingency violation. The fictitious generator guarantees that the subproblem is always feasible. If the post- contingency violations are eliminated without fictitious gneration, then IV(~Y<,) = 0. Each time, the total generation at a bus i is given by the sum x;+f;, where xi is the real generation andf; the fictitious generation.

For load buses, si will be zero and/; is interpreted as the load shedding at bus i required to make the problem feasible.

5.2 Optimisation algorithm The optimisation process is based on the dual algorithm proposed by Scott [6] and extended here to include security constraint and post-contingency rescheduling.

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In Scott's algorithm the fictitious generators are handled in the same way as the real generators (bounded variables). At the beginning, all the generations are null. The associated cost of x at PV buses is zero, and forJ'is assumed to be I. In order to prioritise the use of PV generation, adequate high cost values are assumed at PQ buses. Once these resources are finished, fictitious generators associated with PQ buses are activated, representing the required load shedding for feasibility.

Note that in Stott's algorithm, the subproblem can be solved without creating explicitly fictitious generators associated with the PV buses. They are represented by using multi-segment logic. Each generator has a cost curve given by Fig. 3 , where the segment with no null cost represents the fictitious generation.

The potential of the proposed method with respect to the complete Benders method lies in the fact that it explores each contingency characteristic, making the job effortless when the case is easier to reschedule. That is, the proposed relaxed approach avoids treating all contingencies equally.

6 Test results

Test results obtained from the implementation of the proposed algorithm are presented. The parallel machine used in this work was 64node hypercube computer. Tests have been performed using two Brazilian electric networks with 725 and 1663 nodes. respectively. The system features are summarised in Table 1. The target task is to lower the minimum quadratic shift of a specified active power operating point. The rescheduling margins d is given as the generator capacity percentage. Due to space limitation, results of a serial version with the 725-bus system have omitted.

Table 1: Brazilian systems used in tests

System Branches Controllable Contigency aenerators list

725-bus 1212 76 900

%€&bus 2349 99 1555

In Table 2 are presented results of the algorithm executed in a single processor (serial version) as a function of the allowed rescheduling. The Table begins with d = 0, which corresponds to the classical SCD (rescheduling not allowed) and consequently to the case of the highest objective function value. The processing time has been normalised based on this case. Columns 2 and 3 ( M and S-P) show the number of times the master problem and subproblems are invoked during the process. respectively. Column 4 (N'J gives the number of constraints that remain active at the optimal solution. Column 5 (N,) gives the total number of performed contingency analyses. Column 6 and 7 show the objective function and processing time normalised to the case d = 0, respectively. In order to obtain realistic time processing for d = 0, rescheduling routines are skipped.

Note that a high processing time is observed when a small rescheduling margin is allowed. In fact, there is little impact on violation reduction, leaving a large number of scenarios with violation to be solved. As soon as the rescheduling margin increases, the feasible cases are detected more quickly, saving processing time.

Tables 3 and 4 show the performance of the parallel algorithm as a function of the number of processors

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Table 2: Serial version: performance of the algoiihm using the 1663-bus system as a function of the allowed rescheduling; total CPU time normalised by case A = 0

1663-bus system A . % M S-P Nha N, t pu Time,

DU

0.0 9 0 7 1588 1.000 1.00 1.0 12 32 6 1625 0.915 1.40 2.0 8 40 4 1620 0.802 1.42 4.0 4 20 2 1568 0.662 1.21 6.0 4 27 2 1586 0.542 1.20 8.0 3 27 1 1586 0.440 1.17

10.0 3 28 1 1586 0.365 1.17 12.0 3 28 1 1586 0.289 1.17

15.0 3 28 1 1586 0.192 1.17 20.0 3 28 1 1586 0.073 1.17

25.0 3 28 1 1586 0.008 1.17 30.0 1 14 0 1555 0.000 1.02

Table3: CPU time processing and speed-up for the Brazilian 725-bus system using an nCUBE computer, allowed rescheduling d = 8%

Speed-up725-bus system Nodes CPU time, s Speed-up

1 130.24

2 70.83 4 37.93 8 23.86

16 14.22

32 13.73 64 12.07

1 .oo 1.84 3.43

5.46 9.15

9.58 10.80

(nodes), for A = 8.0% and A = 1.0%, respectively. The asynchronous paradigm used in similar to those proposed in [ I , 71, extended by the inclusion of post-contingency rescheduling. The machines used was an nCUBE-2 computer with 64 processors. The CPU time reported here is given as a reference, because now the CPU time can he drastically reduced using recent and more efficient parallel computers.

6.1 Comments In the tests illustrated in Table2, for simplicity, a contingency list was selected, where eventual violations can he eliminated through rescheduling or by means of preventive actions. Note that even when cases are included in which the postcontingency emergency remains steady, our approach will provide a solution by load-shedding, represented by the fictitious generators’ output level.

Tables3 and 4 show the speed-up for both Brazilian system used in the tests. These results show the beneficial impact of parallel processing on the CPU time. For instance, for the 1663-bus system, the CPU time has been reduced from 4.88minutes (serial case) to 23 seconds with 64 processors.

The load balancing is improved by the decomposition of a post-contingency problem in several parallelisable sub-

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Table4 CPU time processing and speed-up for the Brazilian 1663-bus system using an nCUBE computer; allowed rescheduling A = 1.0%

Speed-up-1663-bus system Nodes CPU time, s Speed-up

1 292.69 1.00 2 156.33 1.87

4 122.53 2.40 8 47.84 6.12

16 38.19 7.70

32 25.64 11.41 64 23.00 12.70

problems. As more parallel tasks with similar computa- tional grain size are obtained, better load balancing is achieved.

Additional grain size uniformity can be obtained from approach 2. Each post-contingency scenario is parcelled into subscenarios and allocated to a subset of processors. However, in this approach the number of subscenarios varies from case to case. Thus, in order to achieve real gain, dynamic task allocation to available processors must be adopted, demanding careful programming.

7 Conclusions

Two approximate decompositions based on the linearised loadflow model for solving the security-constrained dispatch with post-contingency rescheduling problem in a multi- processing environment have been presented. These approaches exploit practical characteristics commonly observed in power systems, allowing the derivation of relaxed versions of Benders cuts. In computational terms, these algorithms can potentially improve the load balancing in parallel processing environments, once they lead to a more uniform decomposition. One of these approaches has been successfully implemented and tested on a hypercube computer. Validation tests have been performed using two real-life Brazilian systems.

8 Acknowledgments

This project was supported by CNPq, Conselho Nacional de Desenvolvimento CientXco e Tecnolhgico, and BNB Banco do Nordeste do Brasil, Brazil.

9 References

POW kyir., 1994,9, (4jpp 2021-2027 TEIXEIRA, M.J., PINTO, H.J.C.P., PEREIRA, M.V.F.. and MC COY, M.F.: ’Developing concurrent Processing Applications to Power system^ Planning and Operation’, IEEE T r m . Power Sysl., 1990, 5, pp. 659464 PINTO, H.J.C.P., PEREIRA. M.V.F., and TEIXEIRA, M.J.: ‘New Parallel Algorithms for the Security-Constrained Dispatch with Post- Contingency Corrective actions’. Proc. 10th Pawer systems computa- tion Conf, (PSCC) Cod. G r z , Austia, August 1990 MONTICELLI,,A., PEREIRA. M. V. F., and GRANVILLE. S.: Security-Constrained Optimal Power Flow with Post-Contingency Corrective Rescheduling’, IEEE Trum. P o w r Sxt . , 1987. 2, (1) pp 17S-182

5 PEREIRA. M.V.F., MONTICELLI, A., and PINTO. L.M.V.G.: ‘Security-Constrained Dispatch with Corrective Rescheduling’. Proc. IFAC Symp. on Planning and operation of electric energy systems. Rio de Janeiro, July 1985, pp 387-394

2Y5

6 STOTT, B., and MARINHO, J.L.; 'Linear Programming for Power System Network Security AppPcatms', IEEE Tram Power Appor. Syrl., 1979, 98, pp. 837448 SAAVEDRA, O.R.: 'Solving the w r i r y constrained optimal power Row problem in a distributed corilputing environment'. JEE hoc., Gene?. T r a m . Disrrib.. 1996, 143, (6). pp. 593-598

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10 Appendix: Incremental linearised formulation

Formulated as an incremental linear problem, the power dispatch can be stated as:

Min f = c'AP

s.t. P'AP = 0

A& 5 ALdP 5 A F h

AFh 5 A g A p 5 Fk A p i < A I A P < A P , i = l , . . . :NG

( k , m) E Nh

(k, m) E N ,

Where AP is the base case incremental power injection vector; F& and Fb, are the lower and upper limit for the power flow at branch km. respectively; &,, and Fh are the lower and upper contingency power flow limit at branch km respectively; Ax, and Ai are distribution factor vectors; Nh is the system branches set; N,. is the contingency set ; B is the vector of incremental transmission factors, which can be obtained in [6]. Only a small subset of constraints is dealt with at any stage. Consequently, the number of binding constraints in the optimal solution is not excessive.

Contingency construint The post-contingency power flow as a function of the

case base incremental active power injection is given by:

A f & = A k A P

where Abn is computed as:

A h = -bbn[B'l-'(eh - a h e i j ) ]

The distribution factors can be calculated easily from triangular factors of [B'I-'. as follows [6]:

a h = Y P with

y = eL[B']-'e,

and

P = I / ( ( l / A b y ) + ( l / b y ) )

where Ab, is the branch susceptance of the outage branch and b y is the equivalent susceptance among nodes i and j , that can be quickly calculated as follows:

beq = l/z? 'I

where

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