MULTI-STAGE TRANSMISSION EXPANSION PLANNING CONSIDERING MULTIPLE DISPATCHES AND CONTINGENCY CRITERION

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1 MULTI-STAGE TRANSMISSION EXPANSION PLANNING CONSIDERING MULTIPLE DISPATCHES AND CONTINGENCY CRITERION GERSON C. OLIVEIRA, SILVIO BINATO, MARIO V. PEREIRA, LUIZ M. THOMÉ PSR CONSULTORIA LTDA R. VOLUNTARIOS DA PÁTRIA 45/1402, RIO DE JANEIRO, RJ S: Abstract This paper presents an optimization procedure for transmission expansion planning that takes into account multiple dispatch scenarios, contingency constraints and multiple stages. The resulting model is a very large nonlinear integer programming problem, which is solved by the combination of the following techniques: (i) a disjunctive formulation transforms the non linearities into linear integer constraints; (ii) an heuristic ranking procedure is used to select the most critical combinations of dispatch and contingency scenarios, to be incorporated into the global optimization model; afterwards, a greedy optimization scheme produces the required reinforcements for the remaining scenarios; (iii) a horizon year /backward optimization approach is used to decompose the multi-stage problem into a sequence of one-stage problems. The application of these techniques is illustrated for two real life planning studies for the systems of El Salvador and Venezuela. Keywords transmission planning, integer programming. Notation n number of buses (nodes) m number of circuits (branches) K number of circuit contingencies Ω i set of circuits connected to bus i f 0 vector of circuit flows (base case) f_ k circuit flows (k-th contingency) _ f vector of normal circuit capacities f c vector of emergency circuit capacities g vector of bus generations d vector of active bus loads θ 0 vector of bus voltage angles (base case) θ k voltage angles (k-th contingency) x decision vector for building/not circuits c vector of circuit construction costs S 0 node-branch incidence matrix (base case) S k incidence matrix (k-th contingency); [γ 0 ] circuit susceptance matrix (base case) [γ k ] circuit susceptances (k-th contingency); M penalty matrix ( big M) 1 Introduction As the 2003 blackout in the East Coast of the United States and the 2002 country-wide blackout in Brazil illustrate, the transmission network is a critical component for the adequate functioning of electricity sectors all over the world. With the reestructuring of those sectors in many countries, which includes the separation of transmission and generation, the planning activity has become even more complex, because the networks must now be able to accommodate a wider range of generation dispatches (due, for example, to power exchanges among regions or countries) and are operated much closer to the limits (which makes security constraints very important). This paper presents an optimization procedure for transmission expansion planning that takes into account multiple dispatch scenarios, contingency constraints and multiple stages. The resulting model is a very large nonlinear integer programming problem, which is solved by the combination of the following techniques: (i) a disjunctive formulation transforms the non linearities into linear integer constraints; (ii) an heuristic ranking procedure is used to select the most critical combinations of dispatch and contingency scenarios, to be incorporated into the global optimization model; afterwards, a greedy optimization scheme produces the required reinforcements for the remaining scenarios: (iii) a horizon year /backward planning approach is used to decompose the multi-stage problem into a sequence of one-stage problems. The application of these techniques is illustrated for two real life pla n- ning studies for the systems of El Salvador and Venezuela. The paper is organized as follows: Sections 2 through 4 present successive formulations of the planning problem, each incorporating additional levels of detail: (i) base case problem, with one stage; one dispatch scenario; and no contingency constraints; (ii) incorporation of contingency constraints and multiple dispatch scenarios; and (iii) incorporation of multiple stages. For each formulation, we describe the adopted solution technique. Finally, Sections 5 and 6 present the case studies and the conclusions. 2 Base case formulation 2.1 Network model Planning studies of high voltage meshed networks usually adopt the so-called linearized power flow model, which represents Kirchhoff s first and second laws as linear equations relating bus volt-

2 age angles, generations and loads to circuit flows. These laws are written in matrix notation as follows: Sf = d - g (1) f = [γ]s θ (2) Equation (1) represents Kirchhoff s first law, which is the power flow balance in each node (sum of circuit flows f entering/leaving the node equal to the difference d - g between generation and load). The k-th column of the n m incidence matrix S is zero in all positions except for bus terminals of circuit k, which have values 1 and 1 respectively for the circuit s origin and terminal nodes. Equation (2) represents Kirchhoff s second law: the flow in each circuit is proportional to the difference of the voltage angles θ at its terminal buses. The proportionality factor is the circuit susceptance, represented by the diagonal matrix [γ]. In addition, circuit flows are bounded by their capacities, expressed as follows: 2.2 Planning problem -f _ f f _ (3) The base case transmission expansion problem (one stage, one dispatch scenario, no contingency constraints) is formulated as the following nonlinear mixed integer mathematical pr ogramming problem (Pereira et al, 1985) Min {x,f,θ} c x subject to (4) Sf = d g (4a) f - [γx]s θ = 0 (4b) -f _ x f _ f x (4c) x {0,1} m where x represents the binary (build/no build) investment decisions and c is the investment cost. For simplicity, we suppose that all circuits are candidates for expansion (the x variables for existing circuits are made = 1). The notation [γx] indicates pair-wise products between _ diagonal components and decision variables; f x is also a pair-wise product on the vector elements. 2.3 Solution approach The main difficulty in (4) is the product of the binary decision variable x and the voltage angle variables θ in constraint (4b). We solve this problem using a mixed integer disjunctive formulation (Granville et al, 1988), where the nonlinear constraints (4b) are replaced by the following disjunctive form: -M(1-x) f - [γ]s θ M(1-x) (5) Constraint (5) is interpreted as follows: when the circuit is not constructed (x = 0), the flow limit constraint (4c) makes f equal to zero; and the disjunctive constraint (5) is relaxed (no Kirchhoff s second law). Conversely, when the circuit is constructed (x = 1), constraint (4c) enforces the flow limits and the disjunctive constraint (5) enforces Kirchhoff s second law. The disjunctive inequalities can be strengthened on both sides (Bahiense et al, 2001). The big M penalty coefficients in (5) are calculated to minimize ill conditioning (Binato et al, 2001). The resulting mixed linear integer programming (MILP) problem is solved by a commercial Branch-and-Bound (B&B) code. 3 Security criterion and dispatches 3.1 Security criterion In actual transmission planning studies, supply should be ensured not only under base-case conditions, but also in the case of failure of any circuit. This security criterion is modeled by repeating all constraints for each circuit contingency, indexed by k=1,, K (note that the incidence and suspectance matrices for the k-th contingency are different from the base case matrices, because the circuit has been removed). Min {x,f,θ} c x subject to (6) S 0 f 0 = d g -M(1-x) f 0 [γ 0 ]S 0 θ 0 M(1-x) -f _ x f 0 _ f x S k f k = d g -M(1-x) f k [γ k ]S k θ k M(1-x) -f _ k x f k _ f k x x {0,1} m, k=1,...,k The dimension of (6) grows linearly with the number of contingencies K (continuous variables f and θ, as well as network constraints). Note, however, that the binary variables x are the same, independently of the number of circuit contingencies. 3.2 Multiple dispatch scenarios As mentioned in the Introduction, the transmission network should be able to accommodate multiple dispatches, where each dispatch involves distinct load level and inflow scenarios (and therefore bus generations and loads). This is handled by creating, for each dispatch scenario s = 1,..., S, the corresponding node voltage angle and circuit flow variables and repeating the network constraints. As in the multiple-contingency formulation (6), the number of continuous variables and network constraints grows linearly with the number of

3 scenarios S, but the binary investment variables x do not change. Figure 1 illustrates the combined multiple-contingency and multiple-dispatch formulation. Investment decision x Minimize cost Dispatch scenario #1 case #1 for a given stage. In this section, we address the timing aspect, that is, when to construct each of the reinforcements along the study period. Differently from the previous section, where the representation of multiple contingencies and dispatch scenarios increased the number of continuous variables, but did not affect the binary variables, the extension to multiple stages increases the number of the continuous and the binary variables, as well as the number of network constraints. As a consequence, the planning problem rapidly becomes intractable by integer programming techniques. Dispatch scenario #S case # K case #1 case # K Figure 1 Multiple Contingencies and Multiple Dispatches 3.3 Solution approach Because the problem size increases with the product K S, we adopted the following relaxation scheme: (i) select a subset of critical dispatch scenarios and contingencies to be incorporated to the optimization problem (we used a ranking scheme based on circuit loading); (ii) solve the relaxed planning problem with only the selected scenarios/contingencies; (iii) verify whether the optimal solution of the relaxed problem in step (ii) is feasible for the remaining (not selected) dispatch and contingency scenarios. If yes, the planning problem is solved; otherwise, go to step (iv) (iv) re-solve the planning problem with the violated constraints, but already incorporating the reinforcements of the previous solution ( incremental planning); return to step (iii). 4 Multiple stages 4.1 Problem formulation In the previous sections, we represented the siting (where) and sizing (how much) decisions 4.2 Solution approach If there are no economies of scale (for example, if all candidate reinforcements have the same voltage level), an effective simplified approach is to solve the expansion one stage at a time ( forward planning ). However, this approach is less effective if there are potential economies of scale, for example, it may be better to construct a 500-kV circuit today, instead of a 230-kV, because this circuit will be more intensively used in the future. In this case, we apply the backward heuristic solution procedure proposed by (Binato and Oliveira, 1995), which takes into account economies of scale by polarizing the decisions towards the last-year expansion: (i) solve the transmission planning problem for the last year of the study period (the so-called horizon year ), this is the target solution; (ii) given the target solution of step (i), go backwards in time, solving the transmission problem in each stage, initially only considering as candidates the reinforcements previously made for the horizon year (in other words, it is only allowed to anticipate the investment decisions made for the horizon year); (iii) if no feasible solution results in any intermediate stage during step (ii), this step is aborted and we solve the problem for this stage allowing all candidates, and a forward planning is then applied from this stage until the horizon year so as to provide a new target solution, and step (ii) is repeated; (iv) compare the cost of the solution obtained with backward procedure and the one provided by the forward procedure, choose the cheaper. 4.3 Solving the horizon year One aspect remains to be discussed, which is the solution of the horizon year sub-problem (step (i) of the above procedure). In actual planning studies, usually a few circuits are constructed in each year. In the horizon year case, however, we are jumping ahead and solving the final problem without the intermediate-year reinforcements.

4 Because the optimal solution now requires several circuits, the computational effort may be substantial. We alleviate this problem by using an upper bound for the investment cost, which in turn prunes the B&B search tree. This upper bound is produced either by a GRASP-based heuristic search (Binato and Oliveira, 2001), or by applying the forward planning scheme mentioned above. (note that the potential drawback of not represen t- ing economies of scale is not a problem in this case, because we are only using the forward planning to provide an upper bound). 5 Case studies In this section, we illustrate the application of the previously described methodologies, incorporated to the OPTNET planning software [PSR, 2001], to planning studies in Venezuela and El Salvador. 5.1 Venezuela The reduced transmission network of Venezuela has 134 buses, 271 circuits and five voltage levels, ranging from 765 kv to 69 kv. The northeast region is to be planned from 2004 until 2008, with the following planning criteria: (i) one dispatch scenario per year (yearly peak load and the most severe hydrological condition); (ii) 36 critical contingencies (single circuit outages). There are 125 candidate circuits. Because there are potential economies of scale, we adopted the backward planning scheme discussed in the previous section. In order to reduce the computational effort for the horizon year solution, we used an upper bound produced by a forward planning procedure (solve the planning problems year by year). Each one-stage MILP problem has on the order of 43 thousand rows, 31 thousand columns and 150 thousand non-zero elements, and is solved in about one minute on a 2 GHz Pentium IV computer with 512MB, running the XPRESS B&B solver (Dash, 1999). Six 69 kv reinforcements were identified by the forward planning procedure (three in 2004, and one for 2006, 2007 and 2008), with a total investment cost of US$ million. There is a cheaper target solution for 2008 costing US$1.504 million, which also uses six 69 kv candidates, taking 2 minutes on the same computer when the previous solution value is used as an upper bound. Only three of the reinforcements are common to both solutions. Next, the backward approach was applied to find the reinforcements for each of the intermediate years, where only reinforcements chosen for the target solution are allowed as candidates. For year 2007, due to the contingency criterion, no feasible solution was found when considering as candidates only the ones chosen in the target solution. By the end of this procedure an additional reinforcement was necessary, and therefore this expansion plan has a higher cost (US$1.619 million) than the one provided by the forward procedure. 5.2 El Salvador El Salvador s high-voltage network has 58 buses and 80 circuits (115 kv). In addition, there are 230 kv interconnections with Guatemala and Honduras. The study period was , with the following planning criteria: (i) no network violations for all monthly dispatch scenarios; for all 4 load levels; and for all 50 hydrological situations; (ii) no network violations for 34 selected single contingencies. The 200 monthly dispatch scenarios (combinations of 4 load levels and 50 hydrological conditions) were produced by the SDDP hydrothermal dispatch model (PSR, 1993). For each of the 200 dispatch scenarios, one base-case plus 34 contingency situations were simulated, for a total of 7,000 combinations. The relaxation scheme described in Section 3 was applied, by selecting a month (December) and ranking the dispatch scenarios by severity (network loading) with respect to inflow and load levels (note that the peak load levels are not necessarily the most severe scenarios, because substantial interchanges with the neighboring countries take place during the intermediate load levels). Because all 47 candidate circuits have the same voltage level (115 kv), the problem of economy of scale is less significant. As a consequence, the forward planning procedure, in which the optimal solution is successively found for each year, was applied. Only two circuit reinforcements were needed, both for year The solution time of the relaxed problem (second step of the relaxation scheme) for this year, considering the critical dispatch scenarios and all 34 contingencies, was 50 seconds (same computer used in the previous case). For the remaining scenarios, no reinforcements were necessary. 6 Conclusions and future work In this work, the consideration of the contingency criterion in static transmission planning was shown to be handled by a natural extension of the mixed integer disjunctive model. The combinatorial complexity of the model is unaltered (the number of discrete variables related to investment decisions remain the same if one considers contingencies or not), since only the number of con-

5 straints and the number of continuous variables grow linearly with the number of contingencies. In order to take into account multiple dispatches, the relaxation scheme combined with incremental planning was used so that the problem size would not grow out of hand. Ranking scenarios by decreasing severity (network loading) allowed an early identification of relevant rei n- forcements and therefore incurred in small overhead when processing the model for the remaining scenarios. Finally, the concept of horizon year planning was used to devise a target configuration without the need to tackle with the timing planning decisions jointly with choice of siting and sizing of candidates, while taking into account possible economies of scale. By means of the same incremental planning scheme, the timing decisions are sought by initially focusing on the reinforcements chosen for the target configuration, therefore allowing to accomplish all the multi-stage multiple dispatch planning requirements by sequentially using the static disjunctive multiple contingency transmission expansion model. The real world application of the proposed approach to plan the network expansion of Salvador, considering nearly half a million dispatches over a period of five years, illustrated the adequacy of the proposed approach. The Venezuelan utility example showed that even for larger networks the contingency constrained disjunctive model combined with the dynamic heuristic forward and backward procedures was successively used to provide fiveyear expansion plans within affordable time. For this example, the contingency criterion was against the potential benefit due to the effect of economic of scale dealt by the backward procedure, so for this case the solution provided by forward (greedy) procedure was better. This shortcoming will be the subject of future research. The authors also intend to strengthen the multiple contingency mixed integer disjunctive formulation by means of additional constraints based on relaxed network flow duality cuts devised for each contingency case, and therefore reduce the search tree of the B&B algorithm. Using modern commercial solvers, these cuts can be automatically generated, pooled and incorporated when needed to the LP relaxations solved for the nodes of the B&B search. Mathematical Decomposition Techniques for Expansion Planning Vol 2: Analysis of the Linearized Power Flow Model Using the Bender Decomposition Tecnique. EPRI Report EL S. Binato and G. Oliveira (1995). A Heuristic Approach to Cope with Multi-year Transmission Expansion Planning. Proceedings of the IEEE Stockholm Power Tech Conference. S. Binato, M. Pereira, S. Granville (2001). A new Benders decomposition approach to solve power transmission network design problems. IEEE Transactions on Power Systems, Vol 16, #2. L. Bahiense, G. Oliveira, M. Pereira, S. Granville (2001). A mixed integer disjunctive model for transmission network expansion. IEEE Transactions on Power Systems, Vol 16, #3. S. Binato, G. Oliveira, J. Araújo (2002). A GRASP for transmission expansion planning. IEEE Transactions on Power Systems, Vol 16, #2. PSR Consulting (2001). OPTNET Reference Guide, DASH Optimzation (1999). XPRESS-MP Reference Guide, PSR Consulting (1993). SDDP Reference Guide, References M. Pereira, L. Pinto, S. Cunha, G. Oliveira (1985). A decomposition approach to automated generation-transmission expansion planning. IEEE Transactions on PAS, Vol. 104, #11. S. Granville, M. Pereira, G. Dantzig, B. Ivi-Itzhak, M. Avriel, A. Monticelli, L. Pinto (1988).