Recent advances in stochastic operations research II
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The ain idea of this approach is to establish a benchmark for a comparison with the other methods, which make use of decomposition strategies: the ND and the PH. Both decomposition algorithms break the MTOP into smaller subproblems and therefore greatly reduce memory requirements. To obtain reliable results, a realistic hydrothermal configuration extracted from the Brazilian hydrothermal power system was used. Additionally, sensitivity analyses 4 were carried out considering different piecewise linear models that describe the hydro plant production function. The remainder of the paper is organized as follows.
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In Section 2, a brief description of the multistage stochastic optimization problem features is shown. The list of symbols used in this paper is presented in Section 3. The three problems, related to each solution strategy, are detailed in Section 3.
The test problem and the results on solving this problem are shown in Section 4. Finally, conclusions are presented in Section 5. The MTOP is a multistage stochastic problem  and, therefore, a difficult problem to be solved.
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In general, computational methods for multistage stochastic programming problems can be divided into two main groups . The first group takes advantage of the special features of stochastic problems to improve data structures and solution strategies . The second group uses special decomposition methods which exploit the problem structure to split it into manageable pieces and coordinate their solution .
As presented in the previous section, in this work three methods are used to solve the MTOP Brazilian problem: i the implicit DE: a large-scale LP is given to a commercial package to solve which belongs to the first group of the computational methods ; ii the ND: a Benders decomposition method; iii the PH: an AL-based method.
In a multistage stochastic optimization problem, decisions are to be made in stages, and the uncertainties can be modeled by means of a scenario tree that can be generated by sampling techniques . Figure 1 gives an example of a scenario tree for a three stage problem, in which the water inflow to the hydro plants is uncertain.
In this structure, each node filled circle represents a specific random realization for the water inflow system state. A branch represents the relationship between two water inflow realizations state transition. Thus, a scenario consists of a complete path from the node at stage one to a node at stage three. As a consequence, this tree has nine possible scenarios and 13 nodes. Now, consider that a decision must be taken for some node at stage t in Figure 1. This decision should take into account two available information on the scenario tree: the full path up to stage t from the past and the possible realizations for the following nodes to the future.
Since it is not possible to anticipate which scenario will happen, this decision must be unique for all scenarios passing by this node. This condition is called nonanticipativity . For instance, the decisions of nodes n 1 and n 2 must be identical for the scenarios s 1 , s 2 and s 3. The Figure 2 shows a different representation for the same scenario tree as in Figure 1.
At this representation, the scenarios are represented by the full lines while the dotted lines link the decisions for different scenarios, representing the nonanticipativity concept. The problem can be modeled in different ways according to the chosen solution method. Therefore, the ND decomposes the problem into stages of decision, in which subproblems correspond to nodes and are linked through the temporary joining constraints, so that the mathematical model corresponds to that shown in Figure 1.
Given that we intend to use the PH method, introducing scenario decomposition, the mathematical model must use the nonanticipativity condition to link the subproblems, like in Figure 2. The mathematical modeling of the problem for each solution method will be detailed on the next sections. In this section, the problem formulation model and some theoretical properties of the methods studied in this paper are presented. Aiming to make easier reading of the remaining of this paper, the notations are shown bellow. The notations describe fixed parameters of the thermal and hydro plants, indices and variables.
E total number of subsystems;. S total number of scenarios;. I total number of thermal plants;. R total number of hydro plants;. M r set of upstream reservoirs from reservoir r ;. N number of linear constraints used in piecewise linear function of the hydro plant;. J number of linear constraints used in the piecewise future cost function;.
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X s t set of all scenarios related to scenario s at stage t by the nonanticipativity, including itself;. Int le power interchange from subsystem l to subsystem e [MWh];. L e energy demand at subsystem e [MWh];. In this strategy, the stochastic linear problem is solved by means of a standard Linear Program solver. Given that the size of the problem increases sharply as more stages and scenarios are considered, it is important to exploit the particular matrix structured and its sparsity as much as possible.
In this paper, commercial software, CPLEX, was used , which uses an advanced optimization algorithm to improve the performance. Indeed, according to the literature , multistage stochastic optimization problems, in general, need to be solved by using decomposition techniques, even considering the recent advances in the computing technology.
Therefore, given that we are interested in developing efficient decomposition algorithms to solve MTOP problem in an acceptable time, we make use of the implicit DE approach applied to a small problem 5 in order to obtain a benchmark solution to other methods.
The problem formulation is presented as follows. The objective function aims to minimize the system's operation cost. Then, it can be written as:. Notice that the hydrothermal system is composed of subsystems that are interconnected. In this context, power plants are located in different subsystems defined by the indexes I e and R e , respectively. Additionally, the demand was considered constant through all stages.
Here, it is important to discuss the successors and the ancestor nodes in the scenario tree. So, observe the Figure 3. This idea can be expanded for a bigger scenario tree Fig. The hydro production function depends on the discharged outflow, the volume of water in the reservoirs and the spillage. Furthermore, this function may be neither convex nor concave, which requires some a careful approach in the linearization process, similar to the presented in , with the purpose to guarantee the construction of a convex function.
Therefore, the following steps, based on , are required:. For all plants, choose N , which is dependent on the precision required. Else return to step 3. As a result of this approach, the hyperplanes 4 form a wrap function tangent to the hydro plant production, maintaining, in this way, the problem's convexity.
This function is given by the longer-term planning model, such as  and estimates the expected future cost. It is a piecewise linear function depending on the volume of water in the reservoir at the end of the planning horizon, T. It is worth to notice that these constraints 5 are only included in the subproblems associated with the last stage. The variable that describes the power interchange can assume a negative value if the interchange occurs from subsystem e to subsystem l. The ND is a decomposition method based on the Bender's decomposition principle.
This method solves the first stage problem and deals with the remaining stages as other subproblems, solving them recursively. In other words, this solves Problem 11 in a recursive manner. In summary, ND is an iterative process divided into two steps: forward , in which the operative cost for each stage is calculated and passed forward to later stages as input to the right hand side; and backward , in which the approximations costs are fulfilled and passed back from later stages in the form of optimality cuts, also called a Bender's cut, to the ancestor problem.
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In this way, by the application of the single-cut version , the Problem 11 is decomposed into subproblems by nodes, which represent a possible water inflow in each stage of the planning horizon. The objective function for the subproblems has the expression:. The subproblems's constraints are similar to those presented in the previous section; however, in the Eq. In this decomposition scheme, the current stage decision depends on the previous stage decision and, therefore, denotes the ancestor node solution.
As aforementioned, the cuts created are passed back to the ancestor node subproblem to represent an outer linearization of the recourse function. The expected value of the Lagrange multipliers dual prices is used to form the optimality cuts:. Thus, as the subproblems size is increased iteration after iteration and there are finitely many optimality cuts, the ND method is finitely convergent. Therefore, the initial iterations of this method are very inefficient. The fundamental difference between ND and PH is the way that the two algorithms address the nonanticipativity constraints.
As shown, ND handles these constraints by having a master problem generating proposals to the subproblems further down the tree scenario; proposal are affected by "futures" nodes by optimality cuts. In PH a different approach is taken: nonanticipativity constraints are relaxed by expressing the large-scale problem in terms of smaller subproblems that are discouraged from violating the original constraints.
Each scenario subproblem is a deterministic problem and has a separate set of variables. These subproblems are coupled by the nonanticipativity constraints which, as discussed above, stipulate that nodes sharing the same stochastic history up to and including that stage must make the same sequence of decisions.
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More precisely, the PH models the nonanticipativity by the average value of these scenarios decisions, such as shown in Observe that in this work the nonanticipativity constraints are represented by the water stored in the reservoirs. This is possible because the storage volumes are the decision variables, which couple constraints in different time steps.
Thus, once the water volume is known, the values of all other decision variables are easily obtained. Notice that in the AL function there exists an additional parameter, , for each iteration, which is used to uncouple these subproblems for all scenarios . Volume 11 , Issue 1 January Pages Related Information. Close Figure Viewer.
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