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. 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 [26].

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The solution algorithm of the PH consists in an iterative process which solves the subproblems 17 and the dual problem Since the dual problem 18 is differentiable, it can be solved by applying of the gradient method:. This method has two parameter sets that are iteratively updated: the additional parameter and the Lagrange multipliers.

## Stochastic Linear Programming - Models, Theory, and Computation | Peter Kall | Springer

Besides, the gradient method only uses information from the last iteration. Owing to these features, the PH is also sensitive to the use of warm start [27]. This class of methods has some advantages when compared to other methods: the primal solution feasibility is guaranteed and it has a weak link between scenarios.

The weak link means that in case of computational parallelization, only little information needs to be shared by the processors. The only communication requirement is a share of scenario solutions to enforce nonanticipativity constraints.

## SIAM Journal on Applied Mathematics

Thus, this algorithm is suitable for paralleling processing. To observe the efficiency of the methods, a performance analysis of each method facing test systems with different complexities are shown. For that, a real hydrothermal system is modeled into two different LP problems, differing from each other by the precision of the hydro production model 6. In the first case, the hydro plant production is modeled by a piecewise linear function only depending on the discharge outflow, resulting in three linear functions. In the second case, the piecewise linear function is more accurate and it depends on the reservoir storage, discharged outflow and spillage.

In this paper, with the purpose of presenting a coherent analysis of the methods, the computational time was used as the stop criterion. Before showing the study cases, a brief description of the hydrothermal system is presented. The hydrothermal system is based on the Brazilian system and made up of 21 hydro and 20 thermal plants. The hydro plants are physically connected in cascades, as depicted in Figure 4.

This hydro configuration has approximately 47 GW installed. The thermal configuration is composed by 20 plants using different fuels. The total thermal installed capacity is equal to 9. The stochastic model has a study period made up of five weeks plus two months, as depicted in Figure 5 , similar to the Brazilian MTOP. The first three stages three weeks are modeled as deterministic and the remaining stages as stochastic, each one with seven possible realizations.

Tests were performed on an Intel Core2 Duo 2. In this case, the hydro plant production is represented by a piecewise linear function with three constraints.

In addition, this function depends only on the discharged outflow of the hydro plant. This is a simplified model that does not consider the impact of the water head on the production function. Additionally, the maximum numbers of iterations was fixed in five, once the cpu time requested is significantly higher when compared with ND. Thus, the comparative study is presented in the Table 1. It is important to say that in this case the DE problem has , variables and , constraints , are equality constraints and , are inequality constraints.

The deviations shown in refer to the solution deviation of each methodology objective function with respect to the DE solution, which is the exact solution. Notice that the DE method presents the best cpu performance when compared with other methods. Additionally, it can also be observed that ND has reached a better solution faster than PH. The aim of this case is to demonstrate that problem's size increases, the use of DE can be infeasible and, at the same time, to analyze the performance of the other methods against this benchmark. In contrast to Case I, the hydro plant production is modeled by a piecewise linear function depending on the reservoir storage, the discharged outflow and the spillage.

Moreover, this function has five linear constraints. In others words, five points are used to construct the piecewise linear approximation. For both decomposition methods, the computational time was considered as the stop criterion and both should be similar for propitiating a suitable comparison. Additionally, the PH uses the solution of the expected value problem [19] to set additional parameter 7 ; therefore this approach works as a warm start. The comparative analysis between each algorithm is shown in Table 2. The DE problem has , variables and , constraints, with , inequality constraints.

Each ND subproblem node subproblem in its first iteration has variables and constraints, which are inequality constraints, except in the last stage where it has variables and constraints due to the expected future cost function, summing up inequality constraints. It is worth pointing out that in the ND method, the number of constraints increases at each iteration. On the other hand, each quadratic programming subproblem resulting from the PH decomposition scenario subproblem has a fixed set of variables and constraints, which are inequality constraints.

First of all, notice that the production cost resulting from this case is approximately two times greater than the obtained in the previous case. This occurs because in Case II we used a more realistic model regarding the volume variation 8 and spillage 9. It is important to model this characteristic because the volume variation in some reservoirs is significant and the spillage is quite common in some plants. Also, according to Table 2 , observe that ND and PH have obtained satisfactory results in a small cpu time in comparison to the DE approach 10 , although the PH did not already find a feasible solution Figure 6 illustrates the convergence process of Case II.

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Observe that if both algorithms were limited to almost one hour of processing, the smallest deviation would be given by the PH. On the other hand, if the time limit was close to two hours and thirty minutes, the PH presents a deviation slightly bigger than the ND. Finally, if the stop criterion was extended to close to four hours, the ND presents the best solution. The Table 3 illustrates these comments. The ND subproblems are linear and refer to nodes while the PH subproblems are quadratic and refer to scenarios. Therefore, each iteration in the PH method requires more time than the ND method.

These results show that the precision of the mathematical modeling can interfere with the method's performance. In other words, the increase of the problem size which in this paper depends on the hydro plant's production accuracy affects the convergence for each method. Additionally, operation cost in Case II is higher than in Case I as expected , given that a more accurate modeling is used for the hydro plant production.

As a consequence, in Case II a larger computational effort is required. Also, although the ND and PH methods require more computational time than the Case I, now they require less time than the DE to reach lower deviations. This proves that as the size of the problem increases the DE requires more computational cpu time, making its application prohibitive for large systems We can also observe that ND is more sensitive than PH with respect to the problem's size; this may be an indicative that for bigger problems PH will work better than ND.

The objective of the hydrothermal systems MTOP is to calculate an operation strategy for the purpose of minimizing the operation cost over a time horizon. Owing to the uncertainties associated with the inflows scenarios, this problem is quite complex and addressed as a multistage stochastic programming problem. In this paper we compare three solution methods: a Deterministic Equivalent, b Nested Decomposition, and c Progressive Hedging.

## Linear programming

Although the Deterministic Equivalent can offer the exact solution, its application in a large-scale problem may be infeasible due to the huge memory requirements and high computational burden. Thus, alternative approaches like Nested Decomposition and Progressive Hedging methods constitute interesting paths to be investigated. Additionally, it is important to emphasize that the accurate representation of the hydro plant production inserts an additional complexity into this particular problem but, as benefit, it improves the quality solution.

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Certainly the results obtained in the Case I cannot implemented in real life. This fact can be observed when it is compared the computational time and the operation cost of the cases I and II. Studies presented demonstrated that the Equivalent Deterministic is not useful to solve huge problems and that the Progressive Hedging is competitive when compared to the Nested Decomposition.