An Improved Lagrangian Relaxation Algorithm for the Robust Generation Self-Scheduling Problem

The robust generation self-scheduling problem under electricity price uncertainty is usually solved by the commercial solver, which is limited in computation time and memory requirement. This paper proposes an improved Lagrangian relaxation algorithm for the robust generation self-scheduling problem where the quadratic fuel cost and the time-dependent exponential startup cost are considered. By using the optimal duality theory, the robust generation self-scheduling problem, which has a max-min structure, is reformulated as aminimizationmixed integer nonlinear programming (MINLP) problem.Upon the reformulation, the Lagrangian relaxation algorithm is developed. To obtain a solvable relaxed problem, the variable splitting technique is introduced before the relaxation. The obtained relaxed problem is decomposed into a linear programming-type subproblem and multiple single-unit subproblems. Each single-unit subproblem is solved optimally by a two-stage backward dynamic programming procedure. The special cases of the problem are discussed and a two-stage algorithm is proposed. The proposed algorithms are tested on test cases of different sizes and the numerical results show that the algorithms can find near-optimal solutions in a reasonable time.


Introduction
In the competitive electricity market, the generation selfscheduling problem plays a key role in the planning and operation of electric power systems.An effective generation selfscheduling can help the power generation company to decide the bidding strategies and make generation plans for the subsequent time horizon.The generation self-scheduling problem is to determine the operation of generation units, including the on/off statuses of generation units and the output level of committed generation units, according to electricity prices over the scheduling horizon.The objective is to maximize the total generation profit.Because the formulation of the generation self-scheduling problem includes quadratic fuel cost function, exponential startup cost function, and integer variables indicating the on/off statuses of generation units, the problem is a nonconvex MINLP problem.As there could be a large number of generation units and a long scheduling horizon, the generation self-scheduling problem can become large-scale and computationally challenging.This motivates us to study the solution method of the generation self-scheduling problem.
The decision-making of the generation self-scheduling is affected by electricity prices, which are volatile and full of uncertainty in the practical spot market.The uncertainty may be caused by the volatility of fuel prices and the changes in government subsidies, industry regulations, and local weather conditions.Therefore, electricity prices cannot be forecasted accurately and need to be treated as uncertain parameters.Two approaches are usually applied in dealing with uncertain parameters: stochastic programming and robust optimization.The stochastic programming approach relies on the probability distribution of uncertain parameters and typically expresses the possible realizations of uncertain parameters by a finite number of scenarios according to the probability distribution.Examples of using the approach can be seen in [1][2][3][4][5][6][7].The drawback of the approach is that it is 2 Mathematical Problems in Engineering difficult to obtain the exact distribution of uncertain parameters in the practice.The robust optimization approach is distribution-free.The approach represents uncertain parameters with a deterministic uncertainty set that contains all possible realizations or an adequate realization range of uncertain parameters.The obtained decision is robust against the variation of uncertain parameters within the uncertainty set.Examples of using the approach can be seen in [8][9][10][11][12][13].To make the decision more practical, in this paper, we use the robust optimization approach to model uncertain electricity prices in the generation self-scheduling problem.
The solution method of the generation self-scheduling problem has been widely investigated, with the majority concentrated on the deterministic generation self-scheduling problem and the stochastic generation self-scheduling problem.For the deterministic generation self-scheduling problem, the mixed integer linear programming approach was used in [14][15][16], a Lagrangian relaxation algorithm was proposed in [17], the Lagrangian relaxation method and the MIP method were compared in [18], a particle swarm optimization algorithm was proposed in [19], a genetic algorithm was proposed in [20], a new memetic algorithm was proposed in [21], an ant colony optimization algorithm was proposed in [22], an evolutionary algorithm was proposed in [23], and a survey of solution methods was presented in [24].For the stochastic generation self-scheduling problem where uncertain electricity prices are modeled by the stochastic programming approach, the mixed integer linear programming approach was used in [1][2][3][4], the typical Lagrangian relaxation algorithm was applied in [5], and a sample average approximation algorithm was proposed in [6].
For the robust generation self-scheduling problem where uncertain electricity prices are modeled by the robust optimization approach, the solution algorithm is not widely investigated.A critical review of the robust generation selfscheduling was presented in [25].To provide a better sketch for the research of the robust generation self-scheduling problem, a comparison of some relevant literature is presented in Table 1.From Table 1, we can observe that the robust generation self-scheduling problem was usually solved by commercial solvers.In order to make the problem solvable by the solvers, the model was simplified by piecewise linearly approximating the quadratic fuel cost function [8,10,11], reducing the time-dependent exponential startup cost function as a constant [8,10], omitting the startup cost [9,12], or omitting the unit commitment decision [12].The solution approach has two disadvantages.First, the use of commercial solvers is limited in computation time and memory requirement [18].When the robust generation self-scheduling problem is large-scale, it is not practical to solve the problem by commercial solvers.Second, the simplification of the model will reduce the effect of the solution.Therefore, an effective solution algorithm should be developed to solve the robust generation self-scheduling problem.
In this paper, we develop a Lagrangian relaxation algorithm for the robust generation self-scheduling problem.Compared with the existing robust generation self-scheduling literature, this paper provides the following contributions: (1) We consider the quadratic fuel cost and the exponential startup cost in the robust generation self-scheduling problem and develop an effective Lagrangian relaxation algorithm to solve the problem.The algorithm decomposes the problem into a linear programming-type subproblem and multiple single-unit subproblems.Each single-unit subproblem is solved by a two-stage backward dynamic programming procedure.The feasible solution is constructed by a heuristic algorithm.
(2) Unlike the typical Lagrangian relaxation algorithm that may lead to an unbounded relaxed problem for the considered problem, we introduce the variable splitting technique to improve the algorithm.
(3) A numerical comparison between the proposed algorithm and the MILP solver is reported.Numerical results demonstrate the effectiveness of the proposed algorithm.
The rest of the paper is organized as follows.The mathematical formulation of the robust generation self-scheduling problem is presented in Section 2. The solution methodology is proposed in Section 3. Numerical experiments are carried out in Section 4. The research is concluded in Section 5.

Problem Description and Formulation
We consider the robust generation self-scheduling problem under electricity price uncertainty from the viewpoint of a price-taking power producer.The objective is to maximize the total generation profit under the worst-case scenario within the considered range of electricity prices.The fuel cost is a quadratic function of the power generation level and the startup cost is an exponential function of the down time of the generation unit.The schedule needs to satisfy generation unit operation constraints and system electricity demand constraints.We take an hour as the scheduling period in the generation self-scheduling problem under consideration.
The remainder of the section is organized as follows.The notations used in the problem formulation are presented in Section 2.1.The modeling of uncertain electricity prices is described in Section 2.2.An MINLP model for the robust generation self-scheduling problem is formulated in Section 2.3.

Notations
Constraints (1a) restrict the electricity price for each hour between a lower bound and an upper bound, which can be set to the 2.5% and 97.5% quantiles of the electricity price forecast, respectively.Constraint (1b) presents a budget lower bound Γ for the sum of the electricity price   over the scheduling horizon.The budget value Γ is used to control the level of conservatism of the robust optimization approach.The smaller the value of Γ is, the more conservative the approach is.System operators can choose the budget value according to their requirements.
Within the above uncertainty set, there are infinite electricity price scenarios.For a fixed generation self-schedule, the best-case electricity price scenario is the scenario corresponding to the maximum electricity sales revenue, while the worst-case electricity price scenario is the scenario corresponding to the minimum electricity sales revenue.It can be observed that the best-case electricity price scenario is at   =    for all  for any fixed power generation level, while the worst-case electricity price scenario is related to the fixed power generation level and the budget value Γ [26].

The Robust Generation
Self-Scheduling Model.The robust generation self-scheduling problem for a price-taking power producer, denoted by (RSS), is formulated as follows: ≥ 0,  = 1, . . ., ,  = 1, . . ., , V  ∈ {0, 1} ,  = 1, . . ., ,  = 1, . . ., : nonzero integer,  = 1, . . ., ,  = 1, . . ., In the above formulation, the objective function ( 2) is to maximize the generation profit under the worst-case scenario within the uncertainty set of electricity prices.Under the max-min decision rule, the obtained generation schedule is robust against the variation of electricity prices within the uncertainty set.The generation profit is determined by the electricity sales revenue, the fuel cost, and the startup cost over the scheduling horizon where the fuel cost is [27].Equations ( 3) and ( 4) show the impact of the initial statuses of units on decision-making.Inequalities ( 5) and ( 6) represent the minimum up-time and down-time requirements, respectively.Inequalities (7) represent the startup status.Equations (8) represent the relationship between the state variables in adjacent hours.Inequalities (9) represent the power generation capacity of units.Inequalities (10a) and (10b) represent the ramping-up and ramping-down rate limits.Inequalities (11) represent the electricity demand constraints.Constraints ( 12)-( 14) show the value field of the decision variables.

Solution Methodology
As the objective function ( 2) is max-min-type and includes nonlinear cost functions, (RSS) is a max-min MINLP problem.To solve the problem, we first reformulate (RSS) as a minimization model using the idea provided in [26] and then develop an improved Lagrangian relaxation algorithm upon the reformulation.

The Lagrangian Relaxation Algorithm
(1) Variable Splitting-Based Lagrangian Relaxation.When the Lagrangian relaxation algorithm is used to solve a generation scheduling problem, constraints that couple different units are typically relaxed to make the relaxed problem separable in units.In this paper, constraints that couple different units include constraints (11) and (20).If constraints (20) are relaxed, the resulting subproblem that contains dual variables  0 and   ,  = 1, . . ., , will be as follows: min s.t.constraints (21) and (22) where   ,  = 1, . . ., , are nonnegative Lagrangian multipliers.Note that the above subproblem is unbounded when either ∑  =1    − Γ − ∑  =1   (   −    ) or 1 −   is negative.Consequently, using the typical Lagrangian relaxation algorithm may lead to an unbounded relaxed problem.
Each (RP2 i ) corresponds a single-unit subproblem.According to the feature of the subproblem, we solve it using a two-stage backward dynamic programming procedure [34,35] In the second stage, determine the on/off statuses of the unit using the backward dynamic programming.The state transition equations are where   is subject to constraint (8) and   ( ,−1 ) is the optimal value function.
The obtained relaxation solution can provide a lower bound for the optimal objective function value of (RSS3).
(3) Construction of the Feasible Solution.Because constraints (10b) and ( 26) are relaxed, the optimal solution to the relaxed problem is generally infeasible for (RSS3).To obtain a feasible solution, we propose the following heuristic method based on the current relaxation solution.
Step 1.For each unit i, adjust   according to  ,−1 to meet constraints (10b) corresponding to hour t without violating constraints (9).
The obtained feasible solution can provide an upper bound for the optimal objective function value of (RSS3).
(4) Updating of the Lagrangian Multipliers.We initialize the Lagrangian multipliers by zero and update them according to the subgradient algorithm [36].The iteration is stopped when the maximum number of iterations is reached or the relative duality gap (  −   )/|  | × 100% is smaller than a certain threshold, where   and   are the best lower bound and upper bound obtained so far for (RSS3), respectively.

Discussions on Special
Cases.(RSS) can be simplified in the following two special cases: Γ = ∑  =1    and Γ = ∑  =1    .In the case of Γ = ∑  =1    , we have   =    for all  according to constraints (1a) and ( 1b) and (RSS) can be simplified as a deterministic multi-unit generation self-scheduling problem.Similar result can be obtained in the case of Γ = ∑  =1    .Therefore, we propose the following two-stage algorithm for the two special cases: Since both the variable splitting-based Lagrangian relaxation algorithm and the two-stage algorithm proposed above can solve (RSS) in the two special cases, we will choose the more effective one for each special case based on the numerical test in Section 4.

Numerical Results
In this section, we consider test cases of different sizes to implement the numerical experiments.For convenience, the variable splitting-based Lagrangian relaxation algorithm is denoted by A1 and the two-stage algorithm for the two special cases is denoted by A2.The organization of the section is as follows.First, the generation of the test cases is described in detail.Second, we discuss the impact of the weight  on the performance of algorithm A1 and determine the value of  in the experiments.Third, we test the performance of algorithms A1 and A2, respectively.Finally, we discuss the effect of the budget value Γ on the self-scheduling.
We use Visual C++ to implement the proposed algorithms on a PC with 2.83 GHz and 3.25-GB memory.The linear programming involved is solved by calling CPLEX 12.5.discussed in Section 3.3.The convergence threshold of the relative duality gap is set to 1%.

Computational Results
(1) Discussion on Weight .We allow  to vary within the set {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1} and discuss the impact of the weight  on the performance of algorithm A1 as follows.First, we carry out the algorithm for each test case under various  and  settings.Then, corresponding to each  setting, we calculate the average relative duality gap and computation time over all test cases and report the results in Table 3. From Table 3, we can make the following observations: (1) The average relative duality gap varies between 0.90% and 0.99%.The results show that algorithm A1 can meet the convergence requirement under all  settings.
(2) The average computation time varies between 0.61 s and 0.99 s and reaches the minimum when  is set to 0.4.The results indicate that algorithm A1 shows the fastest convergence behavior with  setting to 0.4.
Based on the above discussion, we set  to 0.4 for algorithm A1 in the following experiments.
(2) Performance of the Algorithms.We use the relative duality gap, the computation time, and the number of cases, denoted by , in which the algorithm stops before reaching the maximum number of iterations to measure the performance of the proposed algorithms.The numerical results of algorithm A1 for all problem sizes and  settings are reported in Tables 4 and 5.The numerical results of algorithm A2 for the two special cases are reported in Table 6.The reported relative duality gap and computation time are both the average of ten test cases in the same problem size.
From Tables 4 and 5, we have the following observations: (1) The average relative duality gap is 0.92% and the maximum relative duality gap is 0.96%.The computation time increases linearly with the increase of the problem size.The average computation time is 0.61 s and the maximum computation time is 3.74 s.
The results demonstrate that algorithm A1 can find a solution very close to the optimal one in a reasonable time for the cases in all sizes.
(2) The average number of cases in which algorithm A1 stops before reaching the iteration limit is 10.The result shows that algorithm A1 can meet the convergence requirement within the iteration limit for almost all cases and indicates good convergence behavior of the algorithm.
(3) The computation time shows a decreasing trend when  increases from 0 to 1.This is because the conservatism of the solution decreases as the budget lower bound increases.
From Table 6, we can observe that algorithm A2 can meet the convergence requirement before reaching the iteration limit for all test cases.For  = 0, the average relative duality gap is 0.89% and the average computation time is 0.81 s.For  = 1, the average relative duality gap is 0.95% and the average computation time is 0.22 s.A comparison between the results of algorithms A1 and A2 indicates that algorithm A2 converges faster than algorithm A1 for the special cases.This is because (RSS) is simplified as a deterministic generation self-scheduling problem in algorithm A2.
Based on the numerical results in Tables 4-6, we suggest using algorithm A2 to solve (RSS) in the special cases and algorithm A1 to solve the problem in other cases.
As a comparison, we also solve (RSS2) by using the MILP approach where the quadratic fuel function is approximated by a ten-piece piecewise linear function, the exponential startup cost function is linearized, and the resulting MILP model is solved by calling CPLEX MILP solver.For each test case, the time limit is set to 1800 s.The numerical results are reported in Table 7.In Table 7, columns 2-7 report the numbers of solvable cases in which the MILP approach can reach the optimality within the time limit, denoted by , under various problem sizes and  settings and columns 9-14 report the average computation times over the solvable cases.For the unsolved cases, the MILP approach either cannot be optimally solved within the time limit or runs out of memory.If all cases cannot be optimally solved, the average computation time is not given.
From Table 7, we can have the following observations: (1) For the problem in small size, all cases can be optimally solved.For the problem in medium size, only partial cases can be optimally solved.For the problem in large size, none of the cases can be optimally solved.
The largest size of the cases that the MILP approach can solve within the acceptable time is 100 × 24.
(2) For the solvable cases, the computation time of the MILP approach grows exponentially as the problem size increases.
The observations show the computation limit of the MILP approach and imply the importance of proposing an effective solution algorithm.Because the proposed algorithm can find the near-optimal solutions within the acceptable time for the cases in all sizes, it is more effective than the MILP approach, especially for the problem in medium or large size.
(3) Effect of the Budget Value on the Self-Scheduling.To show the effect of the budget value on the self-scheduling, we compare the generation levels and the objective function values under various  settings, respectively.The numerical results are reported in Tables 8 and 9. From Tables 8 and 9, we can observe that given the problem size, both the generation level and the generation profit increase as  increases.This is because the problem becomes less conservative with the increase of .

Conclusions
In this paper, we propose an improved Lagrangian relaxation algorithm for the robust generation self-scheduling problem under electricity price uncertainty in the deregulated electricity market.The problem includes quadratic fuel cost, exponential startup cost, unit operation constraints, and electric demand constraints.To avoid obtaining an unbounded relaxed problem, variable splitting is introduced into the algorithm.For the special cases of the budget value, we also propose a two-stage algorithm.Numerical results demonstrate the good performance of the proposed algorithms.Future research can be focused on generalizing the proposed algorithm to solve other robust generation self-scheduling problems.

Table 1 :
A comparison of some relevant literature.
: Binary variable to indicate the on/off status of generation unit i in hour t V  : Binary variable to indicate if generation unit i is started up in hour t   : State variable to indicate the number of hours generation unit i being up or down at the end of hour t   : Power generation level of generation unit i in hour t  0 ,   : Dual variables   : Introduced variable in the solution method   : Artificial variable  1 ,  2 ,   ,   : Lagrangian multipliers 2.2.Modeling of Uncertain Electricity Prices.According to the robust optimization approach, we model uncertain electricity prices with the following uncertainty set: Λ fl {( 1 , . . .,   ): . In the first stage, define   (  ) as the generalized generation cost of unit  in hour  if unit  is on in hour  in (RP2 i ) and determine the corresponding optimal generation level (  ) :  ,−1 −   ≤   ≤  ,−1 +    ,−1 +   (1 −  ,−1 ) ,    ≤   ≤    } , if  =    , arg min {  (  ) :    ≤   ≤    } , if  >    .

Table 2 :
Range of values for parameters associated with units and demands.
[27]meters for the test cases are presented as follows.The number of units is set to 10, 50, and 100, respectively.The scheduling horizon is set to 24, 96, and 168 hours, respectively.The combination of the two parameters forms nine problem sizes, in which 10 × 24, 10 × 96, and 50 × 24 are corresponding to small-sized problems, 10 × 168, 50 × 96, and 100 × 24 are corresponding to mediumsized problems, and 50 × 168, 100 × 96, and 100 × 168 are corresponding to large-sized problems according to the number of decision variables and the number of constraints included in the problem.For each problem size, ten test cases are generated randomly and tested.Therefore, a total of ninety cases are tested in the experiments.For each test case, we let   =   +    ,   =   +    , and   =   for convenience.Value ranges for parameters associated with units are partially based on those in[27]and shown in Table2.We use Pennsylvania-New Jersey-Maryland (PJM) Interconnection Real Time data from 2005 to 2006 to forecast price ranges and set    and     is the sample mean,   is the sample standard deviation,  is the sample size, and  0.975 ( − 1) is the 97.5% quantile of Student's t-distribution with  − 1 degrees of freedom.The forecasted price data for 168 hours are provided in Figure1.The budget lower bound Γ is set to ∑  =1    +  ∑  =1 (   −    ) where  is allowed to vary within the set {0, 0.2, 0.4, 0.6, 0.8, 1}.Note that  = 0 and  = 1 are corresponding to the two special cases 4.1.Test Cases.

Table 3 :
Average relative duality gap and computation time under various  settings.

Table 5 :
Numerical results of algorithm A1.

Table 6 :
Numerical results of algorithm A2 for the two special cases.