Wind-hydrothermal power system dispatching has received intensive attention in recent years because it can help develop various reasonable plans to schedule the power generation efficiency. But future data such as wind power output and power load would not be accurately predicted and the nonlinear nature involved in the complex multiobjective scheduling model; therefore, to achieve accurate solution to such complex problem is a very difficult task. This paper presents an interval programming model with 2-step optimization algorithm to solve multiobjective dispatching. Initially, we represented the future data into interval numbers and simplified the object function to a linear programming problem to search the feasible and preliminary solutions to construct the Pareto set. Then the simulated annealing method was used to search the optimal solution of initial model. Thorough experimental results suggest that the proposed method performed reasonably well in terms of both operating efficiency and precision.
Due to the increasingly serious energy and environmental problems, renewable energy has become an important research topic, and extensive work has been conducted to advance the technologies of power generation systems based on various renewable sources, such as solar energy, geothermal, biomass, fuel cell, and industrial waste heat [
Firstly, in the aspect of establishing model, as the prediction for hydroplant runoff and wind power output and power load is inaccurate, deterministic model [
For the past few years, another modeling method using interval programming [
Secondly, in the other aspect, methods for solving the model have always been the hot spot in the current study. According to the type of the objective function, solving the optimization problem can be divided into two categories: linear and nonlinear. A series of methods represented by linear programming [
This paper proposed a method using the interval to describe the uncertain number in wind-hydrothermal power system. A multiobjective optimization scheduling decision model that contains interval number is established to assist real-time scheduling. In order to reduce the complexity of multiobjective model with interval parameters, a 2-step optimal method is established by utilizing the linear programming and simulated annealing.
The structure of the paper is organized as follows. In Section
If the upper and lower bounds of an uncertainty variable
In the wind-hydrothermal power system, it is necessary to predict hydroplant runoff, wind power output, and power load. Influenced by prediction accuracy, deterministic predicting results often deviate from the actual value. Therefore, deterministic model would fail to meet the needs of the reasonable modeling. The result described by interval number reflects the uncertainty of results more objectively than the deterministic number and this method has usually been applied in the prediction field [
In the scheduling period, hydroplant runoff, wind power output, and power load can be simulated by interval number as follows:
Deterministic model will generally produce the same solutions from a given initial condition. In actual operation, in order to deal with the deviation of wind power prediction output and the load prediction results, power system dispatching departments must adjust deterministic solving result from deterministic model in real time so as to ensure safe and stable operation of the power system. So it is unreasonable to model the output of a power plant in the system by deterministic number. In this paper, we simulate the output of a power plant by interval number as it is shown in the following function:
The wind-hydrothermal power system is simulated by a deterministic multiobjective model.
Through interval simulation of uncertain value, function (
Through interval simulation of uncertain value, (
Superiority of interval model is as follows: output of the unit plan and standby output are established in one model. They meet the power system constraints, so it can avoid invalid standby that is checked by the power system constraints. Objective function values calculated through interval algorithm are interval numbers. Interval number reflects the uncertainty of empirical function values that are due to uncertainty of power plant output. An interval number quantizes uncertainty of quantitative function values. Thus, the foundation for the scheduling is established.
The existing heuristic search algorithm that solves multiobjective model is based on the Pareto-dominant relationship to compare the performance of different solutions. The gist of Pareto dominant is the objective function value of solutions. Compared to the deterministic value of traditional model, the objective function value, which is calculated by the model of this paper, is an interval number. So we use interval number dominant relationship to compare the performance of different solutions.
In order to further develop an efficient and precise method for solving the Pareto solution set, we use improved simulated annealing that adds the idea of NSGA-II [
For the two interval numbers
At present, there are many ranking methods of interval numbers [
For the two interval numbers
Crowding distance among the interval numbers with the same order value needs to be compared. So these solutions can have a good distribution, diversity, and ductility [
For the two interval numbers
Due to the complexity of the multiobjective problem, the method using directly simulated annealing algorithm to solve the problem will lead to the quite different results because of different initial solutions. In addition, the variables exist in a complex, high-dimensional, and nonlinear space, and this causes serious difficulty to solve the problem precisely.
In order to solve the problem, a 2-step method is proposed in this paper. Firstly, the nonlinear objective function is simplified into a rectilinear objective function, as shown in
After getting the feasible result of linear programming, we order that Pareto solution set is equal to this approximate solution. Then we use simulated annealing algorithm to solve the problem precisely. Process of simulated annealing algorithm to solve the problem in this paper is listed in details as follows. Step 1: to initialize the parameters of simulated annealing algorithm, initial temperature Step 2: generate the Pareto solution sets by repeatedly solving linear programming problems and calculate the objective function value of solution. Step 3: construct roulette with 0-1 range of Pareto set. Step 4: bet a solution from the roulette according to the probability and randomly disturb it. Step 5: replace and rank if the new solution is better than the original one or satisfies the Metropolis criterion. Otherwise, the solution is abandoned. Step 6: update the parameter Step 7: terminate the algorithm if no further better results are achieved after certain iterations. Finally, output optimal solution and sort the Pareto solution set.
In order to further clearly express the 2-step optimization algorithm, we illustrated the flowchart as shown in Figure
Flowchart of our algorithm.
We choose a power system with five thermal power plants, five hydropower stations, and a large wind farm in a province of China as an example to perform our experiment. Scheduling period is 24 hours, divided into 24 sessions, all the basic parameters of the thermal power plants are shown in Table
Basic parameters of thermal power plants.
Index | Capacity (MW) | Min output (MW) |
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1 | 1200 | 500 | 2.8 × 10−6 | 0.274 | 13.7 | 4.692 | −6.374 | 5.065 | 48 |
2 | 1200 | 500 | 1.39 × 10−5 | 0.259 | 14.5 | 5.472 | −6.812 | 5.723 | 60 |
3 | 600 | 280 | 6.11 × 10−5 | 0.279 | 6.35 | 4.312 | −5.764 | 5.365 | 36 |
4 | 1200 | 480 | 8.3 × 10−4 | 0.269 | 14.1 | 5.836 | −6.735 | 4.926 | 50 |
5 | 720 | 330 | 3.33 × 10−5 | 0.312 | 4.64 | 6.548 | −6.264 | 5.569 | 40 |
Basic parameters of hydropower stations.
Index | Capacity (MW) | Min Output (MW) | Max storage C |
Min storage C |
Initial storage C |
Initial head (m) |
|
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1 | 1200 | 240 | 102.57 | 25.99 | 83.95 | 110 | 0.85 |
2 | 1320 | 300 | 0.88 | 0.076 | 0.46 | 115 | 0.86 |
3 | 560 | 110 | 2.78 | 1.842 | 2.21 | 85 | 0.87 |
4 | 600 | 120 | 164 | 50.6 | 111.1 | 80 | 0.88 |
5 | 800 | 160 | 33.5 | 10.4 | 22.00 | 100 | 0.84 |
The predicted output ranges of the wind power plant at each time are shown in Figure
Prediction of the wind power output range.
Prediction of the load interval value.
In order to briefly show the effectiveness of our method, the runoffs of five hydropower plants are set at the same predicted internal value, as shown in Figure
Prediction of hydropower plant runoff interval value.
In the experiment, all of wind power output is received by the system. Hydroelectricity price is 0.25 RMB kW/h. Thermal power electricity price is 0.33 RMB kw/h. Wind power electricity price is 0.45 RMB kw/h. Interval dominant credibility is greater than 0.5. Initial parameters of simulated annealing are set as follows:
Consider
Pareto cube with three objects
Considering
The optimal output range of all power plants.
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The optimal output range of Hydropower 1.
The optimal output range of thermal power plant 1.
The average computational time for the problem is around 50 ms per iteration for the interval programming. In fact, we hardly get the results that satisfy the constraints if we simply use the simulated annealing algorithm without any strategy. In particular, the time complexity of nonlinear programming is presented with the increase in exponential growth trend.
After the optimal output range of each power plant in the every scheduling time was obtained, real-time scheduling only needed to consider the constraint of output range, load balancing, and the scheduling goal because the output range was satisfied with the all constraint of power system. Thus, the real-time scheduling model is simplified and convenient for real-time solution. The solution of real-time scheduling can satisfy the optimal decision in scheduling period because the output range of each power plant is the optimal decision in interval model.
In this paper, we presented an interval program model for wind power scheduling system. The model utilizing the interval theory is able to reasonably simulate the problem of wind-hydrothermal power system dispatching. The 2-step optimization method can solve the complex models efficiently. Experimental results showed that our method has a high precision and speed. Therefore, it is suitable to solve large-scale interval programming model. Further work will focus on the following: (i) combined model including probability and interval should be established and (ii) the solving method should be further improved. How to search the new solutions based on the Pareto set to enhance the performance of our approach will be a further study.
The authors declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work; there is no professional or other personal interests of any nature or kind in any product, service, and company that could be construed as influencing the position presented in, or the review of, the paper.