Hydraulic turbine governing system (HTGS) is essential equipment which regulates frequency and power of the power grids. In previous studies, optimal control of HTGS is always aiming at one single operation condition. The variation of operation conditions of HTGS is seldom considered. In this paper, multiobjective optimal function is proposed for HTGS under multiple operation conditions. In order to optimize the solution to the multiobjective problems, a novel multiobjective grey wolf optimizer algorithm with searching factor (sMOGWO) is also proposed with two improvements: adding searching step to search more no-domain solutions nearby the wolves and adjusting control parameters to keep exploration ability in later period. At first, the searching ability of the sMOGWO has been verified on several UF test problems by statistical analysis. And then, the sMOGWO is applied to optimize the solutions of the multiobjective problems of HTGS, while different algorithms are employed for comparison. The experimental results indicate that the sMOGWO is more effective algorithm and improves the control quality of the HTGS under multiple operation conditions.
With the increase of people’s consciousness of environmental protection, more and more renewable energy has been applied to replace the traditional energy, such as wind power [
Hydraulic turbine governing system (HTGS) which mostly contains PID controller is important automatic control equipment of hydroelectric unit [
Many research works have been done to improve the control quality; these works can be mainly divided into two categories: proposed new control models instead of PID controller and optimized control parameters of HTGS. In [
Parameter optimal control of HTGS under multiple operation conditions actually refers to the multiobjective optimization problems. Some multiobjective optimization algorithms have been proposed and successfully applied in various applications in decades [
The main contribution of this paper is the design and optimization of HTGS under multiple operation conditions by using an improved MOGWO algorithm. Firstly, the problem in optimal control of HTGS is explained and multiobjective optimal function is proposed. Secondly, a novel MOGWO algorithm based on searching factor (sMOGWO) is proposed to optimize the multiobjective problem. The sMOGWO is expected to improve the searching ability of MOGWO from two aspects: adding searching step to search more no-domain solutions nearby the wolves and adjusting control parameters to keep exploration ability in later period.
The remaining part of this paper is organized as follows: the HTGS model and its control problem are discussed in Section
The HTGS is essential equipment of the hydropower station. The HTGS contains PID controller, electrohydraulic servo system, hydroturbine, and hydrogenerator.
The PID controller has been widely used in HTGS. In this paper, the structure of the PID controller model and electrohydraulic servo system is set as in Figure
PID control device and electric-hydraulic servo system.
The hydroturbine is a complex system with strong nonlinear and time-varying character. Generally, the hydroturbine model can be descripted as follows:
In this paper, the little fluctuation in transient of hydraulic power station is researched; the model of hydraulic turbine can be described as liner model, as follows:
In water diversion system, the water hammer and pipe wall can be considered as rigid under small fluctuations. The transfer function of rigid water diversion system can be expressed as
In this paper, the core problem is the dynamic response of the HTGS. Hence, the dynamic speed of the generator is considered. The model of the generator can be described as the following transfer function:
As in the above description of the mathematical model, the hydrogenerator unit model can be described as the block diagram in Figure
Hydroturbine unit model with rigid water hammer.
The performance indexes of the optimal control of the HTGS are the most direct measure to evaluate the control quality. Therefore, the optimal control of the HTGS is the optimal control of performance indexes. The objective functions of optimal control are formed by a certain performance index or several performance indexes in most researches. The performance indexes in HTGS can be mainly divided into the following two categories [
(i) Performance indexes for the transition process mainly include adjusting time, rising time, steady state error, and overshoot.
(ii) Performance indexes of error functional integral mainly include integral time absolute error (ITAE), integral absolute error (IAE), integral square time absolute error (ISTAE), integral square time square error (ISTSE), integral time square error (ITSE), and integral square error (ISE).
The performance indexes of error functional integral are comprehensive indexes which make optimal control easier to meet the control requirements. The ITAE is one of the most widely used which has the characteristics of steady adjustment and small overshoot.
Many researches have been done for optimal control of HTGS by optimizing ITAE. However most of the researches are for one single operation condition. As the hydrogenerator usually plays the roles of generating electricity, peak modulation, frequency modulation, and voltage modulation, there is a variety of operation conditions and frequent changes. The optimal control parameters may not be suitable for all the operation conditions.
Here, we will give an example to explain this problem. Parameters of two classical operation conditions of the above HGTS model are given as follows:
No-load:
On-load:
5% step disturbance is set for HGTS. The optimal control parameters of HTGS under no-load condition are acquired by optimizing ITAE. The parameters are
Transient process by different control strategies.
The result indicates that the optimal control parameters of one single operation condition cannot be suitable for all the operation conditions. Accordingly, the optimal control of HTGS is a multiobjective optimization problem in fact. In this paper, we have proposed a novel control parameters optimization objective function. Different operation conditions are considered for optimal control. Because the no-load and on-load are the extreme operating conditions, the ITAE of both conditions are the objective functions. The objective function can be described as follows:
MOGWO algorithm as a new intelligent optimization algorithm has a better convergence rate than other intelligent optimization algorithms. Therefore, it has received great attention and wide application since it has been proposed.
The grey wolf optimizer (GWO) algorithm proposed by Mirjalili et al. in 2014 is a new intelligent optimization algorithm mimicking the hierarchies and hunting strategies of wolves [
In 2016, Mirjalili proposed a multiobjective GWO algorithm named ‘MOGWO algorithm’ for multiobjective problems. There are two major changes in MOGWO: using the external population Archive to store the current nondominated solutions and proposing a selection strategy for multiobjective optimization.
Although the MOGWO algorithm has a better convergence rate, it is easy to fall into local optimum and has poor stability. The main reasons are summarized as follows:
(i) The algorithm has great randomness only when initializing the position of wolves. Even though there is random factor in algorithm, when the position of wolves is updated, the effect of the guidance of lead wolves is much greater than the random factor. Thus the algorithm is highly dependent on the initial value, and self-regulation ability is weak.
(ii) The MOGWO algorithms select
As the guidance of lead wolves has too much influence, the grey wolves always blindly follow the lead wolves and the nondominated solutions around the lead wolves. The grey wolves always ignore the nondominated solutions beside them. Actually, the grey wolves often pass nearby other nondominant solutions when following the lead wolves. If the grey wolves have the ability of independent searching, the global optimization ability of the algorithm will be greatly improved.
Consequently, in this paper, we proposed to add the searching factor in MOGWO named ‘sMOGWO algorithm’. After the position updating of wolves, the wolves will have the searching step. Each wolf will search the nearby position randomly; if the nearby position is better than the current position, the wolf will move to the new position. The step can be described as the following mathematical expressions:
In the original algorithm, the control parameter
The higher the value of
The flow chart of the sMOGWO algorithm.
The steps of the sMOGWO algorithm are as follows.
Archive size is
The objective function values of each individual wolf are calculated. The Archive set is established, and the Archive population is grouped into iterative process.
The
The position of wolves is updated according to (
The wolves get into the searching step according to (
The objective function values of each individual in wolves are calculated. The nondominant solutions of wolves are compared to the individuals in the Archive population one by one, and the Archive set is updated.
The updated Archive set is regrouped and the number of individuals in the Archive population is checked. If the number of individuals exceeds the maximum of population, the extra solutions will be deleted.
The algorithm ends when the maximum number of iterations is reached. All individuals in the Archive population are the optimization results of the algorithm. Otherwise, the algorithm returns back to Step
The computational complexity of MOGWO and sMOGWO is analyzed in this part.
The computational complexity of each iteration (Step
In Step
In Step
In Step
In Step
In Step
The computational complexity of sMOGWO in each iteration is O(
In this section, simulation verification is present to demonstrate the advantage of the proposed sMOGWO algorithm. The proposed algorithm is compared to the MOGWO algorithm [
UF series test problems in CEC 2009 are multimode test functions [
UF3 problem,
The principles and the operation modes of the algorithms are different. In order to make the algorithm contrast, the same maximum number of iterations, population size, and Archive size are set up. The parameters of different multiobjective optimization algorithms are shown in Table
The parameters of different multiobjective optimization algorithms.
Description | sMOGWO | MOGWO | MOPSO | NSGA-III | MOEA/D | SPEA2 |
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Maximum number of iterations | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 |
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Population size | | | | | | |
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Archive size | 100 | 100 | 100 | 100 | 100 | 100 |
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Inflation rate | | | | / | / | / |
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Number of grids per dimension | | | | / | / | / |
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Leader selection pressure | | | | / | / | / |
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Deletion selection pressure | | | | / | / | / |
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Inertia weight | / | / | | / | / | / |
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Inertia weight damping rate | / | / | | / | / | / |
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Mutation rate | / | / | | | | |
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Crossover rate | / | / | / | 0.5 | 0.5 | 0.5 |
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Number of Neighbors | / | / | / | / | 10 | / |
The Generational Distance (GD) and the Spacing (SP) [
In order to eliminate the contingency, each algorithm is run 30 times independently. The statistical results of the evaluation indexes are shown in Tables
Statistical results of GD for different algorithms.
GD | sMOGWO | MOGWO | MOPSO | NSGA-III | MOEA/D | SPEA2 | |
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UF3 | Mean | 0.0210 | 0.0542 | 0.0853 | 0.0268 | 0.0561 | 0.0254 |
Med | 0.0211 | 0.0540 | 0.0737 | 0.0255 | 0.0571 | 0.0252 | |
Max | 0.0253 | 0.0900 | 0.2322 | 0.0716 | 0.0829 | 0.0342 | |
Min | 0.0180 | 0.0315 | 0.0425 | 0.0166 | 0.0299 | 0.0116 | |
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UF4 | Mean | 0.0239 | 0.0331 | 0.0361 | 0.0516 | 0.0409 | 0.0485 |
Med | 0.0241 | 0.0330 | 0.0363 | 0.0521 | 0.0439 | 0.0475 | |
Max | 0.0257 | 0.0394 | 0.0406 | 0.0622 | 0.0526 | 0.0618 | |
Min | 0.0234 | 0.0259 | 0.0324 | 0.0415 | 0.0320 | 0.0340 | |
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UF7 | Mean | 0.0078 | 0.0267 | 0.0468 | 0.0319 | 0.0802 | 0.0313 |
Med | 0.0077 | 0.0278 | 0.0424 | 0.0316 | 0.0584 | 0.0301 | |
Max | 0.0091 | 0.1012 | 0.0881 | 0.0726 | 0.1751 | 0.0608 | |
Min | 0.0072 | 0.0148 | 0.0202 | 0.0140 | 0.0182 | 0.0082 |
Statistical results of SP for different algorithms.
SP | sMOGWO | MOGWO | MOPSO | NSGA-III | MOEA/D | SPEA2 | |
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UF3 | Mean | 0.5719 | 1.0807 | 0.9091 | 1.0130 | 1.0233 | 0.9992 |
Med | 0.53456 | 1.1129 | 0.9073 | 1.0000 | 1.0196 | 0.9993 | |
Max | 0.7215 | 1.4434 | 1.0411 | 1.0850 | 1.0747 | 1.0002 | |
Min | 0.4490 | 0.6002 | 0.7810 | 0.9991 | 1.0000 | 0.9971 | |
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UF4 | Mean | 0.6098 | 0.8207 | 0.7923 | 0.9991 | 1.2974 | 1.5026 |
Med | 0.6217 | 0.8055 | 0.7707 | 0.9940 | 1.2924 | 1.4754 | |
Max | 0.6551 | 1.0963 | 0.9160 | 1.0616 | 1.4337 | 1.9870 | |
Min | 0.5428 | 0.6724 | 0.6908 | 0.9464 | 1.1553 | 1.1513 | |
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UF7 | Mean | 0.6646 | 0.8731 | 0.8649 | 1.0961 | 1.0697 | 0.9995 |
Med | 0.6646 | 0.8592 | 0.8386 | 1.0850 | 1.0015 | 0.9996 | |
Max | 0.7011 | 1.2232 | 1.0611 | 1.2400 | 1.5393 | 1.0760 | |
Min | 0.6186 | 0.5954 | 0.7292 | 0.9997 | 1.0000 | 0.9553 |
The boxplots of GD value of three algorithms.
The boxplots of SP value of three algorithms.
The best optimal results of the algorithms in repeated experiments are shown in Figures
The best optimal results of the different algorithms in repeated experiments on UF3.
The best optimal results of the different algorithms in repeated experiments on UF4.
The best optimal results of the different algorithms in repeated experiments on UF7.
For UF3 test problem, the sMOGWO, NSGA-III, and SPEA2 are the first class in GD value which means the solutions of these methods are closer to the real Pareto front. The GD values of MOPSO, MOGWO, and MOEA/D are high and fluctuating. The sMOGWO method has a better SP value which means the solutions are well distributed. Although the NSGA-III and SPEA2 are closer to the real Pareto front, the SP values are high, which means the solutions of these methods tend to fall into local optimum. We can discover that the NSGA-III and SPEA2 methods fall into local optimum on UF3 test problem intuitively according to Figure
For UF4 test problem, the sMOGWO has the best GD value. The MOPSO, MOGWO, and MOEA/D are in the second class in GD value. The sMOGWO, MOGWO, and MOPSO are in the first class in SP value. From Figure
For UF7 test problem, the GD values of sMOGWO, MOGWO, NSGA-III, and SPEA2 are in the first class, but the SP value of sMOGWO is better than those of other methods. From Figure
Accordingly, the proposed sMOGWO algorithm has good stability under all test functions, and the proposed algorithm performs well in repeated experiments with few poor results.
In this section, a case study is presented to show the effect of the proposed algorithm on optimal control of HTGS under multiple operation conditions.
The no-load and on-load operation conditions are considered. The model of HTGS and its parameters are provided in Section
The Pareto front of different algorithms.
The solutions of MOPSO method are far away from Pareto front. The solutions of sMOGWO, MOGWO, MOPSO, MOEA/D, NSGA-III, and SPEA2 are similar to the Pareto front, but the solutions of sMOGWO are more evenly distributed than those of other methods.
In order to analyze the detailed control transient process of the optimization results, some representative control strategies are chosen from the Pareto optimal solution set as shown in Table
Some representative control strategies are chosen from the Pareto optimal solution set.
Control strategy | Parameters of HGTS | Performance indexes | |||
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| | | ITAE under no-load | ITAE under on-load | |
1 | 8.750 | 5.173 | 1.921 | 0.2352 | 0.0176 |
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2 | 9.352 | 4.612 | 1.602 | 0.2156 | 0.0220 |
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3 | 9.791 | 3.864 | 1.216 | 0.1925 | 0.0311 |
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4 | 10.171 | 3.514 | 1.028 | 0.1752 | 0.0402 |
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5 | 10.473 | 3.191 | 0.759 | 0.1585 | 0.0517 |
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6 | 10.967 | 2.595 | 0.271 | 0.1284 | 0.0823 |
Transient process by different control strategies.
All the three control strategies have good control stability under multiple operation conditions. Strategy 1 is most stable under on-load operation condition, but it has bigger overshoot than the other two strategies under no-load operation condition. Strategy 3 has the smallest overshoot among the three strategies under no-load operation condition, but its stability is the worst among the control strategies under on-load operation condition. Strategy 2 is a compromise control strategy.
Thus, a Pareto solution set can be got after one optimization which can be suitable for multiple operation conditions. And the solutions have different emphasis on objective functions. According to the specific requirements of the HTGS, the decision maker can select one or some satisfactory solutions from this solution set as the final solution, so that the HTGS can obtain better control quality.
The control strategies of hydraulic turbine governing system need to consider the multiple operation conditions. A multiobjective optimal function under different operation conditions is proposed in this paper to solve this control problems.
In order to optimize the multiobjective problems more effectively, a novel MOGWO algorithm based on searching factor (sMOGWO) is proposed. The sMOGWO method is verified with several UF test problems compared to MOGWO, MOPSO, MOEA/D, NSGA-III, and SPEA2. The sMOGWO provided better solution compared with its competitors. And the proposed algorithm has good stability under all test functions, and the proposed algorithm performs well in repeated experiments with few poor results.
A case study has been designed to test the control quality of the control strategies which are got by the proposed method. The experimental results have confirmed that the control strategies perform well under multiple operation conditions.
The simulation data used to support the findings of this study are available from the corresponding author upon request.
We declare that we have no conflicts of interest regarding the publication of this manuscript.
This paper is supported by the National Natural Science Foundation of China (no. 51709121, no. 51709122) and Six Talent Peaks Project of Jiangsu Province of China (no. RJFW-028).