Water Resour Manage (2015) 29:5671–5687 DOI 10.1007/s11269-015-1140-6
Optimal Operation of Multi-Reservoir System Based-On Cuckoo Search Algorithm Bo Ming 1 & Jian-xia Chang 1 & Qiang Huang 1 & Yi-min Wang 1 & Sheng-zhi Huang 1
Received: 9 June 2015 / Accepted: 10 September 2015 / Published online: 20 September 2015 # Springer Science+Business Media Dordrecht 2015
Abstract Efficient utilization of water resources by reservoir operation in arid and semi-arid regions has been a traditional and important approach to mitigate water and energy scarcity. As the complexity of water resources system increases, there are urgent needs to apply effective optimization techniques to derive better operation rules. The meta-heuristic algorithms have been an alternative to the optimal operation of multi-reservoir system (OOMRS) for their parallel search capabilities. This paper presents the application of the cuckoo search (CS) algorithm to OOMRS with the objective to maximize the energy production. The penalty method was used to handle the physical and operational constraints. To make the CS more efficient, the optimized parameters of the CS were selected based on the parameter sensibility analysis. Finally, the performance of the CS was accessed by comparison with the genetic algorithm (GA) and the particle swarm optimization (PSO) under the same objective function evaluations (FEs). A case study of China’s Wujiang multi-reservoir system reveals that the CS can provide better and more reliable optimal results with average energy production of 12.31 billion kW h, 10.43 billion kW h, and 10.02 billion kW h for three different scenarios, which are approximately 0.52, 0.32, and 1.64 % higher than that of the GA, respectively. Meanwhile, the convergence performance of the CS is also satisfying. Therefore, it can be concluded that the CS is quite promising in handling complex reservoir operation optimization problem in terms of its simple structure, excellent search efficiency, and strong robustness. Keywords Reservoir operation . Multi-reservoir system . Cuckoo search algorithm . Energy . Parameter sensibility analysis
* Jian-xia Chang
[email protected] 1
State Key Laboratory Base of Eco-hydraulic Engineering in Arid Area, Xi’an University of Technology, Jinhua Road 5, Xi’an 710048 Shaan xi, China
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1 Introduction Human societies and economic entities are seriously threatened by diverse factors, such as water and energy shortages, environmental degradation and global climate change (Bai et al. 2015a). Consequently, it has become extremely important to mitigate water and energy scarcity by efficient utilization of water resources, especially for arid and semi-arid regions. Reservoirs are one of the most efficient key infrastructure components in integrated water resources development and management (Haddad et al. 2015; Liu et al. 2015). By the rational allocation of water resources among various purposes like flood control, power generation, irrigation, river navigation etc., reservoirs can increase the social and economic benefits notably. However, the optimal operation of the reservoirs is a high-dimensional, non-linear, multi-stage and stringent constraint optimization problem (Fu et al. 2011). The operation decision of current stage will influence the future outcomes all over the system (Zhang et al. 2013a). Furthermore, with continuous increase in the demand for water and the complexity of systems related to limited water resources, reservoir operation is becoming more complicated and urgent than ever before. Application of efficient optimization techniques coupled with effective constraint handling skills to optimize water resources system according to practical problems have become one of the most important approaches. Reservoir operation optimization (ROO) can be used for formulating, analyzing, and solving operation optimization problems in water resources planning and management (Zhang et al. 2014). During past several decades, numerous optimization methods have been widely proposed to solve the ROO problem. A comprehensive review on these methods used for reservoir operation can be found in Labadie (Labadie 2004). To sum up, there are three major modeling methods that have been conventionally used, namely Linear Programming (LP) (Yoo 2009), Nonlinear Programming (NLP) (Basu 2004) and Dynamic Programming (DP) (Bhaskar and Whitlatch 1980; Hall et al. 1968; Labadie 2004; Li et al. 2014; Zhang et al. 2013b). However, the aforementioned traditional methods have some obvious shortcomings. For instance, when LP is used, the nonlinear and unsmooth characteristic of ROO problems are often ignored during linearization, thus generating large errors in the optimization process. As for NLP, some approximations must be made in the formulation when the objective function are noncontinuous or non-differentiable, which often results in local optimal solutions and poor convergence performance. Since the reservoir operation problem is a sequential decisionmaking process, DP has been successfully used for ROO problems for it can easily handle the nonlinearity and non-convexity of the problem. However, DP suffers from the shortcomings of exponential increase in computational and memory requirement as the scale of the problem increases, which is known as Bthe curse of dimensionality^. To alleviate the dimensionality problem, various improved versions of DP have been proposed, such as incremental dynamic programming (IDP), dynamic programming successive approximations (DPSA), and discrete differential dynamic programming (DDDP), etc. However, these variants all require a reasonable initial trajectory for each state variable (Li et al. 2014). To overcome the limitations of classical methods, many evolutionary and meta-heuristic algorithms have been developed and applied to reservoir operation with the development of the computer science, such as genetic algorithm (GA) (Bai et al. 2015b; Ballester and Carter 2007; Chang et al. 2013; Chang et al. 2010; Elferchichi et al. 2009; Ngoc et al. 2014), particle swarm optimization (PSO) (Afshar 2012, 2013; Ostadrahimi et al. 2012; Zhang et al. 2013a, 2015), ant colony algorithm (ACO) (Afshar et al. 2015; Jalali et al. 2007; Kumar and Reddy 2006), honey-bee mating optimization (HBMO) algorithm (Afshar et al. 2007),water cycle
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algorithm (WCA) (Haddad et al. 2015), imperialist competitive algorithm (ICA), and cuckoo optimization algorithm (COA) (Hosseini-Moghari et al. 2015). The broad applicability, ease of use, and global perspective of metaheuristics prompt their extensive application and success as optimization tools for ROO problems. Recently, an efficient meta-heuristic, named cuckoo search (CS), was proposed by Yang and Deb. Since the CS is simple in structure, easy in application and effective in optimization, it has been widely applied to many fields, such as hybrid flow shop scheduling (Marichelvam et al. 2014), reliability optimization (Kanagaraj et al. 2013), structural optimization (Gandomi et al. 2011), etc. The previous study have indicated that the CS is very promising and can outperform existing popular algorithms (Civicioglu and Besdok 2011; Patwardhan et al. 2014; Yang and Deb 2010). However, few scholars have put such high efficiency algorithm to solve ROO problems to the best of our knowledge. Since the application of effective optimization techniques to reservoir operation will contribute to getting better and more reliable operation rules, which will further increase the social and economic benefits in context of water and energy shortages. Therefore, it is meaningful to verify whether the new algorithm has great potential to solve complex ROO problems. To make the CS more efficient for ROO problems, the optimal parameters of the CS were selected based on the parameter sensibility analysis, and the impacts of each parameter on the final optimal results was analyzed, which is one motivation of this study. Based on the optimal parameters, the CS was implemented to assess its performance in solving an optimal operation problem of four-reservoir system in China with the objective to maximize the energy production, which is another motivation of this study. The remainder of the paper is organized as follows. Section 2 presents the problem formulation of OOMRS. Section 3 introduces the cuckoo search algorithm and its application to OOMRS problems. The case study is given in Section 4. In Section 5, the results of the study are presented and discussed. Finally, conclusions are provided in Section 6.
2 Problem Formulation The OOMRS is aimed at maximizing the water resources benefits as much as possible by determining an optimal release for each reservoir over the whole operation periods, while satisfying all kinds of physical and operational constraints (Papageorgiou 1985). Generally, the objective function and associated constraints of OOMRS problems can be formulated as follows.
2.1 Objective Function The goal in this paper is to maximize the energy production of the multi-reservoir system, and the water level is taken as the decision variable. Due to the fact that the water storage is reflected by the water level, therefore, the fluctuation of the water level represents the variation of the outflow. The power output is related to turbine release and hydraulic head, and both of them are dynamically determined by the water level by period. Once the upstream water level is determined, the outflow can be calculated through the water balance equation shown in Eq. (4), which involves the turbine release and spillway release. Simultaneously, the downstream water level is connected with the outflow of the reservoir. The increase of the outflow will result in a higher downstream water level, thus leading to a lower hydraulic head. The objective function can be expressed as follows:
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8 > < > :
F ¼ max N m;t ¼
T M X X
t¼1 m¼1 k m qm;t hm;t
N m;t Δt ; ∀m∈M; ∀t∈T
ð1Þ
where F is total energy production of all plants for the entire operation period (kW h); T is the number of operation intervals; M is the number of the hydro plants; Nm,t is the power output of m-th plant in t-th operation interval (MW); Δt is the operation interval (h); km is the power coefficient for m-th plant; qm,t is the release passing turbines of m-th plant in t-th operation interval (m3/s); hm,t is hydraulic head of m-th plant in t-th operation interval (m).
2.2 Constraints In the multi-reservoir system, each individual reservoir is subject to its own set of constraints, while the whole system is subject to the system constraints brought by the interconnection of reservoirs and hydro plants (Li et al. 2014). Specifically, the following constraints are considered: (1) Power output constraint
&
For each hydro plant N m;t ≤N m;t ≤N m;t
&
ð2Þ
For entire plant system
Nt ≤
M X
N m;t ≤N t
ð3Þ
m¼1
where N m;t and N m;t are, respectively, the lower and upper limits of the power output for m-th hydro plant in t-th interval; N t and N t are, respectively, the lower and upper limits of the power output for the system in t-th interval; (2) Continuity equation constraint 8 < V m;tþ1 ¼ V m;t þ η I m;t−Qm;t V ¼ V mþ1;t þ η I mþ1;t −Qmþ1;t : mþ1;tþ1 I mþ1;t ¼ IQmþ1;t þ Qm;t−τ m
ð4Þ
where Vm,t is the initial storage volume of m-th reservoir in the beginning of t-th interval; Im,t is the inflow of m-th reservoir in t-th interval; Qm,t is the outflow of m-th reservoir in t-th interval, which includes the power release and spillway release; η is the conversion coefficient for unit; IQm+1,t is the inter-zone inflow into m+1-th reservoir in t-th interval; τm is the flow routing time from m-th reservoir to downstream m+1-th reservoir, which is often ignored in middle and long-term reservoir operation; (3) Water level constraint Z m;t ≤Z m;t ≤Z m;t
ð5Þ
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where Z m;t and Z m;t are, respectively, the lower and upper limits of water level for m-th reservoir in t-th interval, (4) Outflow constraint Qm;t ≤Qm;t ≤Qm;t
ð6Þ
where Qm;t and Qm;t are, respectively, the lower and upper limits of outflow for m-th reservoir in t-th interval (5) Boundary constraint Z m;1 ¼ Z m;b ; Z m;Tþ1 ¼ Z m;e
ð7Þ
where Zm,b is the water level of m-th reservoir at the beginning of the operation period. Zm,e is the water level of m-th reservoir at end of the operation period. Note that water levels, releases and power outputs are variables to be solved, the others are known input data in the optimization, which involve the historical inflow, water level-storage curve, outflow-downstream water level curve, and parameters for reservoir-hydropower. Without loss of generality, the evaporation loss balanced by precipitation is assumed.
3 Methodology 3.1 Cuckoo Search Algorithm Similar to other meta-heuristics, the CS is also a population-based and evolutionary algorithm. The cuckoo population consists of Npop nests, and each nest denotes a solution, which is called a chromosome in the GA and a particle in the PSO, respectively. For a d-dimensional optimization problem, the nest is an array of d decision variables with the size of 1×d, which is defined as follows: nest ¼ ½x1 ; x2 ; x3 ; ⋯; xd
ð8Þ
Then the population is formed as: 2
3 n e s t1 6 n e s t2 7 6 7 7 pop ¼ 6 6 n e s t3 7 4 ⋮ 5 nest Npop
ð9Þ
After the initial population is generated, it is updated by two evolutionary operators. The first one is the Lévy flight operator, which is described as: ð10Þ nesti ðtþ1Þ ¼ nest i ðtÞ þ a ⊕ LðλÞ; i∈ 1; N pop where nesti(t) denotes the i-th solution of t-th generation; a>0 denotes the step size, which should be related to the search space of practical problems. According to Yang (Yang and Deb 2010), a=0.01∗(xu −xb) is recommended, where xu and xb are, respectively, the higher and
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lower band of the search space; The product ⊕ denotes entry-wise multiplication; L(λ) is the step size drawn from a Lévy distribution, which is defined as: i ð11Þ LðλÞ∼u ¼ t‐λ ; λ∈ 1; 3 The use of the Lévy flight operator makes the CS more efficient mainly for two reasons. For one thing, the mean and variance of Lévy flight step can approach infinite, which is more suitable for large-scale optimization problems (Marichelvam et al. 2014). For another, the occasional larger steps drawn from a Lévy distribution enable the algorithm to escape from the local optima more easily. The characteristics of the Lévy flight have been discussed intensively in the literatures (Valian et al. 2011; Wang et al. 2012). To make full use of the information provided by the current best solution of t-th generation, a complete Lévy flight operator is formulated as: i u ðtþ1Þ ðtÞ ðtÞ ðtÞ ; β∈ 1; 2 ð12Þ ¼ nest i þ α nest −nest nesti i best jν j1=β where β=λ−1, the default value is equal to 1.5; u and ν are both drawn from normal distribution; That is u~N(0,σu2) ν~N(0,σv2). With ( )1=β Γ ð1 þ β Þsinðπβ=2Þ 2 ; σv 2 ¼ 1 ð13Þ σu ¼ Γ ½ð1 þ β Þ=2β2ðβ−1Þ=2 Another way of generating new solutions is by the detection operator, which can be represented as: h i ð14Þ nesti ðtþ1Þ ¼ nest i ðtÞ þ γ H ðpa ‐εÞ⊗ nest j ðtÞ −nest k ðtÞ where nesti(t), nestj(t), and nestk(t) are three different solutions of t-th generation. H(pa ‐ε) is a Heaviside function; pa denotes the probability of worse nests to be abandoned, the default value is equal to 0.25; γ and ε are random numbers draw from the uniform distribution. The cuckoo nests are updated by the above two operators, and the quality of them are evaluated by the fitness function, which should be designed properly according to practical problems. After the evolution, the best nest of the population will be recorded and transmitted to the next generation.
3.2 CS for OOMRS Problem 3.2.1 Design of Fitness Function Evolutionary algorithms are basically developed for unconstrained optimization problems (Afshar 2013). The key issue of solving constrained optimization problems with evolutionary algorithms is the construction of a reasonable and effective fitness function, which should obviously distinguish the quality of different types of solutions. According to the feature of the hydroelectric operation of multi-reservoir system, the fitness function is designed by using the penalty method as follows: ! T M M X X X 2 Δt ð15Þ N m;t −δ N t − N m;t F¼− t¼1
m¼1
m¼1
Optimal Operation of Multi-Reservoir System Based-On Cuckoo
With
δ¼
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8 M X > > > N m;t ≤N t 1; >
X > > > N m;t >N t : 0; m¼1
The power output of m-th reservoir in t-th operation interval is calculated as follows:
N m;t
8 > ; p q −q > 1 m;t m;t > < ¼ k m qm;t hm;t ; > > > : p q −q 2 m;t m;t ;
qm;t ∈ −∞; qm;t qm;t ∈ qm;t ; qm;t qm;t ∈ qm;t ; þ∞
ð17Þ
With qm;t ¼ I m;t −
V m;t −V m;t−1 η
ð18Þ
where p1 >0 and p2 >0 are both penalty parameters. By using the designed fitness function, it can be ensured that the feasible solutions are always better than unfeasible solutions. Besides, when the solutions are not in feasible regions, the function can effectively measure the distance between unfeasible solutions and the boundary of feasible regions, which contributes to making the search move rapidly to feasible regions.
3.2.2 Procedure of CS for OOMRS Problem (1) Structure of individuals For the optimal operation problem of M-reservoir in T-operation period, the nesti consists of the water levels of all reservoirs in the whole operation period with the size of M×(T+1), which is defined as: nest i ¼ z1;1 ; z1;2 ; ⋯;
z1;T þ1 ; ⋯ ; zM;1 ; zM;2 ; ⋯; zM ;T þ1 ; i∈ 1; N pop
ð19Þ
(2) Initialization The individuals are initialed randomly while satisfying the constraints based on Eq. (5), which is given by nest i ¼ zmin þ r⋅ðzmax −zmin Þ ;
i∈ 1; N pop
ð20Þ
where zmin and zmax are, respectively, the lower and upper limits of the water levels for all reservoirs during the whole operation periods; r is a random array drawn from the uniform distribution with size of M×(T+1). (3) Updating the cuckoo population Firstly, the nests are updated by the Lévy flights operator based on Eq. (12). The ðtþ1Þ
fitness function is used to find the best nest nest b1
of the t+1-th generation. Then, go
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on updating the current nests by the detection operator based on Eq. (14) and record the ðtþ1Þ
best nest nest b2 generation.
. Finally, the best nest nest(tb + 1) can be found and transmitted to the next
ðtþ1Þ nestb
8 < nest ðbtþ1Þ ; if F nest ðbtþ1Þ ≤ F nest ðbtþ1Þ 1 1 2 ¼ : nest ðtþ1Þ ; if F nest ðtþ1Þ > F nest ðtþ1Þ b2 b1 b2
ð21Þ
(4) Stopping criteria The CS is terminated while the maximum iterations is reached. Otherwise, the evolution process will be continued until the terminate condition is reached.
4 Case Study 4.1 Wujiang Multi-Reservoir System As the largest tributary in the right bank of Upper Yangtze River, the Wujiang river flows through the central of Guizhou Province in Southwest China, with a drainage area of 87,900 km2 and a mainstream length of 1,037 km. The precipitation is abundant and the multi-year average precipitation ranges from 900 to 1,400 mm. With the typical continental monsoon climate, the precipitation has a strong seasonality with more than 80 % of annual precipitation concentrated between April and August. The Wujiang multi-reservoir system is one of the thirteen major hydropower bases in China. Four hydropower-reservoirs (HJD, DF, SFY, and WJD) as shown in Fig. 1 are selected as the case study, which is a typical multi-reservoir system in China. Since the SFY is a run-ofthe-river hydropower station with small regulation storage, the operation model are only applied to optimize the operation of the HJD, DF and WJD reservoirs Fig. 1. The HJD reservoir, located in the upstream of Wujiang River, has a normal pool level of 1140 m and dead water level of 1076 m. It is a multi-purpose reservoir that prioritizes hydropower generation with a maximum turbine release of 491 m3/s and a maximum capacity of 600 MW. To satisfy the water demand for downstream in dry years, the water level of the HJD cannot be lowered to 1095 m of each operation interval under normal circumstances. The main task of the DF reservoir is also power generation, which is situated in the 65 km downstream of the HJD. The normal pool level and dead water level of the DF are 970 m and 936 m, respectively. The installed capacity is 695 MW, and the maximum turbine release is 632.1 m3/s.
Fig. 1 Sketch of cascade reservoir in WJ River basin Upstream
Dowstream q1
HJD Reservoir
q2
DF Reservoir
q3
SFY Reservoir
WJD Reservoir
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The active regulation storage of the SFY reservoir is only 0.07 billion m3, thus, it is termed here as a run-of-the-river reservoir, which is located in the downstream of the DF. The distance between the two reservoirs is 35.5 km. Due to a small storage volume of the DF, its hydraulic head is considered as a constant of 69 m. Hence, the power generation of the SFY is only related to the turbine release, which has a maximum value of 994.5 m3/s. The WJD reservoir is located in 74.9 km downstream of SFY, with a normal pool level of 760 m and dead water level of 720 m. The installed capacity of the WJD reaches up to 1250 MW, which is the largest of the cascade reservoirs. In order to ensure the navigation safety, the outflow of WJD should be higher than 100 m3/s. The four reservoirs in the system are operated as a whole. The guaranteed power output of the system should be higher than 680 MW. Due to the existence of the abundant inter-zone inflow, the ecological base flow is not considered. The main features of the WJ multi-reservoir system are also presented in Table 1.
4.2 Simulation Scenarios The optimal operation of Wujiang multi-reservoir system is herein simulated for three scenarios, with a monthly time step. The operation periods range from May of the current year to April of the following year. According to hydrological frequency analysis of the monthly inflow data, three typical years are selected as the input of the operation model, which are called, respectively, the wet year (1951.5–1952.4), the normal year (1985.5–1986.4), and the dry year (1963.5–1964.4). The corresponding empirical probabilities of the water volumes are nearly 75, 50, and 25 %, respectively. The monthly inflow and inter-zone inflow data of reservoirs are given in Table 2.
5 Results and Discussion The CS along with the standard GA and an improved version of the PSO (Yuhui and Eberhart 1998) were applied to maximize the energy production of the Wujiang multi-reservoir system. The maximum iterations of 1,000 for all algorithms were found suitable to solve the problem. To make an intensive comparison, the objective function evaluations (FEs) of all algorithms
Table 1 The salient features of the Wujiang multi-reservoir system Reservoir items
Units
HJD
Average inflow
m3/s
150
345
385
502
Normal water level
m
1140
970
837
760
Dead water level Total storage
m billion m3
1076 4.5
936 0.9
822 0.2
720 2.1
Regulation storage
billion m3
1.4
Regulation ability Installed capacity
MW
Annual generation
billion kW·h
Power coefficient
DF
SFY
WJD
3.4
0.5
0.07
multi-year
seasonal
daily
seasonal
600
695
600
1250
1.6
2.4
2.0
4.1
8.5
8.35
8.5
8.17
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Table 2 Inflow and inter-zone flow of three typical years for the reservoirs Interval
Wet year
Normal year
HJD
H-D
D-S
S-W
1
140
179
89
2
291
393
113
3
438
602
150
250
4
467
643
60
190
154
149
5
198
182
34
45
206
204
6
186
191
106
65
84
86
7 8
94 76
72 59
37 30
79 51
74 46
75 47
Dry year
HJD
H-D
D-S
S-W
HJD
H-D
D-S
S-W
50
88
159
196
482
538
111
57
143
331
93
137
240
230
123
256
61
444
616
180
82
320
330
720
290
210
53
25
171
230
35
60
40
63
86
146
18
23
14
16
143
136
16
72
20 11
37 39
81 58
123 100
62 46
33 39
9
58
37
18
26
37
39
10
6
47
44
19
39
10
45
30
13
15
41
34
9
34
52
44
19
43
11
39
25
11
11
42
31
5
12
46
39
14
48
12
51
27
10
12
48
44
9
24
114
136
102
135
173
203
56
82
146
168
58
72
116
192
65
77
Average
Notes: 1. H-D represents the inter-zone flow between HJD reservoir and DF reservoir, the rest may be inferred 2. The units of all values are m3 /s
should be kept the same in principle, which are synchronously determined by the population size and iterations. To maintain approximately the same FEs of the three algorithms, the population size of the CS should be kept half of the GA and the PSO.
5.1 Parameter Sensibility Analysis To obtain the optimal parameters for all algorithms, the parameter sensibility analysis was conducted based on scenario 2. The range of each critical parameter for all algorithms and the corresponding optimal results are shown in Table 3. It can be found that a satisfying energy production of 104.06 (108 kW h) is obtained with a small cuckoo population size of 30, which is about 99.8 % of the maximum energy obtained from a larger population size of 50. With the population size increases, the optimal results do not become better obviously. However, a larger population size prolongs the search time evidently. Therefore, the population size of 50 is chosen for the CS to solve the optimization problem. With regard to the detection probability, the bigger values appear to bring out the worse optimal results. Because a bigger value make the detection operator difficult to generate new solutions, which controls the local search of the CS. The detection probability of 0.15 is found suitable for the problem. As for the step size parameters, when α is equal to 0.4 and β is equal to 1.8, the corresponding best energy production are 104.31 and 104.30, respectively. The value of α is approximately equal to 0.01∗(zu −zl) (zu and zl are the upper and lower limits of the search space, respectively), which is in line with recommended value proposed by Yang (Yang and Deb 2009). For β, a bigger value than 1.5 is likely to make the final optimal results better, and the value of 1.8 is found the best for this study. The parameters of the GA and the PSO are selected in a similar way. Based on the parameter sensibility analysis, the parameters of all algorithms are determined, which are summarized as follows.
104.06
104.01 104.29
103.71
104.11
103.98
104.05
104.20
104.16
104.16
30
40 50
60
70
80
90
100
110
120
0.95
0.85
0.75
0.65
0.55
0.45
0.35
0.15 0.25
0.05
pa
103.81
103.81
103.95
103.14
103.50
103.02
103.85
104.33 103.98
104.33
energy
1
0.9
0.8
0.7
0.6
0.5
0.4
0.2 0.3
0.1
α
100.86
102.24
104.05
103.33
104.06
104.20
104.31
104.28 104.26
104.24
energy
2.0
1.9
1.8
1.7
1.6
1.5
1.4
1.2 1.3
1.1
β
103.74
104.26
104.30
104.25
104.19
103.94
103.13
103.37 103.72
103.83
energy
0.8
0.6 0.7
0.5
pc
GA
104.06
104.01 104.00
104.07
energy
0.110
0.009 0.100
0.008
pm
103.99
103.75 104.16
104.07
energy
2.20
2.10
2.00
1.90
1.80
1.70
1.60
1.40 1.50
1.30
c1 =c2
104.17
104.34
104.34
103.99
103.87
103.93
102.60
103.39 102.20
100.74
energy
2. The unit of the energy is 108 kW h
Notes: 1. When one parameter was analyzed, other parameters were taken the default value, which were from the references
energy
Npop
CS
Table 3 Parameter sensibility analysis of the CS, the GA, and the PSO for the OOMRS problem based on scenario 2
0.9
0.7 0.8
0.6
wmax
PSO
104.34
104.16 104.35
104.26
energy
0.4
0.2 0.3
0.1
wmin
104.24
104.06 104.33
104.25
energy
0.20
0.19
0.18
0.17
0.16
0.15
0.14
0.12 0.13
0.11
vs
104.33
104.22
104.25
104.29
104.23
104.12
104.35
104.25 104.31
104.25
energy
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The Npop =50 for the CS (FEs=50,050), and Npop =100 for the GA and the PSO (FEs= 50,100). The detection probability of the CS is pa =0.15, the step size parameter are α=0.4 and β=1.8, respectively. The crossover and mutation probability of the GA are pc =0.5 and pm = 0.1, respectively. The acceleration constants of the PSO are c1 =c2 =2, the maximum and minimum values of the inertia weight factor are wmax =0.8 and wmin =0.3, respectively. The maximum and minimum velocity are vmax =0.14 (zmax-zmin) and vmin =−vmax, respectively.
5.2 Comparison of Different Algorithms Table 4 shows ten runs of the optimal results of the hydroelectric operation of Wujiang multireservoir system by using the CS, the GA, and the PSO. The average energy production of the CS for the three scenarios are, 123.11, 104.31, and 100.22 (108 kW h), respectively. Compared with the GA, the CS can increase the energy by 0.52, 0.32, and 1.64 %, respectively. It is worth mentioning that the inflow of the scenario 3 is limited, which makes it more difficult to allocate the water resources reasonably than the other two scenarios. Due to the limited search capability, the GA is easy to fall into the local optima. However, the CS performs obviously better than the GA from the aspect of overcoming the Bpremature convergence^. Compared with the PSO, there seems to be no obvious improvement of the energy production. However, the PSO used in this paper is an improved version. Accordingly, the great potential of the CS in solving complex ROO problems can also be exhibited. Furthermore, the standard deviation (SD) of the CS, used to characterize the stability of the optimal results, is significantly lower than that of the GA and the PSO. The more stable optimal results of the CS can also demonstrate its higher search efficiency. Figure 2 indicates the comparison of the average convergence trajectories of the above three algorithms. In scenario 1, the convergence speed of the CS is almost the same as the PSO, but Table 4 Results of 10 different runs for the OOMRS problem by using the CS, the GA, and the PSO Number of run
Energy production (108 kW h) Scenario 1
Scenario 2
Scenario 3
CS
GA
PSO
CS
GA
PSO
CS
GA
PSO
1
123.11
122.58
123.11
104.31
104.15
104.16
100.24
97.84
100.20
2
123.11
122.20
123.11
104.27
104.07
104.25
100.22
98.45
100.24
3
123.11
122.61
123.11
104.30
104.15
104.18
100.23
98.72
100.18
4
123.11
122.57
122.67
104.30
103.58
104.22
100.17
98.37
100.16
5
123.11
122.18
123.11
104.30
103.87
104.24
100.24
98.97
100.23
6
123.11
122.46
122.95
104.32
104.11
104.07
100.23
99.19
100.23
7
123.11
122.69
122.91
104.35
103.71
104.34
100.19
99.29
100.33
8 9
123.11 123.11
122.65 122.53
123.11 123.11
104.32 104.31
103.99 104.16
104.32 104.12
100.21 100.25
99.04 99.10
100.26 100.21
10
123.11
122.22
123.11
104.30
104.04
104.26
100.23
97.00
100.13
Best
123.11
122.69
123.11
104.35
104.16
104.34
100.25
99.29
100.33
Average
123.11
122.47
123.03
104.31
103.98
104.22
100.22
98.60
100.22
Worst
123.11
122.18
122.67
104.27
103.58
104.07
100.17
97.00
100.13
SD
0.002
0.195
0.147
0.019
0.201
0.083
0.024
0.716
0.055
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Fig. 2 Comparison of the convergence trajectories of different algorithms for different scenarios. a Scenario 1, b Scenario 2 and c Scenario 3
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Fig. 3 The optimal trajectories of the water level optimized by different algorithms for scenario 2. a HJD, b DF and c WJD
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it is obviously faster than the GA. In scenario 2 and 3, the convergence speed of the CS is withal faster than the GA, but it is a little slower than the PSO. The PSO appears to find the feasible solutions faster. Though the convergence performance of the CS is not the best among the three algorithms, it is still satisfying for the CS used in this paper is only a standard version and many improvements can be made in the follow-up studies just like the GA and the PSO Fig. 2.
5.3 Rationality of the Operation Results The optimized trajectories of the water level of all reservoirs obtained from scenario 2 are selected as the representative to validate the rationality of the operation results, which are shown in Fig. 3. It can be found that the water levels of the HJD, the DF, and the WJD all vary in the rational ranges, respectively. At the beginning of the operation periods, the inflow of the HJD is very small with only 88 m3/s. To meet the minimum power output requirement of 680 MW, the HJD should increase the outflow to compensate for the other reservoirs, which results in the lower water level of itself. With the inflow increases, the water level of each reservoir rises rapidly to the normal pool level, thus, the reservoir can take the advantages of the higher hydraulic head to generate more energy, and this phenomenon can also be explained by the variation of the hydraulic head shown in Table 5. In dry periods, the HJD continues to play the role of compensation for the other reservoirs, thus making its own water storage gradually decline. However, the water levels of the other reservoirs in downstream will not descend until to the end of the operation periods. The detailed operation results obtained from the CS of scenario 2 are presented in Table 5. It can be clearly seen that the navigation security can be satisfied for the outflows of the WJD are higher than 100 m3/s in all periods. Meanwhile, the power outputs of the whole system can also meet the minimum demand of 680 MW. Moreover, the variation of the hydraulic head of each reservoir except for the SFY generally takes on a rule that rises in wet season and falls in dry season, which is in accordance with the analysis of the Fig. 3. Finally, it should be pointed out that
Table 5 Outflow, hydraulic head and the power output optimized by the CS of scenario 2 Periods Outflow (m3/s) HJD
DF
Hydraulic Head (m) SFY
1
119.9 278.9 389.9
2
0.0 351.5 591.5
3
WJD
HJD
DF
Power of Output (MW)
SFY WJD HJD
DF
SFY
100.5 121.0 102.0 69.0 110.3 123.3 237.5 228.7 680.3 130.9 113.9 69.0 126.1
0.5 616.5 796.5 1086.3 150.5 126.8 69.0 131.2
0.0 334.2 346.9
WJD 90.6
Total Power (MW) 680.1
701.0 1382.1
0.7 652.6 467.2 1164.4 2284.8
4
111.0 260.1 313.1
338.1 161.3 129.3 69.0 133.7 152.2 280.8 183.6
369.2
5
204.1 408.1 448.1
511.1 160.8 128.1 69.0 133.1 278.9 436.5 262.8
555.8 1534.0
985.9
6
109.3 195.3 209.3
226.0 161.7 129.9 69.0 133.7 150.2 211.8 122.8
246.8
731.6
7
149.0 224.0 244.0
280.3 159.4 129.6 69.0 133.4 201.9 242.5 143.1
305.5
893.0
8
207.8 254.8 265.8
305.0 154.6 129.4 69.0 133.1 273.0 275.2 155.9
331.6 1035.7
9
229.9 268.8 278.8
284.9 147.8 129.2 69.0 133.0 288.7 290.1 163.5
309.4 1051.8
10
247.9 281.8 290.8
324.6 139.4 129.1 69.0 132.9 293.8 303.9 170.6
352.4 1120.6
11
303.8 334.8 339.8
12
62.8 293.3 302.3
351.9 127.7 128.7 69.0 132.6 329.8 359.8 199.3
381.3 1270.1
844.0 123.4 114.3 69.0 117.3
809.0 1332.2
65.9 280.0 177.3
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the power outputs of the HJD in operation periods 2 and 3 are near 0. That is because the inflows of the HJD in these two operation periods are very abundant, the reservoir needs to decrease the outflow so as to increase its own water storage rapidly. On one hand, it can make full use of the higher hydraulic head to produce more energy in the following operation periods. On the other hand, it can use the stored water to compensate for the other reservoirs in dry periods.
6 Conclusion The OOMRS is of great importance to integrated water resources development and management. Application of effective optimal techniques will help to acquire more accurate and better operation rules, which will further increase the social and economic benefits in context of water and energy shortages. In this paper, the CS was applied to solve the hydroelectric operation problem of the Wujiang multi-reservoir system in Southwest China. The performance of the CS was compared with the popular GA and PSO. The better and more stable optimal results obtained from the CS demonstrate its excellent search capability. Besides, the convergence performance of the CS is also satisfying. Therefore, it can be concluded that the CS is a practical and valuable optimal technique for complex reservoir operation problems in terms of its simple structure, high search efficiency, strong robustness, and good convergence performance, which can be widely used to solve complex optimization problems of water resources systems. Acknowledgments This research is supported by the National natural science foundation of key projects (51190093), the National Department Public Benefit Research Foundation of Ministry of Water Resources (201501058) and the Key Innovation Group of Science and Technology of Shaanxi (2012KCT-10). Sincere gratitude is extended to the editor and the anonymous reviewers for their professional comments and corrections, which greatly improved the presentation of the paper. Conflict of Interest None.
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