# A Hybrid Search Algorithm for Midterm Optimal Scheduling of Thermal Power Plants

Authors: Shengli Liao; Chuntian Cheng; Jing Wang; Zhongkai Feng
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A hybrid search algorithm consisting of three stages is presented to solve the midterm schedule for thermal power plants (MTSFTPP) problem, where the primary objective is to achieve equal accumulated operating hours of installed capacity (EAOHIC) for all thermal power plants during the selected period. First, feasible spaces are produced and narrowed based on constraints on the number of units and power load factors. Second, an initial feasible solution is obtained by a heuristic method that considers operating times and boundary conditions. Finally, the progressive optimality algorithm (POA), which we refer to as the vertical search algorithm (VSA), is used to solve the MTSFTPP problem. A method for avoiding convergence to a local minimum, called the lateral search algorithm (LSA), is presented. The LSA provides an updated solution that is used as a new feasible starting point for the next search in the VSA. The combination of the LSA and the VSA is referred to as the hybrid search algorithm (HSA), which is simple and converges quickly to the global minimum. The results of two case studies show that the algorithm is very effective in solving the MTSFTPP problem accurately and in real time.


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## Body

## 1. Introduction

Electric power generation in China mainly consists of coal-fired thermal power plants and hydroelectric power plants [1, 2]. At the end of 2013, China’s total installed generation capacity reached 12473.8 GW, with approximately 64.2% derived from coal-fired thermal power plants, 5% from thermal power using other fuels (e.g., oil and gas), 22.5% from hydropower, 6.1% from wind power, 1.2% from nuclear power, and the remainder from other sources [2]. However, coal-fired units are difficult to start up and shut down, and they respond slowly to system demand because preheating and cooling of the stream turbines is required. Moreover, startup and shutdown of a coal-fired unit require considerable amounts of fuel [3]. Therefore, thermal plants are usually scheduled to meet the base demand to ensure that the generation process proceeds smoothly and to minimize the startup and shutdown times of thermal units.Over the past decade, extreme weather events such as storms, droughts, high temperatures, and damaging hail have occurred frequently in China, which has caused the demand on power systems to change substantially from day to day. These large deviations in demand over short periods cause difficulties in the scheduling of the power systems, a problem that is exacerbated by the large number of coal-fired thermal units in use. Consequently, the power dispatch centers in China must determine the optimal midterm schedule for thermal power plants (MTSFTPP), where “midterm” is a period of one to six months, to determine which units to start up and shut down or the boot capacity of every thermal plant each day before determining the short-term and daily schedules.The midterm schedule is important for the economical operation of power systems. The objective of the scheduling problem is to find the optimal set of thermal generating units in a power system that will satisfy the system demand, operational restrictions, reliability constraints, and security requirements for each period considered. Unlike the short-term unit commitment problem associated with power systems, which involves determining a startup and shutdown schedule of units to meet the demand over a period of one day to one week, loads must be dispatched to each plant in the midterm to allow for maintenance and coordination with hydropower plants. Optimal midterm scheduling smoothes the demand on thermal plants, allowing them to operate more efficiently, and makes the best use of hydropower plants. In contrast with the short-term optimal scheduling of thermal power plants, there has been little research on the midterm problem.The MTSFTPP problem is a variation of the unit commitment (UC) problem, which is a large-scale, nonlinear, nonconvex, mixed-integer combinatorial optimization problem. Various techniques [4–8] have been used to solve this computationally expensive problem [5]. Among these techniques are dynamic programming (DP) [9–12], branch-and-bound (BAB), Lagrangian relaxation (LR) [13–15], integer programming (IP) [16, 17], and metaheuristic algorithms such as genetic algorithms (GA) [18–24], particle swarm optimization (PSO), and neural networks [25–27].Many methods such as DP, BAB, and IP suffer from the curse of dimensionality; that is, the solution space increases exponentially with the number of generating units. As a result, the computation time becomes unacceptable. LR, which decomposes the primal problem into a set of single plant or unit optimization subproblems that are easier to solve independently, has shown potential in solving the UC problem. However, this approach requires the conversion of the optimal dual solution into a feasible solution of the primal problem because of the duality gap, which represents the primary difficulty associated with LR. Metaheuristic algorithms have been widely used to solve the UC problem in recent years, but the quality of the solution depends on the choice of the control parameters, and choosing the control parameters of these algorithms is a very difficult task.Essentially, the MTSFTPP problem is a multistage (in this context, the term “stage” refers to time) decision problem with unit constraints (type 1), thermal power load factor constraints (type 2), operating time constraints (type 3), and boundary condition constraints (type 4). Considering that types 1 and 2 are single-stage constraints and independent between any two stages, whereas type 4 constraints can be met by the initial solution, the optimization process mainly involves type 3 constraints, which are time constraints. This study investigates a hybrid search algorithm (HSA) to solve the MTSFTPP problem. The primary objective is to achieve equal accumulated operating hours of installed capacity [25] (EAOHIC) for all thermal power plants over a given period, with extra hours that are given to the plants with lower emissions, higher efficiency, and occasional unit maintenance. A more detailed description of the problem is provided in Section 2. Section 3 presents a detailed explanation of the HSA. A case study is presented in Section 4, and conclusions are presented in Section 5.


## 2. Problem Formulation

### 2.1. Objective Function

Equality, impartiality, and transparency are important in electric power dispatching. Based on power system security, stability, and economics, units should be treated equally in problems involving operations management and information disclosure. In many western countries, the driving force behind deregulation is the belief that competitive markets make the industry more efficient, leading to lower costs. In the case of China, similar to other developing countries, the incentive for restructuring is derived from the desire to provide equal opportunities to all participants. In China, the firms established by the government are responsible for power system operations, and the power systems are operated in an integrated fashion. Therefore, equality is usually the main concern in long-term and midterm scheduling in China. In this study, EAOHIC is the primary consideration in the MTSFTPP problem. Operating hours are accumulated from the beginning of the respective warm-up periods to the end of the computational period (Figure1). The EAOHIC constraint for the MTSFTPP problem is formulated as follows: (1)Hi=Hji≠j,where Hi, Hj are the accumulated operating hours of installed capacity for plants i, j, respectively, in the selected period.Figure 1
Accumulated operating hours defined from the start of the warm-up period to the end of the computational period (horizon).Although the concept captured by (1) is unambiguous and easily understood, the problem is very difficult to solve directly because the formulation is related to all plants and the objective cannot be evaluated by the computation of a single mathematical expression. Considering that the variance can be used to describe the degree of dispersion of data, let H=(H1,H2,…,Hk,…,Hm) be the set of accumulated operating hours of the installed capacity for the plants, and let D(H) be the variance of the set H. Equation (1) can be replaced by the following problem:(2)f=min⁡⁡DH,DH=1m∑k=1mHk-H¯2,H¯=1m∑k=1mHk,Hk=Hkp+HkT-Hke,HkT=1Nk∗C¯k∗T∗24∗r,Ck¯=1T∑t=1TCkt,where the variables are defined as follows: H is the set of Hk, the accumulated operating hours of the installed capacity of plant k during the selected period, in hours (h); m is the total number of plants; D(H) is the variance of set H, in h2; H¯ is the average of set H, in h; Hkp is the operating hours of installed capacity of plant k accumulated during the warm-up period, in h; HkT is the operating hours of installed capacity of plant k accumulated during the computational period, in h; Hke is the extra hours assigned to plant k, in h; T is the total number of steps t in the computational period, where each step is one day; Nk is the installed capacity of plant k, in MW; Ck¯ is the average of the operating capacity of plant k during the computational period, in MW; r is the load factor; and Ckt is the operating capacity of plant k in period t, in MW.


### 2.2. Constraints

Due to the operational requirements, the minimization of the objective function is subject to the following types of constraints.(1) Unit Constraints. Consider(3)ut,kmin⁡≤ut,k≤ut,kmax⁡,where uk,tmin⁡ and uk,tmax⁡ are the minimum and maximum numbers of units, respectively, in plant k in period t and uk,t is the number of active units in plant k in period t.(2) Thermal Power Load Factor Constraints. Consider(4)Cst∗rmin⁡≤Pt≤Cst∗rmax⁡,where Cst is the operating capacity of the system in period t, in MW, and rmin⁡ and rmax⁡ are the minimum and maximum load factors of thermal power, respectively, which are calculated from the long-term real-world operation of power grid. Pt is the demand of system in period t, in MW.(3) Time Constraints. Midterm optimal thermal power plant scheduling only dispatches the system load to the plants and does not distribute the load among the units in each plant. Therefore, we transform the constraints on the response of a plant into minimum-time constraints at minimum or maximum output. Consider (5)τk,jpeak≥Tkpeak,min⁡,τk,jvalley≥Tkvalley,min⁡,where τk,jpeak is the duration of peak j in the output of plant k, Tkpeak,min⁡ is the minimum duration of the peak in the output of plant k, τk,jvalley is the duration of valley j in the output of plant k, and Tkvalley,min⁡ is the minimum duration of the valley in the output of plant k.(4) Boundary Condition Constraints. The planning for the current period depends on the previous states of the units. For the plant k, considering the previous period dk as the maximum in Tkpeak,min⁡ and Tkvalley,min⁡, the states of the units before the starting point affect the operation of the units through the minimum peak and valley time restrictions, as shown in Figure 1. Consider (6)τk,0peak≥Tkpeak,min⁡,τk,0valley≥Tkvalley,min⁡,where τk,0peak and τk,0valley are the first duration of peak and valley in the output of plant k, respectively.Note that the boundary condition constraints are the same as the time constraints (type 3), the time constraints considering the time horizon in[T1,TT], while the boundary condition constraints considering in [Td,TT].


## 2.1. Objective Function

Equality, impartiality, and transparency are important in electric power dispatching. Based on power system security, stability, and economics, units should be treated equally in problems involving operations management and information disclosure. In many western countries, the driving force behind deregulation is the belief that competitive markets make the industry more efficient, leading to lower costs. In the case of China, similar to other developing countries, the incentive for restructuring is derived from the desire to provide equal opportunities to all participants. In China, the firms established by the government are responsible for power system operations, and the power systems are operated in an integrated fashion. Therefore, equality is usually the main concern in long-term and midterm scheduling in China. In this study, EAOHIC is the primary consideration in the MTSFTPP problem. Operating hours are accumulated from the beginning of the respective warm-up periods to the end of the computational period (Figure1). The EAOHIC constraint for the MTSFTPP problem is formulated as follows: (1)Hi=Hji≠j,where Hi, Hj are the accumulated operating hours of installed capacity for plants i, j, respectively, in the selected period.Figure 1
Accumulated operating hours defined from the start of the warm-up period to the end of the computational period (horizon).Although the concept captured by (1) is unambiguous and easily understood, the problem is very difficult to solve directly because the formulation is related to all plants and the objective cannot be evaluated by the computation of a single mathematical expression. Considering that the variance can be used to describe the degree of dispersion of data, let H=(H1,H2,…,Hk,…,Hm) be the set of accumulated operating hours of the installed capacity for the plants, and let D(H) be the variance of the set H. Equation (1) can be replaced by the following problem:(2)f=min⁡⁡DH,DH=1m∑k=1mHk-H¯2,H¯=1m∑k=1mHk,Hk=Hkp+HkT-Hke,HkT=1Nk∗C¯k∗T∗24∗r,Ck¯=1T∑t=1TCkt,where the variables are defined as follows: H is the set of Hk, the accumulated operating hours of the installed capacity of plant k during the selected period, in hours (h); m is the total number of plants; D(H) is the variance of set H, in h2; H¯ is the average of set H, in h; Hkp is the operating hours of installed capacity of plant k accumulated during the warm-up period, in h; HkT is the operating hours of installed capacity of plant k accumulated during the computational period, in h; Hke is the extra hours assigned to plant k, in h; T is the total number of steps t in the computational period, where each step is one day; Nk is the installed capacity of plant k, in MW; Ck¯ is the average of the operating capacity of plant k during the computational period, in MW; r is the load factor; and Ckt is the operating capacity of plant k in period t, in MW.


## 2.2. Constraints

Due to the operational requirements, the minimization of the objective function is subject to the following types of constraints.(1) Unit Constraints. Consider(3)ut,kmin⁡≤ut,k≤ut,kmax⁡,where uk,tmin⁡ and uk,tmax⁡ are the minimum and maximum numbers of units, respectively, in plant k in period t and uk,t is the number of active units in plant k in period t.(2) Thermal Power Load Factor Constraints. Consider(4)Cst∗rmin⁡≤Pt≤Cst∗rmax⁡,where Cst is the operating capacity of the system in period t, in MW, and rmin⁡ and rmax⁡ are the minimum and maximum load factors of thermal power, respectively, which are calculated from the long-term real-world operation of power grid. Pt is the demand of system in period t, in MW.(3) Time Constraints. Midterm optimal thermal power plant scheduling only dispatches the system load to the plants and does not distribute the load among the units in each plant. Therefore, we transform the constraints on the response of a plant into minimum-time constraints at minimum or maximum output. Consider (5)τk,jpeak≥Tkpeak,min⁡,τk,jvalley≥Tkvalley,min⁡,where τk,jpeak is the duration of peak j in the output of plant k, Tkpeak,min⁡ is the minimum duration of the peak in the output of plant k, τk,jvalley is the duration of valley j in the output of plant k, and Tkvalley,min⁡ is the minimum duration of the valley in the output of plant k.(4) Boundary Condition Constraints. The planning for the current period depends on the previous states of the units. For the plant k, considering the previous period dk as the maximum in Tkpeak,min⁡ and Tkvalley,min⁡, the states of the units before the starting point affect the operation of the units through the minimum peak and valley time restrictions, as shown in Figure 1. Consider (6)τk,0peak≥Tkpeak,min⁡,τk,0valley≥Tkvalley,min⁡,where τk,0peak and τk,0valley are the first duration of peak and valley in the output of plant k, respectively.Note that the boundary condition constraints are the same as the time constraints (type 3), the time constraints considering the time horizon in[T1,TT], while the boundary condition constraints considering in [Td,TT].


## 3. Solution Methodology

The MTSFTPP problem is a UC problem, and each approach to the problem has its disadvantages, as mentioned previously. Constraints of types 1 and 2 involve a single period and can be treated independently to reduce the search space, constraints of type 4 can be satisfied in the initial solution, and constraints of type 3 must be satisfied over multiple periods. This study investigates a novel method using three stages to solve the problem.


### 3.1. Producing a Feasible Solution Space Satisfying Constraint Types 1 and 2

Mathematically, the MTSFTPP problem, like the UC problem, suffers from the curse of dimensionality as the number of units and the computational period increases. Thus, it is difficult to solve this problem by considering a simple combination of units. For a power grid with five thermal plants, each having four units, the total number of unit combinations will reach8.67×1041 for a horizon of one month, and the search will be computationally expensive. Fortunately, the actual constraints on the thermal power load factors and the number of units are beneficial because they reduce the search space when searching for feasible solutions. The specific procedure is as follows.Step 1.
For the thermal plants with different values of unit capacity, each plant will be represented by several “virtual plants” whose capacity depends on the number of units of a given type. For instance, for a thermal plant with 5 units, including 2 units with 300 MW of capacity and 3 units with 200 MW of capacity, the plant will be divided into 2 “virtual plants.” By initializing the plants in this manner, the final number of virtual plants will be greater than the number of actual plants and will be denoted asM.Step 2.
Start from the first period; that is, lett=1.Step 3.
Sort the combinationsUt that meet the unit constraints in period t; that is, Ut=ut1,ut2,…,uti,…,utNUt (i=1~NUt), where NUt is the number of combinations that meet constraint 1 (3) in period t, uti=(ut,1i,ut,2i,…,ut,ki,…,ut,Mi), uk,tmin⁡≤uk,ti≤uk,tmax⁡ (k=1~M).Step 4.
Select the combinationsSt that satisfy the thermal power load factor constraints from Ut in period t; that is, St={st1,st2,…,stj,…,stNst} (j=1~Nst), where Nst is the number of combinations that satisfy constraints 1 and 2 ((3) and (4)) in period t.Step 5.
Ift<T, set t=t+1 and return to Step 3. Otherwise, stop and retain the feasible solution space S={S1,S2,…,St,…,ST} satisfying constraints 1 and 2 ((3) and (4)).


### 3.2. Generating an Initial Feasible Solution Using a Heuristic Method

At this point, we have obtained the feasible space satisfying constraints of types 1 and 2. Finding the optimum in the next stage requires a solution that satisfies all of the operational and system constraints. A heuristic method is proposed for this goal. The procedure is as follows.Step 1.
Obtain the operating data ford days before the starting point. These data will be stored in an array and denoted as the scheme S0. The midterm schedule must satisfy the boundary constraints (type 4) between actual operation and future scheduling.Step 2.
Start from the first period; that is, lett=1.Step 3.
Obtain all combinationsSt for period t (t=1~T) using the method described in Section 3.1.Step 4.
Choose a combination that satisfies constraint type 3 in (5) and that is suitable for the boundary constraints, type 4. For each subset stj (j=1~Nst) in combination space St at period t, check the type 3 and type 4 constraints. In particular, the type 4 constraints use the historical data S0 to satisfy the minimum-time constraints at the starting points. Clearly, S0 includes the historical data when t=1. The search is stopped and the current subset appended to S0 when a feasible subset is found for all subsets stj (j=1~Nst). The subset number in the first period is stored to reduce the search in the next cycle. In this case, continue to the next period and set t=t+1. Otherwise, return to the first period, start from the stored number, and repeat the previous procedure. An initial feasible solution will be generated for the optimization.A flow chart and pseudocode for generating an initial feasible solution are shown in Figure2 and Algorithm 1, respectively.Algorithm 1:Algorithm for generating an initial feasible solution.
Heuristic for generating an initial feasible solution
Input actual operation data for d days before the start point S0={S01,…,S0k,…,S0M}
feasible solution spaceS=S1,…,St,…,ST, St=st1,…,stj,…,stNst, stj=(ut1j,…,utkj,…,utMj)
initialNsc=1
Output initial feasible solution I
for all periods from t=1 to T
for all the feasible states stj⊂St from j=b to Nst
if  t=1
b=Nsc
end if
else
b=1
end else
Define num to count, num = 0
for all plants states utkj∈stj from k=1 to M
if τk,jpeak(S0k,utkj)≥Tkpeak,min⁡ & τk,jvalley(S0k,utkj)≥Tkvalley,min⁡
num = num + 1
end if
end for
if num = M
ExpandS0′ by appending stj to S0
Storestj in I
if  t=1
Nsc=j+1
end if
end if
if num < M & j=Nst
t=1, j=Nsc,
end if
end for
end forFigure 2
The procedure for generating an initial feasible solution.


### 3.3. Hybrid Search Algorithm

Numerous operational and system constraints are imposed in the unit commitment problem. The aforementioned initial feasible solution meets only some constraints, and the optimization for unit commitment still must be performed. Sections3.3.1 and 3.3.2 present a two-step procedure for optimization.


#### 3.3.1. Vertical Search Algorithm

Over the past few decades, the progressive optimality algorithm [28] (POA) has been shown to have great advantages over classical optimization methods and has been one of the most widely used techniques for hydroelectric generator scheduling and water resources problems [29–31]. The POA attempts to find the best solution in a given decision space based on Bellman’s Principle and is free of a particular model structure. The advantages of the POA over other optimization techniques are that it can decompose a multistate decision problem into several nonlinear programming subproblems to reduce the dimensionality. In fact, the optimal scheduling of thermal power plants is a multistate and multistage problem, and hence it is suitable for the POA. The POA is defined as follows [27]:(7)f=min⁡⁡∑t=1Tgxt-1,xt,Gxt-1,xt+1=min⁡1≤j≤Nt⁡gxt-1,xtj+gxtj,xt+1,where xtj is the state j in period t, G(xt-1,xt+1) is the objective function in period t, and xt-1 and xt+1 are the decision variables in periods t-1 and t+1, respectively. For the optimal scheduling of thermal power plants, the problem can be mathematically stated as follows:(8)f=min⁡⁡DH,Gst,st+1=min⁡1≤j≤Nst⁡DHstj,st+11≤t<T,Gst-1,st=min⁡1≤j≤Nst⁡DHst-1,stjt=T,where H(stj,st+1) and H(st-1,stj) are functions that give the operating hours of installed capacity when the status is stj and G(st,st+1) and G(st-1,st) are the objective functions for period t. With this formulation, we choose stj from St with the status in the other periods fixed to obtain the optimal objective function value. A diagram of the operating of the POA is shown in Figure 3, and pseudocode for the algorithm is shown in Algorithm 2.Algorithm 2:Algorithm for VSA.
VSA
Input initial feasible solution I=It∣It⊂St,1≤t≤T, It=(it1,…,itk,…,itM)
actual operation data ford days before the start point S0=S01,…,S0k,…,S0M
feasible solution spaceS=S1,…,St,…,ST, St=st1,…,stj,…,stNst, stj=(ut1j,…,utkj,…,utMj)
initial valueV0
Output optimized solution
for  n=1 to N
for all periods from t=1 to T
for all the feasible states stj⊂St from j=1 to Nst
Define num to count, num = 0
for all plants states utkj∈stj from k=1 to M
if  τk,jpeak(S0k,utkj)≥Tkpeak,min⁡ & τk,jvalley(S0k,utkj)≥Tkvalley,min⁡
num = num + 1
end if
end for
if num = M
CalculateV=D(H)
if  V<V0
Replaceitk using utkj; V0=V
end if
end if
end for
end for
Obtain new solutionR
If  I = R
Obtain optimized solution
end if
end forFigure 3
Search principle of VSA using POA.The aforementioned search is a sequential procedure, which we refer to as the vertical search algorithm (VSA), as distinguished from the following search algorithm.


#### 3.3.2. Lateral Search Algorithm

The solution obtained using the VSA is clearly affected by the starting point (i.e., initial guess) and can converge to a local minimum. Hence, another search strategy is presented to improve the solution by enlarging the search space. The search addresses the objective that all plants have equal or approximately equal operating hours of installed capacity. To attain this goal, the plants whose operating hours are greater than or less than the average value will be adjusted. The plant with the greatest bias from the average is selected for adjustment. Generally, we choose the periods of maximum or minimum plant output to perform the adjustments. This state involves situations in which the system load allows for a change in the number of active units. Another state occurs in periods of maximum or minimum output during which a change in the number of active units in other periods is not possible. For the first case, the adjustment is started from the first period among the periods of maximum or minimum output. The increase or decrease in the number of units depends on the difference from the average and the constraints. If the plant’s operating hours of installed capacity are greater than the average, we decrease the number of units for selected periods. This adjustment is a trial procedure because of the complex constraints and because the starting period for the adjustment is related to the constraints. For the second case, the changes in the maximum/minimum output periods will cause a change in the unit status in subsequent periods due to the minimum-time constraints. The adjustment features two aspects: satisfying all of the constraints in the maximum- or minimum-output periods and satisfying the constraints in subsequent periods. In either case, a new feasible solution will be produced. Consequently, an updated optimal solution can be obtained using the VSA. Herein, we refer to the search procedure as the lateral search algorithm (LSA). An optimal solution that satisfies the constraints can be obtained using the VSA and LSA alternately. A flowchart for the hybrid search algorithm that combines the VSA and LSA is shown in Figure4.Figure 4
Search procedure for HSA using VSA and LSA.


## 3.1. Producing a Feasible Solution Space Satisfying Constraint Types 1 and 2

Mathematically, the MTSFTPP problem, like the UC problem, suffers from the curse of dimensionality as the number of units and the computational period increases. Thus, it is difficult to solve this problem by considering a simple combination of units. For a power grid with five thermal plants, each having four units, the total number of unit combinations will reach8.67×1041 for a horizon of one month, and the search will be computationally expensive. Fortunately, the actual constraints on the thermal power load factors and the number of units are beneficial because they reduce the search space when searching for feasible solutions. The specific procedure is as follows.Step 1.
For the thermal plants with different values of unit capacity, each plant will be represented by several “virtual plants” whose capacity depends on the number of units of a given type. For instance, for a thermal plant with 5 units, including 2 units with 300 MW of capacity and 3 units with 200 MW of capacity, the plant will be divided into 2 “virtual plants.” By initializing the plants in this manner, the final number of virtual plants will be greater than the number of actual plants and will be denoted asM.Step 2.
Start from the first period; that is, lett=1.Step 3.
Sort the combinationsUt that meet the unit constraints in period t; that is, Ut=ut1,ut2,…,uti,…,utNUt (i=1~NUt), where NUt is the number of combinations that meet constraint 1 (3) in period t, uti=(ut,1i,ut,2i,…,ut,ki,…,ut,Mi), uk,tmin⁡≤uk,ti≤uk,tmax⁡ (k=1~M).Step 4.
Select the combinationsSt that satisfy the thermal power load factor constraints from Ut in period t; that is, St={st1,st2,…,stj,…,stNst} (j=1~Nst), where Nst is the number of combinations that satisfy constraints 1 and 2 ((3) and (4)) in period t.Step 5.
Ift<T, set t=t+1 and return to Step 3. Otherwise, stop and retain the feasible solution space S={S1,S2,…,St,…,ST} satisfying constraints 1 and 2 ((3) and (4)).


## 3.2. Generating an Initial Feasible Solution Using a Heuristic Method

At this point, we have obtained the feasible space satisfying constraints of types 1 and 2. Finding the optimum in the next stage requires a solution that satisfies all of the operational and system constraints. A heuristic method is proposed for this goal. The procedure is as follows.Step 1.
Obtain the operating data ford days before the starting point. These data will be stored in an array and denoted as the scheme S0. The midterm schedule must satisfy the boundary constraints (type 4) between actual operation and future scheduling.Step 2.
Start from the first period; that is, lett=1.Step 3.
Obtain all combinationsSt for period t (t=1~T) using the method described in Section 3.1.Step 4.
Choose a combination that satisfies constraint type 3 in (5) and that is suitable for the boundary constraints, type 4. For each subset stj (j=1~Nst) in combination space St at period t, check the type 3 and type 4 constraints. In particular, the type 4 constraints use the historical data S0 to satisfy the minimum-time constraints at the starting points. Clearly, S0 includes the historical data when t=1. The search is stopped and the current subset appended to S0 when a feasible subset is found for all subsets stj (j=1~Nst). The subset number in the first period is stored to reduce the search in the next cycle. In this case, continue to the next period and set t=t+1. Otherwise, return to the first period, start from the stored number, and repeat the previous procedure. An initial feasible solution will be generated for the optimization.A flow chart and pseudocode for generating an initial feasible solution are shown in Figure2 and Algorithm 1, respectively.Algorithm 1:Algorithm for generating an initial feasible solution.
Heuristic for generating an initial feasible solution
Input actual operation data for d days before the start point S0={S01,…,S0k,…,S0M}
feasible solution spaceS=S1,…,St,…,ST, St=st1,…,stj,…,stNst, stj=(ut1j,…,utkj,…,utMj)
initialNsc=1
Output initial feasible solution I
for all periods from t=1 to T
for all the feasible states stj⊂St from j=b to Nst
if  t=1
b=Nsc
end if
else
b=1
end else
Define num to count, num = 0
for all plants states utkj∈stj from k=1 to M
if τk,jpeak(S0k,utkj)≥Tkpeak,min⁡ & τk,jvalley(S0k,utkj)≥Tkvalley,min⁡
num = num + 1
end if
end for
if num = M
ExpandS0′ by appending stj to S0
Storestj in I
if  t=1
Nsc=j+1
end if
end if
if num < M & j=Nst
t=1, j=Nsc,
end if
end for
end forFigure 2
The procedure for generating an initial feasible solution.


## 3.3. Hybrid Search Algorithm

Numerous operational and system constraints are imposed in the unit commitment problem. The aforementioned initial feasible solution meets only some constraints, and the optimization for unit commitment still must be performed. Sections3.3.1 and 3.3.2 present a two-step procedure for optimization.


### 3.3.1. Vertical Search Algorithm

Over the past few decades, the progressive optimality algorithm [28] (POA) has been shown to have great advantages over classical optimization methods and has been one of the most widely used techniques for hydroelectric generator scheduling and water resources problems [29–31]. The POA attempts to find the best solution in a given decision space based on Bellman’s Principle and is free of a particular model structure. The advantages of the POA over other optimization techniques are that it can decompose a multistate decision problem into several nonlinear programming subproblems to reduce the dimensionality. In fact, the optimal scheduling of thermal power plants is a multistate and multistage problem, and hence it is suitable for the POA. The POA is defined as follows [27]:(7)f=min⁡⁡∑t=1Tgxt-1,xt,Gxt-1,xt+1=min⁡1≤j≤Nt⁡gxt-1,xtj+gxtj,xt+1,where xtj is the state j in period t, G(xt-1,xt+1) is the objective function in period t, and xt-1 and xt+1 are the decision variables in periods t-1 and t+1, respectively. For the optimal scheduling of thermal power plants, the problem can be mathematically stated as follows:(8)f=min⁡⁡DH,Gst,st+1=min⁡1≤j≤Nst⁡DHstj,st+11≤t<T,Gst-1,st=min⁡1≤j≤Nst⁡DHst-1,stjt=T,where H(stj,st+1) and H(st-1,stj) are functions that give the operating hours of installed capacity when the status is stj and G(st,st+1) and G(st-1,st) are the objective functions for period t. With this formulation, we choose stj from St with the status in the other periods fixed to obtain the optimal objective function value. A diagram of the operating of the POA is shown in Figure 3, and pseudocode for the algorithm is shown in Algorithm 2.Algorithm 2:Algorithm for VSA.
VSA
Input initial feasible solution I=It∣It⊂St,1≤t≤T, It=(it1,…,itk,…,itM)
actual operation data ford days before the start point S0=S01,…,S0k,…,S0M
feasible solution spaceS=S1,…,St,…,ST, St=st1,…,stj,…,stNst, stj=(ut1j,…,utkj,…,utMj)
initial valueV0
Output optimized solution
for  n=1 to N
for all periods from t=1 to T
for all the feasible states stj⊂St from j=1 to Nst
Define num to count, num = 0
for all plants states utkj∈stj from k=1 to M
if  τk,jpeak(S0k,utkj)≥Tkpeak,min⁡ & τk,jvalley(S0k,utkj)≥Tkvalley,min⁡
num = num + 1
end if
end for
if num = M
CalculateV=D(H)
if  V<V0
Replaceitk using utkj; V0=V
end if
end if
end for
end for
Obtain new solutionR
If  I = R
Obtain optimized solution
end if
end forFigure 3
Search principle of VSA using POA.The aforementioned search is a sequential procedure, which we refer to as the vertical search algorithm (VSA), as distinguished from the following search algorithm.


### 3.3.2. Lateral Search Algorithm

The solution obtained using the VSA is clearly affected by the starting point (i.e., initial guess) and can converge to a local minimum. Hence, another search strategy is presented to improve the solution by enlarging the search space. The search addresses the objective that all plants have equal or approximately equal operating hours of installed capacity. To attain this goal, the plants whose operating hours are greater than or less than the average value will be adjusted. The plant with the greatest bias from the average is selected for adjustment. Generally, we choose the periods of maximum or minimum plant output to perform the adjustments. This state involves situations in which the system load allows for a change in the number of active units. Another state occurs in periods of maximum or minimum output during which a change in the number of active units in other periods is not possible. For the first case, the adjustment is started from the first period among the periods of maximum or minimum output. The increase or decrease in the number of units depends on the difference from the average and the constraints. If the plant’s operating hours of installed capacity are greater than the average, we decrease the number of units for selected periods. This adjustment is a trial procedure because of the complex constraints and because the starting period for the adjustment is related to the constraints. For the second case, the changes in the maximum/minimum output periods will cause a change in the unit status in subsequent periods due to the minimum-time constraints. The adjustment features two aspects: satisfying all of the constraints in the maximum- or minimum-output periods and satisfying the constraints in subsequent periods. In either case, a new feasible solution will be produced. Consequently, an updated optimal solution can be obtained using the VSA. Herein, we refer to the search procedure as the lateral search algorithm (LSA). An optimal solution that satisfies the constraints can be obtained using the VSA and LSA alternately. A flowchart for the hybrid search algorithm that combines the VSA and LSA is shown in Figure4.Figure 4
Search procedure for HSA using VSA and LSA.


## 3.3.1. Vertical Search Algorithm

Over the past few decades, the progressive optimality algorithm [28] (POA) has been shown to have great advantages over classical optimization methods and has been one of the most widely used techniques for hydroelectric generator scheduling and water resources problems [29–31]. The POA attempts to find the best solution in a given decision space based on Bellman’s Principle and is free of a particular model structure. The advantages of the POA over other optimization techniques are that it can decompose a multistate decision problem into several nonlinear programming subproblems to reduce the dimensionality. In fact, the optimal scheduling of thermal power plants is a multistate and multistage problem, and hence it is suitable for the POA. The POA is defined as follows [27]:(7)f=min⁡⁡∑t=1Tgxt-1,xt,Gxt-1,xt+1=min⁡1≤j≤Nt⁡gxt-1,xtj+gxtj,xt+1,where xtj is the state j in period t, G(xt-1,xt+1) is the objective function in period t, and xt-1 and xt+1 are the decision variables in periods t-1 and t+1, respectively. For the optimal scheduling of thermal power plants, the problem can be mathematically stated as follows:(8)f=min⁡⁡DH,Gst,st+1=min⁡1≤j≤Nst⁡DHstj,st+11≤t<T,Gst-1,st=min⁡1≤j≤Nst⁡DHst-1,stjt=T,where H(stj,st+1) and H(st-1,stj) are functions that give the operating hours of installed capacity when the status is stj and G(st,st+1) and G(st-1,st) are the objective functions for period t. With this formulation, we choose stj from St with the status in the other periods fixed to obtain the optimal objective function value. A diagram of the operating of the POA is shown in Figure 3, and pseudocode for the algorithm is shown in Algorithm 2.Algorithm 2:Algorithm for VSA.
VSA
Input initial feasible solution I=It∣It⊂St,1≤t≤T, It=(it1,…,itk,…,itM)
actual operation data ford days before the start point S0=S01,…,S0k,…,S0M
feasible solution spaceS=S1,…,St,…,ST, St=st1,…,stj,…,stNst, stj=(ut1j,…,utkj,…,utMj)
initial valueV0
Output optimized solution
for  n=1 to N
for all periods from t=1 to T
for all the feasible states stj⊂St from j=1 to Nst
Define num to count, num = 0
for all plants states utkj∈stj from k=1 to M
if  τk,jpeak(S0k,utkj)≥Tkpeak,min⁡ & τk,jvalley(S0k,utkj)≥Tkvalley,min⁡
num = num + 1
end if
end for
if num = M
CalculateV=D(H)
if  V<V0
Replaceitk using utkj; V0=V
end if
end if
end for
end for
Obtain new solutionR
If  I = R
Obtain optimized solution
end if
end forFigure 3
Search principle of VSA using POA.The aforementioned search is a sequential procedure, which we refer to as the vertical search algorithm (VSA), as distinguished from the following search algorithm.


## 3.3.2. Lateral Search Algorithm

The solution obtained using the VSA is clearly affected by the starting point (i.e., initial guess) and can converge to a local minimum. Hence, another search strategy is presented to improve the solution by enlarging the search space. The search addresses the objective that all plants have equal or approximately equal operating hours of installed capacity. To attain this goal, the plants whose operating hours are greater than or less than the average value will be adjusted. The plant with the greatest bias from the average is selected for adjustment. Generally, we choose the periods of maximum or minimum plant output to perform the adjustments. This state involves situations in which the system load allows for a change in the number of active units. Another state occurs in periods of maximum or minimum output during which a change in the number of active units in other periods is not possible. For the first case, the adjustment is started from the first period among the periods of maximum or minimum output. The increase or decrease in the number of units depends on the difference from the average and the constraints. If the plant’s operating hours of installed capacity are greater than the average, we decrease the number of units for selected periods. This adjustment is a trial procedure because of the complex constraints and because the starting period for the adjustment is related to the constraints. For the second case, the changes in the maximum/minimum output periods will cause a change in the unit status in subsequent periods due to the minimum-time constraints. The adjustment features two aspects: satisfying all of the constraints in the maximum- or minimum-output periods and satisfying the constraints in subsequent periods. In either case, a new feasible solution will be produced. Consequently, an updated optimal solution can be obtained using the VSA. Herein, we refer to the search procedure as the lateral search algorithm (LSA). An optimal solution that satisfies the constraints can be obtained using the VSA and LSA alternately. A flowchart for the hybrid search algorithm that combines the VSA and LSA is shown in Figure4.Figure 4
Search procedure for HSA using VSA and LSA.


## 4. Case Study

The proposed method was applied to the MTSFTPP problem for an actual power system, the Yunnan Power Grid (YPG) in China. Currently, it is used as the primary tool to determine the midterm optimal scheduling of thermal power plants by the operators of YPG. Yunnan is among the richest in hydropower resources of all of the provinces in China. Its potential hydropower capacity is 103.6 GW, which contributes 15.3% of the national capacity and ranks third in the nation. The exploitable hydropower capacity is 90 GW, which contributes 17.9% and ranks second in the nation. By the end of 2013, the total installed capacity of the YPG had reached 26.1 GW, with hydropower capacity accounting for 16.7 GW and thermal capacity accounting for 8.935 GW. Proper scheduling of thermal power plants can make the best use of hydropower systems and improve system security.The thermal power system in the YPG consists of 9 thermal power plants with a total installed capacity of 8.935 GW and 27 units. The dispatching center in the YPG controls the thermal power system. Table1 lists the basic data for these plants. The HSA was implemented in Java and executed on a PC with an Intel Core 2 Duo 2.93 GHz CPU and 2 GB of memory. An actual schedule for 2013 in the YPG was used to test the validity and the computational efficiency of the proposed method.Table 1
Thermal power system in the YPG.
Plant number
Units
Capacity (MW)
(number∗ capacity, MW)
1
4∗ 600
2400
2
6∗ 300
1800
3
4∗ 300
1200
4
2∗ 200 + 2 ∗ 300
1000
5
2∗ 300
600
6
2∗ 300
600
7
2∗ 300
600
8
2∗ 300
600
9
1∗ 135
135Two typical months, May and September, were selected to demonstrate the practicality and the efficiency of the method. May represents the beginning of the flood season, when the hydropower plants can operate at their peak and the demand on the thermal system can be decreased. Conversely, September represents the beginning of the dry season, when the hydropower system has the lowest output and the demand on the thermal system increases.According to the real-world operating experience and users’ actual demands, the basic parameters for the HSA are set as follows:(i)
minimum load factorrmin⁡=0.7;(ii)
maximum load factorrmax⁡=0.9;(iii)
initial load factorr=0.8;(iv)
actual operating hours of installed capacity in the warm-up periodHkp=0 (k=1~9);(v)
extra hoursHke=0 (k=1~9);(vi)
minimum number of unitsuktmin⁡=1 (k=1~8, t=1~T), and uktmin⁡=0 for k=9;(vii)
maximum number of unitsuk,tmax⁡=UAk,tmax⁡ (k=1~9, t=1~T), where UAk,tmax⁡ is the number of available units for plant k in period t;(viii)
minimum duration of peak in plant outputTkpeak,min⁡=7 days (k=1~9);(ix)
minimum duration of valley in plant outputTkvalley,min⁡=3 days (k=1~9);(x)
considering previous periodsdk=7 days (k=1~9).The average cost time of 10 runs for May is 9764 ms and for September is 10241 ms, and the calculation efficiency can be met the actual demand in midterm scheduling.In a comparison of the results obtained by the HSA with those obtained by the heuristic, the VSA and the LSA are shown in Table2. As indicated by the table, the results improve progressively in going from the heuristic to the HSA for most of the evaluation criteria, including the average, the max-min difference, and the objective value, for both May and September. It should be noted that the most improved solutions were achieved by the VSA, which demonstrated that the VSA has the capability to find the optimal solution. The LSA attempts to improve the objective function value and avoid convergence to a local minimum. As shown in Table 2, Plant 1 was changed in May by the LSA. Although the max-min difference remained the same, the objective function was diminished from 50.06 to 36.4. There is little difference in the results obtained between September and May. Due to the adjustments to the generating units in the valley periods by the LSA, the max-min difference was greater, but the objective function decreased from 10.81 to 7.17. Equal values for all three evaluation criteria were attained for May and September using the HSA, which indicates that the HSA has the capability to find the global minimum.Table 2
Comparison of three methods for two typical months.
May
September
Heuristic
VSA
LSA
HSA
Heuristic
VSA
LSA
HSA
Operating hours of installed capacity (h)
Plant 1
475.2
484.8
499.2
499.2
264
288
288
288
Plant 2
499.2
499.2
499.2
499.2
304
284.8
284.8
288
Plant 3
475.2
499.2
499.2
499.2
244.8
278.4
292.8
288
Plant 4
499.2
499.2
499.2
499.2
268.8
282.2
282.2
288
Plant 5
489.6
499.2
499.2
499.2
288
288
288
288
Plant 6
499.2
499.2
499.2
499.2
288
288
288
288
Plant 7
499.2
499.2
499.2
499.2
288
288
288
288
Plant 8
489.6
499.2
499.2
499.2
288
288
288
288
Plant 9
480
480
480
499.2
268.8
288
288
288
Average (h)
489.6
495.5
497.1
499.2
278.04
285.9
287.5
288
Max-min difference (h)
24
19.2
19.2
0
59.2
9.6
10.6
0
Objective value (h2)
97.28
50.06
36.4
0
282.61
10.81
7.17
0The generation requirements for the 9 plants in May are shown in Figure5 and the details of results are listed in Table 3. Meanwhile, the generation requirements and details in September are shown in Figure 6 and Table 4. From Tables 3 and 4, it can be seen that the boot capacity result of each plant can meet all constraints. In Plant 2 in Table 3, for example, the days of first peak in plant output are 8 (from 7th to 14th) which is bigger than minimum duration of peak in plant output Tkpeak,min⁡=7 days, while the days of first valley in plant output are 4 (from 17th to 20th) which is bigger than minimum duration of valley in plant output Tkvalley,min⁡=3 days. Figure 5 shows that the outputs of most of the plants by the end of May were reduced to take advantage of the hydropower system. Plant 4 appears to have been an exception to this trend in May and could not satisfy the peak time constraint by the end of May, but this result is misleading. Table 1 shows that Plant 4 has two types of turbine units, 2 units of 200 MW capacity and 2 units of 300 MW capacity. As described in Section 3.1, Plant 4 was divided into two “virtual plants” for computational purposes. Table 3 and Figure 7 show the generation requirements for “virtual plants” 1 and 2 and demonstrate that these plants exhibit the same performance as the other plants. Conversely, the outputs of all of the plants increased from the beginning of September. Figures 5 and 6 demonstrate that the MTSFTPP results for May and September are reasonable.Table 3
Results of generation requirements for Plants 1 to 9 in May (MW).
Date
Plant 1
Plant 2
Plant 3
Plant 4
Plant 5
Plant 6
Plant 7
Plant 8
Plant 9
Virtual plant 1
Virtual plant 2
Total
1st
2400
1500
1200
400
600
1000
300
300
300
300
135
2nd
2400
1500
1200
400
600
1000
600
300
300
300
135
3rd
2400
1500
900
200
600
800
600
600
600
300
135
4th
2400
1500
900
200
600
800
600
600
600
600
135
5th
2400
1500
900
200
600
800
600
600
600
600
135
6th
2400
1500
1200
400
600
1000
600
600
600
600
0
7th
2400
1800
1200
400
600
1000
600
600
600
600
0
8th
2400
1800
1200
400
600
1000
600
600
600
600
0
9th
2400
1800
1200
400
600
1000
600
600
600
600
135
10th
2400
1800
1200
400
300
700
600
300
300
600
135
11th
2400
1800
1200
400
300
700
600
300
300
600
135
12th
2400
1800
1200
400
300
700
600
300
300
600
135
13th
1800
1800
1200
400
600
1000
600
600
600
600
135
14th
1800
1800
900
400
600
1000
600
600
600
600
135
15th
1800
1500
900
400
600
1000
600
600
600
600
135
16th
2400
1200
900
400
600
1000
300
600
600
600
135
17th
2400
900
1200
200
600
800
300
600
600
600
135
18th
2400
900
1200
200
600
800
300
600
600
300
135
19th
2400
900
1200
200
600
800
300
600
600
300
135
20th
2400
900
1200
400
300
700
600
300
300
300
135
21st
2400
1200
1200
400
300
700
600
300
300
300
135
22nd
2400
1200
1200
400
300
700
600
300
300
300
135
23rd
2400
1200
1200
400
300
700
600
300
300
600
135
24th
1200
1800
300
400
600
1000
600
600
600
600
135
25th
600
1800
300
400
600
1000
600
600
600
600
135
26th
600
1800
300
400
600
1000
600
600
600
600
135
27th
600
1800
900
200
600
800
300
600
600
600
135
28th
1200
1800
900
0
600
600
300
600
600
600
135
29th
1200
1800
900
0
600
600
300
600
600
600
135
30th
1200
1800
900
0
600
600
300
600
600
300
0
31st
2400
900
900
400
300
700
300
300
300
300
0Table 4
Results of generation requirements for Plants 1 to 9 in September.
Date
Plant 1
Plant 2
Plant 3
Plant 4
Plant 5
Plant 6
Plant 7
Plant 8
Plant 9
1st
600
300
300
300
300
300
300
300
135
2nd
600
300
300
600
300
300
300
300
0
3rd
600
300
600
600
300
300
300
300
0
4th
600
300
600
600
300
300
300
300
0
5th
600
600
600
600
300
300
300
300
0
6th
600
600
600
600
300
300
300
300
135
7th
1200
600
600
600
300
300
300
300
135
8th
1200
600
600
600
300
300
300
300
135
9th
1200
600
600
600
300
300
300
300
135
10th
1200
900
600
600
300
300
300
300
135
11th
1200
900
600
600
300
300
300
300
135
12th
1800
900
600
600
300
300
300
300
135
13th
1800
900
600
600
300
300
300
300
0
14th
1800
900
600
600
300
300
300
300
0
15th
1800
900
600
600
300
300
300
300
0
16th
1800
900
600
600
300
300
300
300
0
17th
1800
900
600
600
300
300
300
300
0
18th
1800
900
600
600
300
300
300
300
0
19th
1800
900
600
600
300
300
300
300
0
20th
1800
900
600
600
300
300
300
300
135
21st
1200
900
600
600
300
300
300
300
135
22nd
600
1200
600
600
300
300
300
300
135
23rd
600
1500
600
600
300
300
300
300
135
24th
600
1500
600
600
300
300
300
300
135
25th
1200
1500
600
600
300
300
300
300
135
26th
1200
1500
600
600
300
300
300
300
135
27th
1200
1500
600
600
300
300
300
300
0
28th
1200
1500
600
600
300
300
300
300
0
29th
1200
1500
600
600
300
300
300
300
0
30th
1200
300
1200
900
300
300
300
300
0Figure 5
The process of generation requirements for Plants 1 to 9 in May.Figure 6
The process of generation requirements for Plants 1 to 9 in September.Figure 7
The process of generation requirements for “virtual plants” 1 and 2.Figures8 and 9 show the generation and capacity requirements in May and September, respectively. It can be observed that the generation requirements are between maximum and minimum capacity.Figure 8
Generation and capacity requirements in May.Figure 9
Generation and capacity requirements in September.


## 5. Conclusions

A model for the MTSFTPP problem was proposed. The model includes three stages. The feasible spaces for the MTSFTPP problem are first produced by narrowing the search space based on constraints 1 and 2. Next, an initial feasible solution for the MTSFTPP problem is obtained using virtual plants and considering constraints 3 and 4. The POA is used to solve the MTSFTPP problem, and the optimal search is referred to as the VSA. To avoid convergence to a local minimum, the LSA is used, and the updated solution is used as a new feasible starting point for the next search with the VSA. Thus, the HSA is produced by combining the LSA with the VSA. A complete framework for solving the MTSFTPP problem was established and demonstrated using a case study. The results showed that the proposed method can efficiently produce satisfactory solutions with practical requirements.A novel method focusing on solving strategies for the MTSFTPP problem was proposed, in which the thermal system load was given after the other electric sources were optimized. However, the joint modeling of thermal power system with other source systems in midterm schedule can be considered as a research basis for future studies.



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*Source: 710876-2015-05-13.xml*