# Multiobjective CVaR Optimization Model and Solving Method for Hydrothermal System Considering Uncertain Load Demand

Authors: Zhongfu Tan; Liwei Ju; Huanhuan Li; Chao Qin; Daoxin Peng
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

In order to solve the influence of load uncertainty on hydrothermal power system operation and achieve the optimal objectives of system power generation consumption, pollutant emissions, and first-stage hydropower station storage capacity, this paper introduced CVaR method and built a multiobjective optimization model and its solving method. In the optimization model, load demand’s actual values and deviation values are regarded as random variables, scheduling objective is redefined to meet confidence level requirement and system operation constraints and loss function constraints are taken into consideration. To solve the proposed model, this paper linearized nonlinear constraints, applied fuzzy satisfaction, fuzzy entropy, and weighted multiobjective function theories to build a fuzzy entropy multiobjective CVaR model. The model is a mixed integer linear programming problem. Then, six thermal power plants and three cascade hydropower stations are taken as the hydrothermal system for numerical simulation. The results verified that multiobjective CVaR method is applicable to solve hydrothermal scheduling problems. It can better reflect risk level of the scheduling result. The fuzzy entropy satisfaction degree solving algorithm can simplify solving difficulty and get the optimum operation scheduling scheme.


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## Body

## 1. Introductions

Electric power system scheduling means when load demand is met, units’ power generation assignment is allocated according to minimized energy consumption or power generation cost goal, which takes system operation output, power climbing, backup service, and other constraints into consideration [1, 2]. Thermal power and hydro power are the primary power types in China. The hydrothermal power joint scheduling can achieve energy-saving requirements and reduce system power generation’s average cost [3]. So the study on hydrothermal power joint scheduling has important theoretical value and practical significance.Currently, many researchers have already studied on hydrothermal power joint scheduling optimization problems. Existing literatures mainly focus on 3 aspects, namely, inflow forecast, scheduling model, and solving method building. On aspect of inflow forecast, it generally uses load demand and inflow historical data and classical forecast methods [4, 5], modern forecast methods [6–8], and intelligence forecast methods [9–11] to forecast load demand and inflow in the next period of time. However, during practical applications the methods above can get quite accurate forecast results but their forecast accuracy can hardly meet system scheduling requirements.As for hydrothermal power joint scheduling optimization model, there are two research priorities, namely, uncertain problems and environmental requirements. Because of the uncertainty of inflow and load demand, units’ output should be allocated according to load forecast results. So the core of the scheduling model is prior-decision. Generally, fuzzy decision theory [12], random optimization theory [13], opportunity theory [14], and other theories are introduced to build optimization model and solve prior-decision problems. Influenced by environmental pressure, minimized pollutant emission has become another optimize point [15], which makes system scheduling optimization problems transforming from single objective to multiobjective.The introduction of uncertain factors and new scheduling objectives makes scheduling models becoming more complex. Multivariable, multiconstraint, multiobjective, nonconvexity, and nonlinear problems need corresponding algorithms to get ideal scheduling result. In existing literatures there are conventional solving algorithms and intelligent solving methods, such as calculus of variations, dynamic planning, differential evolution algorithm, genetic algorithms, and artificial neural network algorithm [16–20]. However, conventional algorithms usually have problems in defining parameters, transferring constraints that they could hardly achieve high optimization degree [21]. Intelligent algorithms can overcome those shortcomings. But when individual’s extreme values do not meet multiobjective planning principles, intelligent algorithms are easy to fall into local extreme point and the algorithm’s searching capability is limited [22].According to the above overview, existing researches about hydrothermal joint scheduling are mainly concentrated in model building when uncertain factors are considered, and few on risk analysis, prevention, and control about what the uncertain factors would bring to the system. In China, hydropower resources are mainly concentrated in southern regions. Reservoir storage capacity is quite big and inflow uncertainty is small [23]. Therefore this paper does not take inflow uncertainty into consideration and only considers load demand uncertainty.The rest of the paper is organized as follows. In Section2, the basic mathematic model is construct which take the minimum power generation cost, the minimum pollutant emission, and the maximum storage capacity of the hydropower station as the optimization objectives. Section 3 introduces the multiobjective CVaR method and built a hydrothermal joint scheduling optimization model. In order to solve the proposed model in Section 3, Section 4 linearizes nonlinear constraints, used fuzzy satisfaction degree theory to blur the objective function, and used entropy theory to weight the multiobjective function. The model was transferred into a fuzzy entropy multiobjective CVaR model. In Section 5, six thermal plants and three cascade hydropower stations are taken as the simulation system to verify the effectiveness and feasibility of the proposed model and solving method. Section 6 highlights the main conclusions of the paper.


## 2. Basic Mathematic Model

### 2.1. Objective Functions

Currently, the objectives functions of the hydrothermal system scheduling optimization model consist of minimized power generation cost, minimized pollutant emission, and maximized storage capacity of the hydropower station.(1) Minimized power generation energy consumption objective function is as follows:(1)min⁡f1=∑t=1T∑i=1Iui,tfcgi,t+ui,t1-ui,t-1Di,t,(2)fcgi,t=ai+bigi,t+cigi,t2,wherein f1 is system’s power generation energy consumption cost; I is the number of thermal units; T is the cycle of power generation scheduling, generally 24 hours; ui,t is an 0-1 variable; and ui,t=1 means the unit i is in operating; otherwise it is not in operating. gi,t is the output of unit i at time t; ai, bi, and ci are fuel cost parameters of unit i, which can be calculated according to historical data of the unit. Di,t is the startup-shutdown cost of thermal unit i at time t:(3)Di,t=Dih,Td,imin⁡<Tioff⁡t≤Hioff⁡Dic,Tioff⁡t>Hioff⁡,Hioff⁡=Td,imin⁡+Ts,ic,wherein Dic is the cold-start cost of unit i; Dih is the hot-start cost of unit i; Td,imin⁡ is the minimum allowed continuous downtime of unit i; Tioff⁡t is the continuous downtime of unit i at time t; Ts,ic is the cold-start time of unit i; Hioff⁡ is the summation of minimum downtime and cold-start time of unit i.(2) Minimized pollutant emission objective function is as follows:(4)min⁡f2=∑t=1T∑i=1I∑m=1Mui,tϖim+βimgi,t+γimgi,t2,wherein M is the number of pollutant types and ϖim, βim, and γim are pollutant emission parameters of unit i.(3) Maximized storage capacity of the first-level hydropower station is as follows:(5)max⁡f3=∑t=1T∑j=1JVjt,Vjt=Vj,t-1+Q⌢t-Q⌣t-Sjt+∑h=1HQh,t-τhj+Sh,t-τhj,wherein Vjt is the water storage capacity od hydropower station j at time t; Q⌢t is the inflow volume of the superiors hydropower station at time t; Q⌣t is the discharged flow of hydropower station j at time t; Sjt is the nature abandoned water of hydropower station j at time t; H is the number of the upstream hydropower units that directly connected with hydro unit j; τ is the flow delay between hydropower unit j and upstream hydropower unit h.


### 2.2. Constraints

(1) Load demand and supply balance constraint is as follows:(6)∑i=1Igi,tui,t+∑j=1Jgj,t-gtl=Lt,wherein Lt is the load demand at time t; gj,t is the output of hydropower station j at time t; gtl is system’s total power loss at time t.(2) Thermal units’ output constraint is as follows.Thermal units’ output constraints include upper and lower power generation limit constraint, power climbing limit constraint, the shortest start time constraint, and the shortest downtime constraint. The detailed constraint functions are (7)–(9). Consider the following:(7)uitgimin⁡≤git≤uitgimax⁡,wherein gimax⁡ and gimin⁡ are the power generation output upper and lower limit of unit i:(8)Δgi-≤git-gi,t-1≤Δgi+,wherein Δgci- and Δgci+ are power climbing limits:(9)Ti,t-1on⁡-Mion⁡ui,t-1-uit≥0,Ti,t-1off⁡-Mioff⁡uit-ui,t-1≥0,wherein Mion⁡ is the shortest startup time of thermal unit i; Ti,t-1on⁡ is the continuous operation time of unit i at time t-1; Mioff⁡ is the shortest downtime of unit i; Ti,t-1off⁡ is the continuous downtime of unit i at time t-1.(3) Hydropower units’ output constraints are as follows.Hydropower units’ output constraints mainly include units’ power generation flow constraint, storage capacity restrictions constraint, and dynamic water balance constraint. The detailed constraint functions are (10)–(12). Consider the following:(10)gj,t=c0j+c1jQjt+c2jVjt+c3jQjtVjt+c4jQjt2+c5jVjt2,wherein Qjt is the flow of hydropower unit j at time t; Vjt is the storage capacity of hydropower unit j at time t; c0j, c1j, c2j, c3j, c4j, and c5j are output parameters of hydropower unit j, which can be calculated according to the historical data of the unit:(11)Qjmin⁡≤Qjt≤Qjmax⁡,Vjmin⁡≤Vjt≤Vjmax⁡,wherein Qjmin⁡ and Qjmax⁡ are, respectively, the minimum and maximum power generation flow of hydropower unit j; Vjmin⁡ and Vjmax⁡ are, respectively, the lower and upper storage capacity limit of hydropower unit j:(12)VjT=Vj0+∑t=1TQ⌢t-Q⌣t-Sjt+∑h=1HQh,t-τhj+Sh,t-τhj,Vjtt=0=Vj0,Vjtt=T=VjT,wherein Vj0 and VjT are, respectively, storage capacities at the initial time and end time of hydropower unit j.


## 2.1. Objective Functions

Currently, the objectives functions of the hydrothermal system scheduling optimization model consist of minimized power generation cost, minimized pollutant emission, and maximized storage capacity of the hydropower station.(1) Minimized power generation energy consumption objective function is as follows:(1)min⁡f1=∑t=1T∑i=1Iui,tfcgi,t+ui,t1-ui,t-1Di,t,(2)fcgi,t=ai+bigi,t+cigi,t2,wherein f1 is system’s power generation energy consumption cost; I is the number of thermal units; T is the cycle of power generation scheduling, generally 24 hours; ui,t is an 0-1 variable; and ui,t=1 means the unit i is in operating; otherwise it is not in operating. gi,t is the output of unit i at time t; ai, bi, and ci are fuel cost parameters of unit i, which can be calculated according to historical data of the unit. Di,t is the startup-shutdown cost of thermal unit i at time t:(3)Di,t=Dih,Td,imin⁡<Tioff⁡t≤Hioff⁡Dic,Tioff⁡t>Hioff⁡,Hioff⁡=Td,imin⁡+Ts,ic,wherein Dic is the cold-start cost of unit i; Dih is the hot-start cost of unit i; Td,imin⁡ is the minimum allowed continuous downtime of unit i; Tioff⁡t is the continuous downtime of unit i at time t; Ts,ic is the cold-start time of unit i; Hioff⁡ is the summation of minimum downtime and cold-start time of unit i.(2) Minimized pollutant emission objective function is as follows:(4)min⁡f2=∑t=1T∑i=1I∑m=1Mui,tϖim+βimgi,t+γimgi,t2,wherein M is the number of pollutant types and ϖim, βim, and γim are pollutant emission parameters of unit i.(3) Maximized storage capacity of the first-level hydropower station is as follows:(5)max⁡f3=∑t=1T∑j=1JVjt,Vjt=Vj,t-1+Q⌢t-Q⌣t-Sjt+∑h=1HQh,t-τhj+Sh,t-τhj,wherein Vjt is the water storage capacity od hydropower station j at time t; Q⌢t is the inflow volume of the superiors hydropower station at time t; Q⌣t is the discharged flow of hydropower station j at time t; Sjt is the nature abandoned water of hydropower station j at time t; H is the number of the upstream hydropower units that directly connected with hydro unit j; τ is the flow delay between hydropower unit j and upstream hydropower unit h.


## 2.2. Constraints

(1) Load demand and supply balance constraint is as follows:(6)∑i=1Igi,tui,t+∑j=1Jgj,t-gtl=Lt,wherein Lt is the load demand at time t; gj,t is the output of hydropower station j at time t; gtl is system’s total power loss at time t.(2) Thermal units’ output constraint is as follows.Thermal units’ output constraints include upper and lower power generation limit constraint, power climbing limit constraint, the shortest start time constraint, and the shortest downtime constraint. The detailed constraint functions are (7)–(9). Consider the following:(7)uitgimin⁡≤git≤uitgimax⁡,wherein gimax⁡ and gimin⁡ are the power generation output upper and lower limit of unit i:(8)Δgi-≤git-gi,t-1≤Δgi+,wherein Δgci- and Δgci+ are power climbing limits:(9)Ti,t-1on⁡-Mion⁡ui,t-1-uit≥0,Ti,t-1off⁡-Mioff⁡uit-ui,t-1≥0,wherein Mion⁡ is the shortest startup time of thermal unit i; Ti,t-1on⁡ is the continuous operation time of unit i at time t-1; Mioff⁡ is the shortest downtime of unit i; Ti,t-1off⁡ is the continuous downtime of unit i at time t-1.(3) Hydropower units’ output constraints are as follows.Hydropower units’ output constraints mainly include units’ power generation flow constraint, storage capacity restrictions constraint, and dynamic water balance constraint. The detailed constraint functions are (10)–(12). Consider the following:(10)gj,t=c0j+c1jQjt+c2jVjt+c3jQjtVjt+c4jQjt2+c5jVjt2,wherein Qjt is the flow of hydropower unit j at time t; Vjt is the storage capacity of hydropower unit j at time t; c0j, c1j, c2j, c3j, c4j, and c5j are output parameters of hydropower unit j, which can be calculated according to the historical data of the unit:(11)Qjmin⁡≤Qjt≤Qjmax⁡,Vjmin⁡≤Vjt≤Vjmax⁡,wherein Qjmin⁡ and Qjmax⁡ are, respectively, the minimum and maximum power generation flow of hydropower unit j; Vjmin⁡ and Vjmax⁡ are, respectively, the lower and upper storage capacity limit of hydropower unit j:(12)VjT=Vj0+∑t=1TQ⌢t-Q⌣t-Sjt+∑h=1HQh,t-τhj+Sh,t-τhj,Vjtt=0=Vj0,Vjtt=T=VjT,wherein Vj0 and VjT are, respectively, storage capacities at the initial time and end time of hydropower unit j.


## 3. Multiobjective CVaR Hydrothermal Joint Scheduling Optimization Model

### 3.1. Description of Load Demand Uncertainty

When making hydrothermal power generation scheduling decision, the decision maker should allocate power generation assignment according to the forecast value of load demand. The forecast value is calculated based on historical data. And if load demand uncertainty is considered, the load demand and supply balance can be described as(13)∑i=1Igi,tui,t+∑j=1Jgj,t-gtl≥Lt,wherein L~t is the forecasted value of load demand.The relationship between the actual value and forecasted value is(14)Lt=L~t+et,wherein et is the deviation value between the actual value and forecasted value, which, according to literature [24], obeys normal distribution et~Nθt,σt2. θt and σt are, respectively, the average value and standard deviation of et. Distribution law of et can be calculated according to historical data.When the load demand uncertainty exists, there are two possible scenarios. When load demand forecasted value is higher than the actual value, some units may be forced to shut down. And when the forecasted value is lower than the actual value, the hydropower may not have full access to the grid, which would bring scheduling loss to the system. Load demand uncertainty can directly influence system’s stability and bring corresponding risks. Therefore its necessary to use corresponding risk analysis method to study on the optimized decision method when uncertain factors are took into consideration.


### 3.2. Multiobjective CVaR Method

VaR method has a shortcoming that it can be applied to measure risk situation only when the confidence level is determined. CVaR method can overcome the shortcoming of VaR and describe risk distribution beyond the confidence. However, in practical CVaR method can only solve combinatorial optimization problems with single loss function. When system has multiple loss functions, CVaR need be expanded to multiobjective CVaR method [25]. Assume there is a loss function fkx,ξ∈Rn×Rm→R1 (k=1,2,…,K) that is related with decision variable x∈X⊂Rn. The loss function is a continuous function. ξ is a continuous random variable and its probability density function is pz. Then the distribution function of fkx,ξ is(15)ψkx,yk=Pfkx,ξ≤yk∫fkx,z≤ykpzdz.Defineλii=1,2,…,n as weights of each loss function and α-VaR loss value of decision variable x under confidence level α and weight λ is(16)y∗x,λ=min⁡y∣∑i=1Kλk1-αk-1ψkx,ξ≥∑k=1Kλkαk1-αk-1.Introduce loss functionϕk,αkx,yk:(17)ϕk,αkx,yk=1-αk-1∫fkx,z≥ykfx,zpzdz.Then the optimumα-CVaR loss value ϕx,y∗x,λ of decision variable x under confidence level α and weight λ is(18)ϕx,y∗x,λ=∑i=1nλiϕi,αix,y∗x,λ,wherein αi is the confidence levels of loss function i.To find the minimumα-CVaR loss value, ϕx,y∗x,λ should achieve the minimum value x on X. And the optimization problem can be defined as(19)P1minϕx,y∗x,λ=∑k=1Kλkϕk,αkx,y∗x,λs.t.x∈X.To simplify the solving progress of model (P1), this paper redefined loss functions and their optimization problems. The detailed model is model (P2):(20)Fk,αkx,yk=yk+1-αk-1∫z∈Rmfkx,z-yk+pzdz,P2min∑k=1KλkFk,αkx,ys.t.y∈R,x∈X.The equivalent problem of model (P2) is as follows: for anyx, if min⁡∑k=1KλkFk,αkx,y has a unique and optimal solution y. y meets the following constraint:(21)Pπkx,y=ξ=∫πkx,z=ξpzdz=0.Then, the following conclusion can be gotten:(22)∑i=1nλkϕk,akx,y=minξ∈R⁡∑k=1KλkFk,αkx,y,y=y∗x,λ.For any determined weight valueλ, if x,y could make model (P2) achieve the optimum and (2) is tenable, then x could make model (P1) achieve the optimum.


### 3.3. Multiobjective CVaR Scheduling Model

When load demand uncertainty is considered, load deviation would be undertaken by thermal units and hydro units. Transfer system scheduling objective functions referring to the literature [24] and get the new objective function, which take load demand uncertainty into consideration:(23)min⁡f1=∑t=1T∑i=1Iui,tfcgi,t+φiet+ui,t1-ui,t-1Di,t,min⁡f2=∑t=1T∑i=1I∑m=1Mui,tϖim+βimgi,t+φiet+γimgi,t+φiet2,max⁡f3=∑t=1T∑j=1JVjt′,Vjt′=Vj,t-1+Q⌢t-Qt-Sjt-φjetρj+∑h=1HQh,t-τhj+Sh,t-τhj,wherein φi and φj are, respectively, task proportions of thermal units and hydro units and ρj is the output power coefficient of hydro units.After considering uncertainties, traditional constraints are needed to converse to uncertainty constraints, as follows:(24)φiet+Δgi-≤git-gi,t-1≤Δgi+-φiet,(25)gj,t+φjet=c0j+c1jQjt+c2jVjt′+c3jQjtVjt′+c4jQjt2+c5jVjt′2.Further, multiobjective CVaR approach is used to build the hydrothermal optimization scheduling model. First, define a schedule loss function, as follows:(26)Fk,αkG,yk=yk+1-αk-1∫z∈RmfkG,e-yk+pede,k=1,2,3,wherein G is units output set of the system; αk is the confidence level requirement of objective function fk; yk is decision maker’s expected value of objective function fk.Then build a hydrothermal joint scheduling optimization model which sets minimizedα-CVaR loss value as its objective:(27)min⁡Fk,αkG,yk,k=1,2max⁡F3,α3G,y3s.t.∑i=1Igi,tui,t+∑j=1Jgj,t-gtl≥L~t+et1–14,24,25G=git,gjti=1,2,…,n;j=1,2,…,J.


## 3.1. Description of Load Demand Uncertainty

When making hydrothermal power generation scheduling decision, the decision maker should allocate power generation assignment according to the forecast value of load demand. The forecast value is calculated based on historical data. And if load demand uncertainty is considered, the load demand and supply balance can be described as(13)∑i=1Igi,tui,t+∑j=1Jgj,t-gtl≥Lt,wherein L~t is the forecasted value of load demand.The relationship between the actual value and forecasted value is(14)Lt=L~t+et,wherein et is the deviation value between the actual value and forecasted value, which, according to literature [24], obeys normal distribution et~Nθt,σt2. θt and σt are, respectively, the average value and standard deviation of et. Distribution law of et can be calculated according to historical data.When the load demand uncertainty exists, there are two possible scenarios. When load demand forecasted value is higher than the actual value, some units may be forced to shut down. And when the forecasted value is lower than the actual value, the hydropower may not have full access to the grid, which would bring scheduling loss to the system. Load demand uncertainty can directly influence system’s stability and bring corresponding risks. Therefore its necessary to use corresponding risk analysis method to study on the optimized decision method when uncertain factors are took into consideration.


## 3.2. Multiobjective CVaR Method

VaR method has a shortcoming that it can be applied to measure risk situation only when the confidence level is determined. CVaR method can overcome the shortcoming of VaR and describe risk distribution beyond the confidence. However, in practical CVaR method can only solve combinatorial optimization problems with single loss function. When system has multiple loss functions, CVaR need be expanded to multiobjective CVaR method [25]. Assume there is a loss function fkx,ξ∈Rn×Rm→R1 (k=1,2,…,K) that is related with decision variable x∈X⊂Rn. The loss function is a continuous function. ξ is a continuous random variable and its probability density function is pz. Then the distribution function of fkx,ξ is(15)ψkx,yk=Pfkx,ξ≤yk∫fkx,z≤ykpzdz.Defineλii=1,2,…,n as weights of each loss function and α-VaR loss value of decision variable x under confidence level α and weight λ is(16)y∗x,λ=min⁡y∣∑i=1Kλk1-αk-1ψkx,ξ≥∑k=1Kλkαk1-αk-1.Introduce loss functionϕk,αkx,yk:(17)ϕk,αkx,yk=1-αk-1∫fkx,z≥ykfx,zpzdz.Then the optimumα-CVaR loss value ϕx,y∗x,λ of decision variable x under confidence level α and weight λ is(18)ϕx,y∗x,λ=∑i=1nλiϕi,αix,y∗x,λ,wherein αi is the confidence levels of loss function i.To find the minimumα-CVaR loss value, ϕx,y∗x,λ should achieve the minimum value x on X. And the optimization problem can be defined as(19)P1minϕx,y∗x,λ=∑k=1Kλkϕk,αkx,y∗x,λs.t.x∈X.To simplify the solving progress of model (P1), this paper redefined loss functions and their optimization problems. The detailed model is model (P2):(20)Fk,αkx,yk=yk+1-αk-1∫z∈Rmfkx,z-yk+pzdz,P2min∑k=1KλkFk,αkx,ys.t.y∈R,x∈X.The equivalent problem of model (P2) is as follows: for anyx, if min⁡∑k=1KλkFk,αkx,y has a unique and optimal solution y. y meets the following constraint:(21)Pπkx,y=ξ=∫πkx,z=ξpzdz=0.Then, the following conclusion can be gotten:(22)∑i=1nλkϕk,akx,y=minξ∈R⁡∑k=1KλkFk,αkx,y,y=y∗x,λ.For any determined weight valueλ, if x,y could make model (P2) achieve the optimum and (2) is tenable, then x could make model (P1) achieve the optimum.


## 3.3. Multiobjective CVaR Scheduling Model

When load demand uncertainty is considered, load deviation would be undertaken by thermal units and hydro units. Transfer system scheduling objective functions referring to the literature [24] and get the new objective function, which take load demand uncertainty into consideration:(23)min⁡f1=∑t=1T∑i=1Iui,tfcgi,t+φiet+ui,t1-ui,t-1Di,t,min⁡f2=∑t=1T∑i=1I∑m=1Mui,tϖim+βimgi,t+φiet+γimgi,t+φiet2,max⁡f3=∑t=1T∑j=1JVjt′,Vjt′=Vj,t-1+Q⌢t-Qt-Sjt-φjetρj+∑h=1HQh,t-τhj+Sh,t-τhj,wherein φi and φj are, respectively, task proportions of thermal units and hydro units and ρj is the output power coefficient of hydro units.After considering uncertainties, traditional constraints are needed to converse to uncertainty constraints, as follows:(24)φiet+Δgi-≤git-gi,t-1≤Δgi+-φiet,(25)gj,t+φjet=c0j+c1jQjt+c2jVjt′+c3jQjtVjt′+c4jQjt2+c5jVjt′2.Further, multiobjective CVaR approach is used to build the hydrothermal optimization scheduling model. First, define a schedule loss function, as follows:(26)Fk,αkG,yk=yk+1-αk-1∫z∈RmfkG,e-yk+pede,k=1,2,3,wherein G is units output set of the system; αk is the confidence level requirement of objective function fk; yk is decision maker’s expected value of objective function fk.Then build a hydrothermal joint scheduling optimization model which sets minimizedα-CVaR loss value as its objective:(27)min⁡Fk,αkG,yk,k=1,2max⁡F3,α3G,y3s.t.∑i=1Igi,tui,t+∑j=1Jgj,t-gtl≥L~t+et1–14,24,25G=git,gjti=1,2,…,n;j=1,2,…,J.


## 4. Entropy Fuzzy Satisfaction Degree Solving Method

According to (25), solve hydrothermal joint scheduling optimization model that needs two key elements, namely, objective functions’ weights and the optimum acceptable degree yk of each objective. This section puts forward entropy fuzzy satisfaction degree solving model. Entropy theory is used to get weight for each objective, and fuzzy satisfaction degree is used to simplify objective functions and get the optimum yk.


### 4.1. Linearization Processing

According to (1) and (2), when an objective function is a quadratic function, the objective function should be simplified linearization processing. Divide thermal units’ power limits [gimin,gimax] into N segments. Then function can be expressed as a piecewise function. When gi,t∈[gimin+nΔ,gimin+(n+1)Δ],(28)fc′git=fc′gimin⁡+nΔ+git-gimin⁡-nΔbi+2n+1ciΔ+2cigimin⁡,wherein n=0,1,…,N-1, Δ=(gimax-gimin)/N. According to the literature [26], when N≥5, the maximum relative error of objective function should be less than 1%, generally less than 0.5%, and in segmented approaching situation the error is approaching to 0. Figure 1 shows the linearization progress of a quadratic function.Figure 1
Linear processing of quadratic function.


### 4.2. Blurring Processing

The optimization direction of the first level cascade hydropower station objective function is maximized, while the other two are minimized. Therefore this section introduces declining semigradient membership degree function and rising semilinear membership degree function to blur the objectives. The detailed progress is as follows.(1) Input the initial data and solve each single optimization objective, respectively, to get functions’ value of different optimization objectives. Form a decision properties table of the objective functions, as shown in Table1.Table 1
The input-output tables of objective function.
Objective function
F
1
F
2
F
3
F
1
∗
F
1
min
⁡
F
1
(
2
)
F
1
(
3
)
F
2
∗
F
2
(
1
)
F
2
min
⁡
F
2
(
3
)
F
3
∗
F
3
(
1
)
F
3
(
2
)
F
3
max
⁡
Note: objective with∗ means the objective is the optimization objective.(2) Scale each objective according to decision makers’ wishes and preferences, and defineF1max, F2max, and F3min and their value range. The equations are(29)0≤F1max⁡-F1min⁡≤F12-F1min⁡,F13-F1min⁡,0≤F2max⁡-F2min⁡≤F21-F2min⁡,F23-F2min⁡,F3max⁡-F3min⁡≥F3max⁡-F31,F3max⁡-F32.(3) Use declining semigradient membership degree function to deal with minimized power generation cost objective function and minimized pollutant emission objective function, and use rising semilinear membership degree function to deal with maximized storage capacity objective function. Then membership degree function could be defined:(30)πifi=0,Fi≥Fimax⁡F3max⁡-F3F3max⁡-Fmin,Fimin⁡<Fi<Fimax⁡1,Fi≤Fimin⁡,π3f3=1,F3≥F3max⁡F3-F3min⁡F3max⁡-Fmin,F3min⁡<F3<F3max⁡0,F3≤F3min⁡,wherein fimin⁡ and fimax⁡ are, respectively, the minimum and maximum value of the objective function fi·. πfi is the membership degree function of the objective function fi·.


### 4.3. Weight Allocation

Multiobjective CVaR optimization problems require rational weight allocation to transfer multiobjective optimization problem into single objective optimization problem. Weight allocation methods can be classified into two types, namely, subjective weighting methods and objective weighting methods. Compared with subjective weighting methods, objective weighting methods have an advantage of avoiding influences from subjective experiences by directly calculating weights according to objective information. Entropy method is an objective weighting method. Entropy method calculates evaluation indexes’ entropy and weight according to evaluation objectives’ actual data. Because of its strong objectivity and simple calculation progress, entropy method is wide and used in many comprehensive evaluation problems. Based on Table1, this section uses entropy method to calculate weight of each objective. Detailed introduction of the entropy meth refers to the literature [26], and the following is the calculation progress.(1) Calculate entropyEk of objective function k:(31)Ek=-ψ∑i=1nrijln⁡rij,wherein ψ=1/ln⁡n is a constant related with the specimen, and it was defined to make Ek∈0,1; rij meets 0<rij<1 and ∑i=1nrij=1, when rij=0, rijln⁡rij=0.(2) Information deviation degree of objective function is as follows:(32)dk=1-Ek.(3) Calculate weights of the objectives:(33)λk=dk∑k=1Kdk=1-EkK-∑k=1KEk.


### 4.4. Entropy Fuzzy Multiobjective CVaR Model

Assume the allocated weight and membership degree of each objective areπkfk and λk. If π∗ is the satisfaction degree of πkfk,(34)π∗=min⁡πkfk,k=1,2,3.According to the maximum and minimum principles of fuzzy set theory, multiobjective CVaR model could be transferred:(35)min⁡∑k=13λkπkfks.t.π1f1≥π∗π2f2≥π∗π3f3≥π∗0≤π∗≤1∑i=1Igi,tui,t+∑j=1Jgj,t-gtl≥L~t+et1–14,24,25i=1,2,…,n;j=1,2,…,m.


## 4.1. Linearization Processing

According to (1) and (2), when an objective function is a quadratic function, the objective function should be simplified linearization processing. Divide thermal units’ power limits [gimin,gimax] into N segments. Then function can be expressed as a piecewise function. When gi,t∈[gimin+nΔ,gimin+(n+1)Δ],(28)fc′git=fc′gimin⁡+nΔ+git-gimin⁡-nΔbi+2n+1ciΔ+2cigimin⁡,wherein n=0,1,…,N-1, Δ=(gimax-gimin)/N. According to the literature [26], when N≥5, the maximum relative error of objective function should be less than 1%, generally less than 0.5%, and in segmented approaching situation the error is approaching to 0. Figure 1 shows the linearization progress of a quadratic function.Figure 1
Linear processing of quadratic function.


## 4.2. Blurring Processing

The optimization direction of the first level cascade hydropower station objective function is maximized, while the other two are minimized. Therefore this section introduces declining semigradient membership degree function and rising semilinear membership degree function to blur the objectives. The detailed progress is as follows.(1) Input the initial data and solve each single optimization objective, respectively, to get functions’ value of different optimization objectives. Form a decision properties table of the objective functions, as shown in Table1.Table 1
The input-output tables of objective function.
Objective function
F
1
F
2
F
3
F
1
∗
F
1
min
⁡
F
1
(
2
)
F
1
(
3
)
F
2
∗
F
2
(
1
)
F
2
min
⁡
F
2
(
3
)
F
3
∗
F
3
(
1
)
F
3
(
2
)
F
3
max
⁡
Note: objective with∗ means the objective is the optimization objective.(2) Scale each objective according to decision makers’ wishes and preferences, and defineF1max, F2max, and F3min and their value range. The equations are(29)0≤F1max⁡-F1min⁡≤F12-F1min⁡,F13-F1min⁡,0≤F2max⁡-F2min⁡≤F21-F2min⁡,F23-F2min⁡,F3max⁡-F3min⁡≥F3max⁡-F31,F3max⁡-F32.(3) Use declining semigradient membership degree function to deal with minimized power generation cost objective function and minimized pollutant emission objective function, and use rising semilinear membership degree function to deal with maximized storage capacity objective function. Then membership degree function could be defined:(30)πifi=0,Fi≥Fimax⁡F3max⁡-F3F3max⁡-Fmin,Fimin⁡<Fi<Fimax⁡1,Fi≤Fimin⁡,π3f3=1,F3≥F3max⁡F3-F3min⁡F3max⁡-Fmin,F3min⁡<F3<F3max⁡0,F3≤F3min⁡,wherein fimin⁡ and fimax⁡ are, respectively, the minimum and maximum value of the objective function fi·. πfi is the membership degree function of the objective function fi·.


## 4.3. Weight Allocation

Multiobjective CVaR optimization problems require rational weight allocation to transfer multiobjective optimization problem into single objective optimization problem. Weight allocation methods can be classified into two types, namely, subjective weighting methods and objective weighting methods. Compared with subjective weighting methods, objective weighting methods have an advantage of avoiding influences from subjective experiences by directly calculating weights according to objective information. Entropy method is an objective weighting method. Entropy method calculates evaluation indexes’ entropy and weight according to evaluation objectives’ actual data. Because of its strong objectivity and simple calculation progress, entropy method is wide and used in many comprehensive evaluation problems. Based on Table1, this section uses entropy method to calculate weight of each objective. Detailed introduction of the entropy meth refers to the literature [26], and the following is the calculation progress.(1) Calculate entropyEk of objective function k:(31)Ek=-ψ∑i=1nrijln⁡rij,wherein ψ=1/ln⁡n is a constant related with the specimen, and it was defined to make Ek∈0,1; rij meets 0<rij<1 and ∑i=1nrij=1, when rij=0, rijln⁡rij=0.(2) Information deviation degree of objective function is as follows:(32)dk=1-Ek.(3) Calculate weights of the objectives:(33)λk=dk∑k=1Kdk=1-EkK-∑k=1KEk.


## 4.4. Entropy Fuzzy Multiobjective CVaR Model

Assume the allocated weight and membership degree of each objective areπkfk and λk. If π∗ is the satisfaction degree of πkfk,(34)π∗=min⁡πkfk,k=1,2,3.According to the maximum and minimum principles of fuzzy set theory, multiobjective CVaR model could be transferred:(35)min⁡∑k=13λkπkfks.t.π1f1≥π∗π2f2≥π∗π3f3≥π∗0≤π∗≤1∑i=1Igi,tui,t+∑j=1Jgj,t-gtl≥L~t+et1–14,24,25i=1,2,…,n;j=1,2,…,m.


## 5. Simulations and Analysis

This section chooses 6 thermal plants (T1–T6) and 3 cascade hydropower stations (H1–H3) as simulation base. Thermal plants’ basic parameters and comprehensive emission parameters refer to the literature [24]; cascade hydropower stations’ operation parameters are listed in Table 2. Assume load forecast error value obeys normal distribution e(t)~(0,(30)2), and both of T5 and T6 bear 20% of the forecast error. Choose a typical daily load distribution as load demand of the simulation, as listed in Table 3.Table 2
Basic parameters of cascade hydropower stations.
ID
Regulation characteristics
Normal water level/m
Normal capacity/(108 m3)
Dead water level/m
H1
Many years
780
83.95
731
H2
Run-off
645
0.26
637
H3
Run-off
112
4.02
110
ID
Dead storagecapacity/(108 m3)
Initial capacity/m
Installed capacity/MW
Hydropower outputcoefficient/(MW/(m3/s))
H1
25.99
80.95
300∗ 4
0.9225
H2
0.076
0.25
200∗ 4
1.5361
H3
3.56
4.01
150∗ 4
0.1831Table 3
Demand load in typical daily.
Time
1
2
3
4
5
6
Load/MW
1332
1254
1243
1431
1662
2343
Time
7
8
9
10
11
12
Load/MW
3630
3544
4162
4214
2998
2963
Time
13
14
15
16
17
18
Load/MW
2493
2218
3137
3873
3567
2896
Time
19
20
21
22
23
24
Load/MW
2896
3627
3901
3102
2260
1654In the risk-neutral situation, meeting all minimized power generation cost, minimized pollutant emission, and maximized first-level hydropower station storage capacity objectives, system units’ output assignment is drawn in Figure2.Figure 2
System unit output arrangement.In risk nonneutral situations, to analyse the influence of risk attitude on units scheduling result, this section studies scheduling results under different confidence levels. This section, respectively, calculates scheduling results of each objective in two situations, namely, constant confidence levelα and mutative confidence level α. The detailed information is listed in Table 4.Table 4
Simulation scenarios set.
Objective function
Same confidence levelα
Different confidence levelsα
Scenario 1
Scenario 2
Scenario 3
Scenario 4
Scenario 5
Scenario 6
f
1
0.85
0.9
0.95
0.85
0.9
0.95
f
2
0.85
0.9
0.95
0.9
0.85
0.9
f
3
0.85
0.9
0.95
0.95
0.95
0.85First, optimize the objective function of each single objective scheduling scenario to get system units’ output assignment and function value, and form objective function decision properties table. Each objective’s boundary value can be calculated according to (24)–(26). And used entropy method can get objective functions’ weights. Fuzzy objective membership degree function parameters and weights are listed in Table 5.Table 5
Fuzzy target membership function parameters and weights.
Scenario
Fuzzy objective membership degree function parameters
F
1/ton
F
1/ton
Scenario 1
[5423, 5648]
[6989, 7532]
[3245, 3573]
Scenario 2
[6372, 6819]
[7989, 8533]
[3645, 4274]
Scenario 3
[7572, 8820]
[9689, 11234]
[4245, 5075]
Scenario 4
[5223, 5848]
[7589, 8233]
[4135, 4868]
Scenario 5
[6274, 6652]
[6675, 7143]
[4038, 5260]
Scenario 6
[7342, 8510]
[6649, 7208]
[3428, 4516]
Scenario
Fuzzy objective function weights
λ
1
λ
2
λ
3
Scenario 1
0.324
0.333
0.343
Scenario 2
0.326
0.326
0.348
Scenario 3
0.332
0.329
0.339
Scenario 4
0.305
0.313
0.382
Scenario 5
0.332
0.303
0.365
Scenario 6
0.348
0.334
0.318
Note: in the[,],  before and after the comma are, respectively, the minimum and maximum value of the objective function.Compare different scenarios in Table5 and find when all objective functions’ confidence levels are the same, weights of the objective functions are basically the same, and the deviation is small. Otherwise when confidence levels are different, the higher the confidence level, the bigger the weight. This phenomenon means the entropy method is rational because it could allocate weights that can reflect decision maker’s risk attitude.Use parameters and coefficients in Table5 to build fuzzy multiobjective CVaR model, and do simulation for 6 scenarios. Table 6 lists the simulation result and output allocation in different scenarios.Table 6
Thermal power output arrangement in different scenarios (MW⋅h).
Unit
Scenario 1
Scenario 2
Scenario 3
Scenario 4
Scenario 5
Scenario 6
T1
5378
5404
5430
5457
5483
5512
T2
5047
5125
5204
5285
5367
5470
T3
8249
8344
8440
8507
8547
8565
T4
7755
7857
7961
8065
8172
8279
T5
4689
4706
4708
4725
4757
4774
T6
3189
3298
3411
3528
3649
3774
H1
9088
8945
8804
8665
8529
8394
H2
19203
19025
18848
18673
18500
18328
H3
3633
3528
3426
3326
3230
3136Compare scenario 1, scenario 2, and scenario 3 in Table6 and find that as the confidence level increases thermal units’ output would increase, and hydropower output would decrease. That means sacrifice some thermal power consumption to increase the first-level hydropower station’s storage capacity as backup service for load fluctuations. Table 7 lists scheduling operation results in different scenarios.Table 7
System scheduling results compared in different scenarios.
Scenario
Fuzzy VaR and CVaR value
Output allocation/MW⋅h
F
1/ton
F
2/105 kg
F
3/105 m3
VaR1
VaR2
VaR3
VaR
CVaR
Thermal
Hydro
Scenario 1
0.33
0.308
0.292
0.31
0.330
34307
31925
8239
10538
3468
Scenario 2
0.347
0.322
0.304
0.324
0.337
34734
31498
6508
8334
3896
Scenario 3
0.364
0.336
0.316
0.339
0.344
35154
31078
5569
7239
4669
Scenario 4
0.341
0.351
0.328
0.339
0.351
35567
30665
7845
7845
4682
Scenario 5
0.352
0.332
0.330
0.338
0.362
35974
30258
6428
6852
4517
Scenario 6
0.371
0.354
0.297
0.342
0.385
36374
29858
5432
7024
4174Compare scenario 1, scenario 2, and scenario 3 in Table7 and find that when all objective functions’ confidence levels are the same, the upgrade of confidence level requirement would promote the scheduling result close to the optimum result. However, VaR and CVaR value would also increase, in which the risk is increasing. Therefore, if a decision maker pursues better scheduling results, he must have the capacity to undertake bigger risk. Compare scenario 4, scenario 5, and scenario 6 and find when the confidence levels are different, the higher the confidence level the better the objective function value, and the higher the risk level. Those two conclusions are consistent with each other.Then, longitudinal compared scenarios with the same or different confidence levels, it can be seen, if confidence levels of objective functions are different, system risk level would increase, but the optimization result would be better, for example, scenario 1 and scenario 4. Therefore if the decision maker wants to optimize a specific objective, the system needs to pay more effort to achieve the goal while meeting other objective functions’ requirements and system’s operation risk would be increased. In uncertain load hydrothermal joint scheduling problems, better scheduling scheme means bigger risk, and smaller risk generally corresponds with worse scheduling scheme. Decision makers should balance effectiveness with their actual situation and risk tolerance to choose rational confidence level and get an accessible optimum scheduling scheme.


## 6. Conclusions

To analyse the risk that load demand uncertainty brings to hydrothermal power system joint scheduling, this paper regarded actual load and forecast error as random variable and built a multiobjective CVaR hydrothermal joint scheduling optimization model considering load demand uncertainty. According to the simulation results, the following conclusions can be concluded:(1)
Multiobjective CVaR method could effectively reflect the risk that uncertain factors bring to hydrothermal scheduling and get the optimum scheduling scheme under different risk attitudes.(2)
Fuzzy satisfaction degree theory and entropy theory could simplify CVaR model and transfer multiobjective nonlinear problems into single objective linear optimization problems, which is conducive to get the optimum scheduling scheme.(3)
Decision maker’s risk attitude would directly influence system scheduling optimization scheme. The stronger the system risk tolerance, the less the power generation energy consumption, the less the pollutant emission and the more the storage capacity of the first-level hydropower station.(4)
This paper assumed load forecast error obeys normal distribution. But this assumption is not always real in the actual production situations. Therefore the proposed model could be improved by using load forecast error’s actual distribution function.



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*Source: 741379-2015-06-16.xml*