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

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 systemoperation 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.


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][7][8], and intelligence forecast methods [9][10][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][17][18][19][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 Section 2, 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.

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: wherein  (2) Minimized pollutant emission objective function is as follows: wherein  is the number of pollutant types and    ,    , and    are pollutant emission parameters of unit .
(3) Maximized storage capacity of the first-level hydropower station is as follows: wherein   is the water storage capacity od hydropower station  at time ;    is the inflow volume of the superiors hydropower station at time ;  S  is the discharged flow of hydropower station  at time ;   is the nature abandoned water of hydropower station  at time ;  is the number of the upstream hydropower units that directly connected with hydro unit ;  is the flow delay between hydropower unit  and upstream hydropower unit ℎ.

Constraints.
(1) Load demand and supply balance constraint is as follows: wherein () is the load demand at time ;  , is the output of hydropower station  at time ;    is system's total power loss at time .
(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: wherein  max  and  min  are the power generation output upper and lower limit of unit : wherein Δ −  and Δ +  are power climbing limits: wherein  on  is the shortest startup time of thermal unit ;  on ,−1 is the continuous operation time of unit  at time  − 1;  off  is the shortest downtime of unit ;  off ,−1 is the continuous downtime of unit  at time  − 1.
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: wherein   is the flow of hydropower unit  at time ;   is the storage capacity of hydropower unit  at time ;  0 ,  1 ,  2 ,  3 ,  4 , and  5 are output parameters of hydropower unit , which can be calculated according to the historical data of the unit: wherein  min  and  max  are, respectively, the minimum and maximum power generation flow of hydropower unit ;  min  and  max  are, respectively, the lower and upper storage capacity limit of hydropower unit : wherein  0 and   are, respectively, storage capacities at the initial time and end time of hydropower unit .

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 wherein L() is the forecasted value of load demand.
The relationship between the actual value and forecasted value is wherein () is the deviation value between the actual value and forecasted value, which, according to literature [24], obeys normal distribution () ∼ (  ,  2  ).  and   are, respectively, the average value and standard deviation of ().Distribution law of () 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.

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   (, ) ∈   ×   →  1 ( = 1, 2, . . ., ) that is related with decision variable  ∈ X ⊂   .The loss function is a continuous function. is a continuous random variable and its probability density function is ().Then the distribution function of   (, ) is Define   ( = 1, 2, . . ., ) as weights of each loss function and -VaR loss value of decision variable  under confidence level  and weight  is Introduce loss function  ,  (,   ): Then the optimum -CVaR loss value (,  * (, )) of decision variable  under confidence level  and weight  is wherein   is the confidence levels of loss function .
To find the minimum -CVaR loss value, (,  * (, )) should achieve the minimum value  on .And the optimization problem can be defined as To simplify the solving progress of model (P1), this paper redefined loss functions and their optimization problems.The detailed model is model (P2): The equivalent problem of model (P2) is as follows: for any , if min ∑  =1    ,  (, ) has a unique and optimal solution . meets the following constraint: Then, the following conclusion can be gotten: For any determined weight value , if (, ) could make model (P2) achieve the optimum and ( 2) is tenable, then  could make model (P1) achieve the optimum.

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: wherein   and   are, respectively, task proportions of thermal units and hydro units and   is the output power coefficient of hydro units.After considering uncertainties, traditional constraints are needed to converse to uncertainty constraints, as follows: Further, multiobjective CVaR approach is used to build the hydrothermal optimization scheduling model.First, define a schedule loss function, as follows: wherein  is units output set of the system;   is the confidence level requirement of objective function   ;   is decision maker's expected value of objective function   .Then build a hydrothermal joint scheduling optimization model which sets minimized -CVaR loss value as its objective: min  ,  (,   ) ,  = 1, 2 max  3, 3 (,  3 ) s.t.

Entropy Fuzzy Satisfaction Degree Solving Method
According to (25) wherein  = 0, 1, . . .,  − 1, Δ = ( max  −  min  )/.According to the literature [26], when  ≥ 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.

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 Table 1.
(2) Scale each objective according to decision makers' wishes and preferences, and define  max 1 ,  max 2 , and  min 3 and their value range.The equations are (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: wherein  min  and  max  are, respectively, the minimum and maximum value of the objective function   (⋅).(  ) is the membership degree function of the objective function   (⋅).

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 Table 1, 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.

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 () ∼ (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.
In 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 Figure 2.
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 level  and mutative confidence level .The detailed information is listed in Table 4.
First, 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.
Compare different scenarios in Table 5 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 Table 5 to build fuzzy multiobjective CVaR model, and do simulation for 6 scenarios.Table 6 lists the simulation result and output allocation in different scenarios.
Compare scenario 1, scenario 2, and scenario 3 in Table 6 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.
Compare scenario 1, scenario 2, and scenario 3 in Table 7 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.

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

3 max
Note: objective with * means the objective is the optimization objective.
, is an 0-1 variable; and  , = 1 means the unit  is in operating; otherwise it is not in operating. , is the output of unit  at time ;   ,   , and   are fuel cost parameters of unit , which can be calculated according to historical data of the unit. , is the startup-shutdown cost of thermal unit  at time : 1 is system's power generation energy consumption cost;  is the number of thermal units;  is the cycle of power generation scheduling, generally 24 hours;

Table 1 :
The input-output tables of objective function.

Table 2 :
Basic parameters of cascade hydropower stations.

Table 3 :
Demand load in typical daily.

Table 5 :
Fuzzy target membership function parameters and weights.

Table 6 :
Thermal power output arrangement in different scenarios (MW⋅h).

Table 7 :
System scheduling results compared in different scenarios.