## Optimal scheduling of wind–photovoltaic power-generation system based on a copula-based conditional value-at-risk model

**Abstract**

Abstract Increasing the application of renewable energy in the power system is an effective way to achieve the goal of ‘Dual Carbon’. At the same time, the high proportion of renewable energy connected to the grid endangers the safe operation of the power system. To solve this problem, this paper proposes the application of a copula function to describe the correlation between wind power and photovoltaic power, and reduce the uncertainty of power-system operation with a high proportion of renewable energy. In order to increase the robustness of the model, this paper proposes the application of the conditional value-at-risk theory to construct the objective function of the model and effectively control the tail risk of power-system operation costs. Through case analysis, it is found that the model proposed in this paper has strong practicality and economy. Open in new tabDownload slide renewable energy, ‘Dual Carbon’ targets, copula–CVaR, particle swarm optimization algorithm, optimized scheduling Introduction With the implementation of the ‘Dual Carbon’ targets in China [1], the focus of current research is how to improve the utilization rate of renewable energy [2–4]. Renewable energy is characterized by intermittence and uncertainty [5, 6]. Increasing the proportion of renewable energy in the power system will certainly harm the safe operation of the power system. In this paper, by constructing the copula function of wind–photovoltaic correlation, the wind–photovoltaic complementary grid connection is proposed to reduce the uncertainty of renewable-energy output. The copula is a probability model that describes a multivariate cumulative uniform distribution. Copulas are used to model the correlation or dependence between random variables. At the same time, system security is considered while pursuing system economy. The uncertainty of the output of renewable energy such as wind power and photovoltaic power intermittently affects the security of the operation of the power system. Reducing the uncertainty of the output of renewable energy can reduce the decision-making risk of system planning and scheduling [7]. Li et al. [8] predicted the output of wind power and photovoltaic power based on the existing turbine or photovoltaic output according to the probability method but did not consider the correlation between wind power and photovoltaic power. Gao et al. [9] proposed that Frank copula was used to describe the correlation between wind and photovoltaic power; Han and Chu et al. [10] proposed that Clayton copula could better describe the relationship between wind and photovoltaic power. However, they did not consider that the relationship between wind and photovoltaic power is affected by regional and seasonal factors, so specific analysis should be made according to the specific problems of different regions, and one copula should not be used to represent the whole relationship between wind and photovoltaic power. In this paper, an empirical copula function was applied to select the copula function [11]. Before describing the correlation between wind and photovoltaic power, the empirical copula function was calculated by using the historical data of local wind and photovoltaic power, and then a copula suitable for local conditions was selected by this index. Many scholars have studied the optimal configuration and operation scheduling of power systems [12]. Huang et al. [13] optimized the structure of the system and the capacity of the equipment, and Shahmohammadi et al. [14] optimized the selection of equipment under a given framework. In terms of operation scheduling, several studies [10, 15, 16] have proposed an optimal scheduling model with cost as the target. The traditional system operation optimization model does not consider the cost risk. This paper proposes the application of a risk-measurement tool to control the cost risk of system operation. Value-at-risk (VaR) has been generally recognized as a standard method for risk-management and capital-requirements calculation in the financial industry. However, VaR does not satisfy the sub-additivity in the consistent-risk measurement and cannot control the tail risk. Conditional value-at-risk (CVaR) solves these two problems of VaR and is a measurement tool to measure investment risk in the financial field [17]. Compared with VaR, CVaR has better tail-risk control ability [18]. Therefore, this paper proposes the application of CVaR to control the cost of system operation. The contributions of this paper may be summarized as follows: (i) In this paper, the copula function is proposed to describe the correlation between wind and photovoltaic power, and an empirical copula function is used as an index to select the most appropriate copula function. (ii) In the model, the risk measure CVaR is used as the objective function, which increases the robustness of the model and improves the security of system operation. (iii) At the end of this paper, the effectiveness and economy of the model are proved by case analysis. The structure of the paper is as follows. Section 1 introduces the copula function and gives the selection method of the copula function and the output model of the battery. In Section 2, a model based on copula–CVaR wind-optimal scheduling of a generation system is constructed to minimize the cost of CVaR on the premise of ensuring the robustness of system operation to improve the economy of system operation. In Section 3, the particle swarm optimization (PSO) algorithm is introduced to solve the model. In Section 4, a case study is presented. The model based on copula–CVaR wind–photovoltaic power optimal scheduling of a generation system proposed in this paper not only ensures the robustness of system operation but also ensures the economy of system operation. 1 System structure of distribution network considering wind–solar output Fig. 1 shows the structure of the system. On the power-supply side, there are thermal power units, wind turbines, photovoltaics and energy-storage batteries, which supply power to the demand side through transmission and distribution lines. The demand side includes residential power loads, commercial power loads and industrial power loads. Renewable-energy sources such as wind power and photovoltaics have the characteristics of intermittency and uncertainty. Large-scale integration of wind and solar power grids will threaten the safe and stable operation of the distribution network system—and running costs will therefore have uncontrollable risks. In order to reduce the impact of wind–solar grid connection on the operation of the distribution network system, this paper proposes to apply the copula theory to construct a wind–solar joint output model to reduce the uncertainty of wind and solar power. In order to control the operation cost of the distribution network system, this paper proposes the application of risk-measurement tools, CVaR measurement, management and control considering the system operation cost risk of wind and solar combined grid connection. Fig. 1: Open in new tabDownload slide System structure. 1.1 Wind–photovoltaic scene generation The copula function can be described as the correlation between random variables. It is a function that connects the joint distribution function of random variables with their respective marginal distribution functions. Due to the intermittent and uncertain nature of wind power, photovoltaic power and other renewable energy, the grid connection of wind power, photovoltaic power and other renewable energy jeopardizes the safe operation of the power grid. By studying the correlation between wind power and photovoltaic power, and constructing the complementary model of wind power and photovoltaic power, the uncertainty of renewable energy can be effectively reduced and the safety of power grid operation can be improved. The choice of the copula function is crucial to accurately describe the correlation between wind power and photovoltaic. This paper uses the empirical copula function as the index to select the copula function. Its definition is as follows: Cn(u,v)=1n∑ni=1I{F(xi)≤u)}⋅I{G(yi)≤v)}(1) where (xi,yi)(i=1,2,…,n) represents the sample taken from the 2D population (X, Y); F(⋅) and G(⋅) represent the edge distribution functions that X and Y obey, respectively; and both I{F(xi)≤u)} and I{G(yi)≤v)} are indicative functions. As can be seen from Equation (1), the empirical copula does not contain any parameters. When selecting the function, the appropriate copula function is selected by comparing the Euclidean distance between different copula functions and empirical copula functions and common rank correlation coefficients. After selecting the appropriate scene, the analysis method is used to generate the typical-day output scene. 1.2 Output of battery In the model studied in this paper, the application of a battery energy-storage model is as follows: SES(t)=(1−δΔt)⋅SES(t+1)⋅(pchηch−pdisηdis)⋅Δt(2) where SES(t) represents the electric quantity of the battery at time t; δ represents the self-discharge rate of the battery; SES(t+1) represents the electric quantity of the battery at (t+1) ; pch and pdis represent the charge/discharge power of the battery, respectively; and ηch and ηdis represent the charging/discharging efficiency of livestock batteries, respectively. 2 Model Renewable-energy sources such as wind power and photovoltaics have the characteristics of intermittency and uncertainty, so the large-scale grid connection of wind power and photovoltaics endangers the safe and stable operation of the distribution network system. In order to solve this problem, this paper proposes to apply copula theory to describe the correlation between wind power and photovoltaics, and then constructs a combined output model of wind power and photovoltaics. By combining two uncertain renewable-energy grids, the uncertainty of the overall renewable-energy grid connection can be reduced compared to considering two uncertain energy sources separately. Through the copula-based wind–solar joint output model, N kinds of classic scenarios of wind–solar joint output and their corresponding probabilities can be generated. The operation cost of the distribution network system under different wind and solar combined output scenarios is different. In order to better control the risk of the operation cost of the distribution network system, this paper proposes to apply the risk-measurement tool CVaR in the financial field to control the distribution considering the wind and solar grid connection. 2.1 Risk measurement In the economic field, risk can be interpreted as the possibility of loss or failure; some scholars explain risk as the probability of an accident and the severity of the consequences. According to the literature, risk is defined as ‘the difference between the outcome that is likely to occur and the intended goal under a given situation and within a given time period’. Thus, the risk can be expressed by the probability of deviation and the degree and direction of deviation between the result and the expected target. The basic mathematical expression for risk is R = f(P, S), where R represents the risk, P represents the probability of deviation and S represents the degree and direction of deviation. After a risk event occurs, the work of assessing and quantifying the impact and loss caused by the event is called risk assessment. Due to the influence of many uncertain factors in the power system, such as changes in natural conditions, load fluctuations and unit failures, the risk of power-system operation exists. Risk assessment in the power system requires a quantitative analysis of the probability and loss of operational risks in the power system. CVaR is defined as follows: VaRβ(X)=inf{x|FX(x)≥α}=FX−1(α)(3) CVaRβ(X)=11−β∫1βVaRλ(X)dλ(4) where β represents the confidence level; VaR at level β is the β-quantile of a random variable X (which we often call loss); FX(x) represents the cumulative distribution function of X ; and β is the confidence or the tolerance level 0<β<1 . 2.2 The objective function minCtotal=CVaRβ(Cbattery+Ccoal+Ccopula)(5) Ccopula=∑Ti=1κpv⋅ppv(t)+κw⋅pw(t)(6) Cbattery=∑Ti=1κbattery(pch(t)+pdis(t))(7) Ccoal=∑Ti=1κcoal⋅pther(t)+ρcoal⋅Qcoal(t)(8) where CVaRβ(∗) represents the cost when the confidence is β ; Ccopula represents the joint cost of wind power and photovoltaic; κpv and κw represent the unit photovoltaic operation and maintenance cost and unit wind power operation and maintenance cost, respectively; and ppv(t),pw(t),pther(t) represent the output of photovoltaic power, wind power and thermal power at time t, respectively. 2.3 The constraint The system operating constraints are as follows: pwt(t)+ppv(t)+pdis(t)+pther(t)=L(t)+pch(t)(9) where pwt(t),ppv(t),pther(t),L(t) refers to the output of wind power, photovoltaic power, thermal power and load at time t. 0≤pwind,i(t)≤pwind,imax(10) 0≤ppv,i(t)≤ppv,imax(11) SOCmin<SOC(t)<SOCmax(12) Pther min<Pther(t)<Pther max(13) −rdown⋅Δt≤pther(t)−pther(t−1)≤rup⋅Δt(14) Equations (10) and (11) constrain the installed power-generation capacity of wind power and photovoltaic power; Equation (12) constrains the output of battery storage at time t; and Equations (13) and (14) constrain the thermal power units at time t. 3 PSO algorithm PSO is an evolutionary computing technique that was proposed by Kennedy and Eberhart in 1995 [19]. The basic idea is to adjust the direction and speed of the evolution of the population through its own experience and social experience, so as to obtain the optimal solution. PSO is an iterative optimization algorithm similar to genetic algorithms. The system is initialized to a set of random solutions and the optimal value is searched iteratively. Instead of the crossover and mutation of genetic algorithms, the particle follows the best particle in the solution space. Compared with genetic algorithms, PSO has the advantage of being simple and easy to implement without many parameters needing to be adjusted. It has been widely used in function optimization, neural network training, fuzzy system control and other applications of genetic algorithms. The basic idea of the PSO algorithm is to randomly initialize a group of particles and finally find the optimal solution through continuous iteration. After each iteration, the particle will update the individual extremum and the global extremum by the fitness value, and update the velocity and position according to the formula. The calculation process is shown in Fig. 2. Fig. 2: Open in new tabDownload slide The flow chart. The basic process of the original PSO algorithm is as follows: Step 1: Determine the motion state of the initial particle, including the velocity and position parameters of the initial particle: Xi,j0=Xjmin+rank(0,1)×(Xjmax−Xjmin)(15) where Xi,j0 represents the j-th dimension of the initialized i-th particle; rank(0,1) represents a random number that obeys a uniform distribution on the interval (0, 1); Xjmin represents the minimum value of the population space; and Xjmax represents the maximum value of the population space. Step 2: Calculate the particle population fitness of each search and save the individual extreme value and the population extreme value of the particle. Step3: Update the velocity and position parameters of the next-generation population of particles by linking the historical extremes of individuals and groups to the inertia of the particles themselves: Vi,jt+1=Vi,jt+c1r1(Pi,j−Xi,jt)+c2r2(gj−Xi,jt)(16) Xi,jt+1=Vi,jt+1+Xi,jt(17) where Xi,jt and Vi,jt respectively represent the position and velocity of the i-th (i = 1, 2, 3,..., N) particle in the j-th dimension (j = 1, 2, 3,..., M) after the t-th iteration; Pi,j represents the optimal value of the i-th particle in the j-th dimension; gj represents the global optimal value figure of merit; c1,c2 represents the learning factor; r1,r2 represents the interval (0, 1), a random number that obeys a random distribution. Step 4: Judge whether the termination conditions required by the number of iterations or convergence accuracy are met. If the termination conditions are not met, return to Step 2. 4 Case This paper takes an industrial park in eastern China as an example to analyse and study the model proposed in this paper. The industrial park is expected to build a power-supply centre to meet the power load of the industrial park. The total default investment in the construction of wind power, photovoltaic power, thermal power and energy-storage equipment must be ≤100 000 yuan. 4.1 Combined output of wind–photovoltaic Figs. 3 and 4 show the historical data value of wind power and photovoltaics in an industrial park in eastern China. Fig. 3. Open in new tabDownload slide Photovoltaic power. Fig. 4. Open in new tabDownload slide Wind power. In this paper, copula functions commonly used to describe the correlation between wind power and photovoltaic power are selected as ‘Gaussian copula’, ‘T copula’, ‘Clayton copula’ and ‘Frank copula’. The empirical copula function of wind power and photovoltaic power is calculated based on historical data. The Kendall correlation coefficient and Spearman correlation coefficient under different copula functions were calculated respectively. Table 1 shows that, whether using the Kendall correlation coefficient or the Spearman correlation coefficient, the difference between the correlation coefficient of the Frank copula and that of the empirical copula is the smallest. Table 1: Copula connect selection Classification . Kendall . Spearman . Distance . Norm 0.7011 0.8904 12.4895 T 0.74451 0.9055 11.455 Clayton 0.6145 0.7911 151 512 Frank 0.6850 0.8376 10.4203 Empirical 0.6878 0.8476 – Classification . Kendall . Spearman . Distance . Norm 0.7011 0.8904 12.4895 T 0.74451 0.9055 11.455 Clayton 0.6145 0.7911 151 512 Frank 0.6850 0.8376 10.4203 Empirical 0.6878 0.8476 – Open in new tab Table 1: Copula connect selection Classification . Kendall . Spearman . Distance . Norm 0.7011 0.8904 12.4895 T 0.74451 0.9055 11.455 Clayton 0.6145 0.7911 151 512 Frank 0.6850 0.8376 10.4203 Empirical 0.6878 0.8476 – Classification . Kendall . Spearman . Distance . Norm 0.7011 0.8904 12.4895 T 0.74451 0.9055 11.455 Clayton 0.6145 0.7911 151 512 Frank 0.6850 0.8376 10.4203 Empirical 0.6878 0.8476 – Open in new tab The last column in Table 1 shows the binary Euclidean distance between the different copula functions and the empirical copula functions. From the data in the table, it can be seen that the Frank copula function is closest to the empirical copula function of historical data. Based on the above three indicators, a clear choice can be made: The Frank copula function is selected as the description function of the correlation between wind power and photovoltaic power in the selected region. After selecting the Frank copula function, the joint distribution function of wind power and photovoltaic power can be obtained through calculation. However, this does not mean that the Frank copula function should be selected for wind power and photovoltaic power functions in all regions. For different regions, different historical data should be reanalysed and selected. 4.2 Case result analysis The unit output cost price of wind power and photovoltaic power, considering the parity of wind power and photovoltaic power feed-in prices without policy subsidy, is 0.4 yuan/kWh. The unit output cost of thermal power units is based on the electricity price of residential electricity: 0.5 yuan/ kWh. The unit output cost of the selected battery is 0.45 yuan/ kWh and the charge–discharge efficiency is 80%. The 24-hour load data are shown in Fig. 5. Table 2 shows the installed capacity of each unit in the region. Table 2: Equipment capacity Equipment . Wind . Photovoltaic . Thermal . Storage . Capacity (kW) 4000 2000 3000 4000 Equipment . Wind . Photovoltaic . Thermal . Storage . Capacity (kW) 4000 2000 3000 4000 Open in new tab Table 2: Equipment capacity Equipment . Wind . Photovoltaic . Thermal . Storage . Capacity (kW) 4000 2000 3000 4000 Equipment . Wind . Photovoltaic . Thermal . Storage . Capacity (kW) 4000 2000 3000 4000 Open in new tab Fig. 5: Open in new tabDownload slide The 24-hour load. From the capacity configuration data in Table 2, it can be seen that wind power is the main power-generation equipment in the region, followed by thermal power and finally photovoltaic power. The total capacity configuration of wind power and photovoltaic power is twice that of thermal power, which is higher than that of traditional industrial-park renewable energy. The ratio of capacity allocation to thermal power ensures the utilization rate of renewable energy. The area is equipped with 4000 kilowatts of electric energy-storage equipment and the storage capacity can reach 66.67% of the renewable-energy capacity, which fully guarantees the safe and stable operation of the distribution network. The confidence level is set to 80% with 10% of the upper tail and 10% of the lower tail. The average costs of the upper and lower tails are calculated respectively, as shown in Table 3. Table 3: Cost Classify . Average . 90% CVaR . 80% CVaR . 90% VaR . 80% VaR . Cost (yuan) 51 048.79 52 495.16 52 195.41 52 251.3 51 708.02 Classify . Average . 90% CVaR . 80% CVaR . 90% VaR . 80% VaR . Cost (yuan) 51 048.79 52 495.16 52 195.41 52 251.3 51 708.02 Open in new tab Table 3: Cost Classify . Average . 90% CVaR . 80% CVaR . 90% VaR . 80% VaR . Cost (yuan) 51 048.79 52 495.16 52 195.41 52 251.3 51 708.02 Classify . Average . 90% CVaR . 80% CVaR . 90% VaR . 80% VaR . Cost (yuan) 51 048.79 52 495.16 52 195.41 52 251.3 51 708.02 Open in new tab As can be seen from the data in Table 3, when the appropriate copula function is used to accurately describe the relationship between wind power and photovoltaic power, the operating-cost difference of the power system is small and the tail risk in extreme cases is controllable. 5 Conclusion In this paper, a model based on copula–CVaR wind-optimal scheduling of a generation system is proposed in which a copula function is used to describe the correlation between wind power and photovoltaic power. To reduce the uncertainty of the high proportion of renewable energy and reduce the harm of the high proportion of renewable-energy grid connection on the safe operation of power system, CVaR theory is used in the objective function of the model to control the tail risk of power-system operation costs. There are different correlations between wind power and photovoltaic power in different regions, different climate environments and different seasons. Therefore, a set of indicators is needed to select the copula function that describes the correlation between wind power and photovoltaic power. Based on the empirical copula, the Kendall rank correlation coefficient, Spearman rank correlation coefficient and the binary Euclidean distance to the empirical copula are proposed as the selection indexes to select the appropriate copula function. In the optimization model, this paper proposes the application of CVaR theory to control the tail risk of system operation costs. The data from the case analysis show that there is no significant outlier in the tail of the power-system operating cost and there is little change in the tail of the cost. 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