Multi-objective Optimal Scheduling of Power System Based on Non-dominated Sorting Genetic Algorithm

With the rapid development of green energy, wind power has been applied more and more. In this paper, a multi-objective optimization model of a power system was established based on a non-dominant sorting genetic algorithm to solve the problem of system instability in wind power output, taking maximum wind power consumption and minimum power purchase cost as the objective function, and simulation verification is conducted. The results show that using this algorithm for model solving can get the corresponding Pareto optimal set of extreme solutions, verify the effectiveness of the algorithm, and achieve wind power consumption maximization, electricity purchase cost minimization, and better economic benefits.


Introduction
As the pillar and driving force for the survival and development of a modern country, energy and power systems are closely related to the rapid and stable operation of the social economy and the sustainable development of the ecological environment [1].Due to the large scale and many participants in the power system, the system is extremely complex, with a high degree of nonlinearity and coupling, system cost control and economic operation optimization become very difficult [2].Due to the randomness, volatility, and uncontrollable output, new energy connections will bring a certain impact on the power grid's security and stability [3].With the steady progress of smart grid construction, the means of demand-side response in the system are gradually increasing, thus providing an efficient and cheap dispatching resource to smooth the volatility of new energy output in the system [4].Therefore, studying the optimal scheduling of power systems has great practical significance for improving the power generation efficiency of power generation enterprises, reducing the emission of pollutants, and saving energy.
Wind power is clean and renewable energy, without any pollution, and can reduce oil consumption [5].Wind power resource is very rich on the reserves in our country, and using wind power is getting more and more attention [6].Wind power resources are prone to the interference of climate, environment, region, and other factors.The influence of wind power output has a certain randomness.Therefore, research on the containing wind power generation power system is of great significance [7].Aiming at multi-objective optimization scheduling of power systems including wind farms, this paper takes the maximum power consumed and minimum power purchase cost by wind power as the objective functions, the second generation of NSGA-II is used to solve the model of multi-objective optimization of the power system mentioned above, and the effectiveness of this algorithm is verified through the analysis of a numerical example.

Probabilistic model of the wind power output
The randomness of wind speed and power output has obvious uncertainty and the variation law function of wind speed v [8] is Equation (1).
In the equation, the shape parameter is k; the size parameter is c.
The probability density function of wind speed v is Equation (2).
The changing relationship of Pw and wind speed v of the fan can be expressed [9] as Equation (3).
where the fan-rated output power is Prate; the rated wind speed is vrate, and the cut in and cut out wind speeds are vin and vout.

Objective function
A multi-objective function was constructed for the user-side power purchase cost minimize and the consumption of wind power maximize, without considering the system network loss.

Minimum electricity purchase cost
where CS is power purchase cost; Rw is on-grid price of wind; Rx is transmission price of transmission line x; RG is on-grid price of thermal power; GQ is generating capacity in the purchasing province, , Pit is active power, NG is number.QL indicates the total load electricity quantity of purchasing province, ܳ = σ ௧ୀଵ ் ԝ ܲ ௧ load .

The maximum amount of electricity consumed by wind power is the target
where Qw is the wind power consumed electricity of the purchasing province.

Constraint condition 2.3.1. Power balance constraint
In the equation, ܲ ௧ load is the system load during the period t, which does not consider the system network loss.

Rotation reserve constraint
where Pimin and Pimax are minimum and maximum technical output respectively; ܴ ୪,௧ up and ܴ ୪,௧ down are positive and negative rotation reserve coefficients of the system against the load in time period t, respectively.ܴ ୵,௧ ୳୮ and ܴ ୵,௧ ୢ୭୵୬ are rotation reserve coefficients.

thermal power unit Climbing rate constraint
In the equation, Șidown and Șiup are the lower limits of descending rate and upper limit respectively.

Minimum start-stop time constraint
where ܶ ௧ିଵ is the start-stop time; ܶ on is the minimum continuous running time; ܶ off is the minimum continuous outage time; ܷ ௧ is start-stop variable.

Wind farm output constraints
In the equation, ܲ ௧,max ୵ is maximum wind power output at the time period t.

Wind power penetration limit constraints
where įw is the wind power penetration coefficient.

Line capacity constraint
where Px is the line x transmission power; Pxmax is the maximum transmission power and Pxmin is the minimum transmission power.

Multi-objective optimization model solving algorithm
The solution problem of Equations ( 4) and ( 5) combines the constraint conditions of Equations ( 6) to ( 14), and it can be obtained from Equation (15).
where f=(f1,-f2) is the objective function; x is a vector composed of decision variables; hj(x) is the equality constraint; gk(x) is an inequality constraint.In the multi-objective optimization problem, each objective is in conflict with each other, so there is generally no solution to make each objective reach the optimal solution at the same time, but there is a set of compromise solutions, which is Pareto optimal solution set.NSGA-II is a kind of intelligent algorithm based on genetic algorithm evolution, which can deal with a variety of complex problems, including multidimensional, non-convex, and nonlinear problems [10].The algorithm is robust, computationally efficient, and can obtain evenly distributed Pareto solution sets with good diversity.It has been widely used in solving single objective optimization problems and multi-objective optimization problems.The process of applying NSGA-II is shown in Figure 1.The detailed procedure is as follows.
Step 1: According to the Weibull distribution model, the wind power output value within the dispatching period was randomly generated.
Step 2: Code.The active power output value of each generator is coded by real number coding.
Step 3: We set the population size parameter.Based on the constraint conditions of thermal power units, the initial population of 0 P is generated.
Step 4: According to each individual's power purchase cost CS and wind power consumption Qw, a fast non-dominant ranking was conducted, and the virtual crowding degree distance was calculated.
Step 5: Genetic manipulation.The most critical link of optimization iteration is genetic operation, and selection operation is based on Step 4. By setting the selection rate, recombination rate, and variation rate, the selection, crossover, and variation operations were carried out to get the subpopulation.
Step 6: Elite retention.That is, the parent population and the child population are merged, the fast non-dominated sorting is carried out, the virtual crowding degree distance is calculated, the optimization is realized, and the new parent population is formed.
Step 7: The number of iterations increases by 1, the number of iterations is judged, and if it is not completed, it jumps to Step 4.

Example description
The power system has 5 thermal power units and 1 grid-connected wind farm.In the system, the maximum and minimum transmitted power of Units 1-4 are 620 MW and 240 MW respectively, and the lower limit of descending rate and upper limit of climbing rate are both 2.1 MW/min.The maximum and minimum transmitted power of Unit 5 are 1000 MW and 420 MW, respectively, and the lower limit of descending rate and upper limit of climbing rate are 4 MW/min.The wind farm inlet wind speed is 5 m/s, rated wind speed is 17 m/s, cut out wind speed is 25 m/s, wind farm rated output is 240 MW, shape parameter is 2, size parameter is 10, wind power penetration coefficient is 0.1; The system against load is 0.1.The reserve coefficients of output fluctuation are 0.05.The on-grid electricity price of wind power is 0.36 yuan /(kW•h); System and line losses are excluded.

Result analysis
In this example, the proportion of Pareto non-dominant individuals was 35%, the population size was 100, the maximum number of evolutionary iterations was 1000, the stop algebra was 300, the fitness function value deviation was e-100, and the crossover rate and variation rate were 60% and 0.1%, respectively.Typical daily load and wind power curves of spring, summer, autumn, and winter are shown in Figure 1 and Figure 2 respectively.The daily load value in summer is the largest, and the difference of peak-valley is also the largest; the daily load value in spring is the smallest; the daily load value in autumn and winter is similar, and the difference is small.The variation law of wind power output is opposite to the variation law of load, with more wind power generation in spring and less in summer.Table 1 is Pareto optimal set obtained by NSGA-II.As can be seen the cost of purchasing electricity in summer is the largest, and the consumption of wind power is the smallest.Spring electricity purchase cost is the smallest, and wind power consumption is the largest.This is determined by the load characteristics of the system and wind power output characteristics.Power purchase cost is a primary function proportional to the total power output of the thermal power unit, while wind power consumption is a primary function inversely proportional to the total power output, and the total load power remains unchanged.It is precisely because of the wind power penetration constraint that the wind power consumed in the system is always limited, so there must be an extreme solution, which simultaneously satisfies the minimum power purchase cost and the maximum wind power consumption.In summer, for example, the minimum power purchase cost is 29008, 617 yuan, and the maximum power consumed by wind power is 1196 MW•h, realizing the maximum economic benefit of power purchase and the maximum consumption of wind power.

Conclusion
In order to solve the instability problem of power systems caused by wind power output, this paper established a multi-objective optimization model of power system degree based on NSGA-II and obtained the following conclusions: (1) The variation law of wind power output is contrary to the variation law of load, with more wind power generation in spring and less wind power generation in summer.(2) By using the algorithm in this paper to solve the power system model, the optimal solution with better economic benefits can be obtained.

Table 1 .
Pareto extreme value solution of the optimal set.