Dynamic Economic Dispatch in Thermal-Wind-Small Hydropower Generation System

With the large-scale wind power integration, the uncertainty of wind power poses a great threat to the safe and stable operation of the system. This paper proposes dynamic economic dispatch problem formulation in thermal power system incorporating stochastic wind and small-hydro (run-in-river) power, called thermal-wind-small hydropower system (TWSHS). Weibull and Gumbel probability density functions are used to calculate available wind and small-hydro power respectively. An improved differential evolution algorithm based on gradient descent information (DE-GD) is proposed to solve the dynamic economic dispatch (DED) problem considering uncertainty of wind power and small-hydro power, as well as complicated constraints in TWSHS. Based on the traditional differential evolution algorithm, the gradient information of the objective function is introduced after the mutation process to enrich the diversity of the population, thus increasing the possibility of convergence to the global optimization. Generation scheduling is simulated on a TWSHS with the proposed approach. Simulation results verify feasibility and effectiveness of the proposed method while considering various complex constraints in the thermal-windsmall hydropower system.


Introduction
Classical economic power dispatch problem is formulated with only thermal generators. But importance of reduction in emission is paramount from environmental sustainability perspective and hence penetration of more and more renewable sources into the electrical grid is encouraged. Small hydropower is concentrated in southern China, and the cumulative gridconnected capacity of wind power is also increasing year by year. In 2017, the cumulative installed capacity of Yunnan and Guizhou was 8190MW and 3690MW respectively. Compared with 2016, it increased by 123.9% and 11.1% respectively. Therefore, it is necessary to consider the connection between small hydropower and wind power in the economic dispatch of power system. Dynamic economic dispatch (DED) is a method to schedule the online generator outputs with the predicted load demands over a certain period of time so as to operate an electric power system most economically [1]. It is a dynamic optimization problem taking into account the constraints imposed on system operation by generator ramping rate limits. The DED is not only the most accurate formulation of the economic dispatch problem but also the most difficult to solve because of its large dimensionality.
In recent years, several literature studied DED problem applying evolutionary algorithms. Chaotic bat algorithm (CBA) was implemented to perform the optimization in Ref. [2]. Ref. [3] proposed a new hybrid grey wolf optimizer (HWGO) with addition of mutation and crossover operators into grey wolf optimizer. In Ref. [4], multi-fuel option and valve-point loading effect of stream turbine generators were taken into account. Delshad [5] performed the study using backtracking search algorithm (BSA) with added complexity of generator prohibited operating zone (POZ) to Ref. [4]. Based on the previous studies, this paper proposes dynamic economic dispatch problem formulation in thermal power system incorporating stochastic wind and small-hydro (run-in-river) power, called thermal-windsmall hydropower system (TWSHS). Weibull and Gumbel probability density functions are used to calculate available wind and small-hydro power respectively. An improved differential evolution algorithm based on gradient information (DE-GI) is proposed to solve the dynamic economic dispatch (DED) problem considering uncertainty of wind power and small-hydro power, as well as complicated constraints in TWSHS. Generation scheduling is simulated on a TWSHS with the proposed approach. Simulation results verify feasibility and effectiveness of the proposed method while considering various complex constraints in the thermal-wind-small hydropower system. The objective of DED problem is to realize minimum of the total economic cost, while considering the stochastic availability of wind power and small hydropower with equality and inequality constraints.

Objective function
The fuel cost function of thermal units includes valve point loading effects and can be formulated as the follows: Where C is the total operating cost over the whole dispatch period of n thermal units. T is the length of dispatch period.

1) Active power balance
In the TWSHS, in addition to thermal power units, the output power of wind turbines and small hydropower plants must be considered. So the active power balance can be expressed as Eq.(2).  3) Generator unit ramp rate limits Where ,

Differential Evolution Algorithm Review
Differential evolution algorithm is used to solve such optimization problems, which can be divided into initialization, mutation, crossover and selection.
x and x are the jth dimension lower and upper limits, respectively.

2) Mutation
DE algorithm achieves individual mutation through differential strategy. The common differential strategy is to randomly select two different individuals in the population, and then scale the vector difference and synthesize the vectors with the individuals to be mutated.

3) Crossover
The purpose of crossover is to randomly select individuals, because differential evolution is also a random algorithm, and the method of crossover is as follow.
( ) Where the parameter is called the crossover probability.

4) Selection
In DE, greedy choice strategy is adopted, that is to choose the better individual as the new individual. in the direction of the negative gradient if ( )

Gradient descent
Where γ is constant parameter and small enough.
. In other words, is subtracted from n x to move against the gradient, toward the minimum.

1) Wind power probabilities
The Weibull probability density function (PDF) method is predominantly used to describe the stochastic characteristic of wind speed. Its PDF is given by Eq.(11) as follow [7].
Where and α β are the scale and shape parameters for PDF, respectively. v is the current wind speed.
Wind power output is determined by wind speed, and the relationship between power and wind speed can be expressed as Eq.(12).
Where , in out r v v and v are the cut-in, cut-out and rated wind speed, respectively.

1) Small hydropower probabilities
Small hydropower is often a runoff type power station, so it has no regulating ability, and its output is mainly affected by inflow. The relationship between power and inflow rate can be described as Eq.(13).

( )
Where , and g η ρ are the power generation efficiency of plant, water density and acceleration gravity, respectively. H is the power generation head.
In this paper, the Gumbel distribution is used to describe the runoff randomness of small hydropower stations. The probability of inflow rate Q following Gumbel distribution with parameter λ and scale parameter γ is expressed as follow [8].

The proposed approach
As DE algorithm is a stochastic optimization algorithm, the diversity of population determines the effect of the algorithm in the process of evolution, so this paper introduces gradient descent method to enrich the diversity of population based on traditional DE. The main idea is to make a gradient descent of the individual population after the mutation process to get a new population, and use the population to cross with the population produced by the mutation process in the crossover process.

Frame of the proposed approach
The algorithm flow chart is shown in the following figure Fig.1.

Structure of individuals
In the DED problem with renewable resources, the output power of thermal units, TH it P , is selected as decision variables. The array of the decision variable can be represented as follow.

Initialization of population
The initial population Pop is generated by creating the size of population, NP , solutions randomly within the feasible ranges of constraints (3) and (4)

Constraint handling
Considering the constraint (4), the feasible region of decision variables should be compressed, the specific implementation is shown as Eq.(17).

Gradient processing method of objection
As shown in Fig.2, Line 1 is the cost function with valve point effect, Line 2 is the cost function without considering valve point effect, and Line 3 is the cost component due to the valve point effect alone, Line 3 is based on Line 4 to obtain absolutely worthwhile curves. And Line 1 equals arithmetic sum of Line 2 and Line 3. Point A, B, and C are the non-differentiable points. Because the objective function takes into account the valve point effect of thermal power units, the objective function appears non-differentiable points. Therefore, in this paper, these non-differentiable points are treated separately when the gradient is calculated.

Case Study
In this section, the dispatching period is one day, the length of the period is 24 hours. The case includes ten thermal units, two sites of windfarm and one small hydropower plant. The load demand, the output constraints of thermal power units and the cost coefficient are selected as Ref. [1], and the parameters of the probability density function of wind power and small hydropower output are referred to Ref. [6].

Parameters setting
The parameters of DE-GD algorithm are set as follow: 100 NP = , the maximum number of iterations max 60000 g = , the differential weight 0.5 F = and the crossover probability 0.8 CR = . The learning rate of gradient descent method used in this paper is 0.001, 0.001 γ = .

1) the results of DE and DE-GD
In the experiment, DE algorithm and DE-GD algorithm are used to solve the DED problem. The results are shown in Table 1. The best solution of DE-GD algorithm is shown in Appendix.
It can be seen from Table 1, the cost of DED problem with DE-GD algorithm has the lower value than the DE algorithm, the difference between the two results reaches 1201.1$.

2) the results of different renewable sources distribution
In order to verify that the algorithm is applicable to different inflow of small hydropower plant distribution and different wind power output distribution, the distribution of wind power and inflow is changed respectively, and the validity of the algorithm is verified by experiment. Figure 3 and Figure 4 are different inflow distribution, and Figure 5 and Figure 6 are different wind power disstribution. The diagrams of these figures are obtained by 10000 times Monte Carlo scenarios with selected values of PDF parameters.  Table 2.
It can be seen from the Table 2 that the results of DE-GD algorithm are similar under different inflow distribution.    Table 3.
It can be seen from the Table 3 that the results of DE-GD algorithm are similar under different wind speed distribution.

Conclusions
In this paper, based on the previous studies on DED, the gradient information and differential evolution algorithm are used to solve the DED problem, which combines wind power and small hydropower with strong randomness. In the process of DE algorithm, gradient descent method is introduced to increase the number of new species, thus enriching the diversity of population. Aiming at the non-differentiable point of the cost function in DED problem, this paper adopts the piecewise gradient method to obtain the gradient of the objective function, and considers that the nondifferentiable point of the objective function is just the local optimum. In order to prevent the algorithm from falling into the local optimum, the gradient of the trend information of the cost function is used at the nondifferentiable point. At the same time, the method is compared with the traditional differential evolution, and the results show the effectiveness of the proposed method. Considering the randomness of small hydropower and wind power, the method proposed in this paper is used to solve the DED problem of small hydropower and wind power with different distribution. The results show that the method is also effective for different renewable energy output scenarios.