Abstract
Abstract
In the modern power system analysis, the investigation on the impacts of growing wind power generation can be mentioned as an inseparable part of power grids. The development of wind-thermal coordination economic dispatch leads to an optimal dispatch scheme and a mature wind power efficiency. Adding further uncertainties to the study of power systems is one of the effects of intermittent nature of wind energy which is apart from random nature of realistic power grids. Now in order to inspect these uncertainties, an efficient probabilistic method is needed to comfort the issue. In this paper a novel optimization method based on Social Spider Optimization (SSO) is presented to meet large non-convex Economic Dispatch (ED) problems and uncertainties in the network caused by wind turbines, simultaneously. To provide a more realistic model, following constrains have been included in the proposed Probabilistic ED (PED) formulation: ramp rate limits, Prohibited Operating Zones (POZs), system spinning reserve, valve loading consequences, and multiple fuel choice. Unscented Transform (UT) approach is utilized in this paper to model the uncertainty in wind speed and thereupon solve the PED. In order to show the efficiency of the presented algorithm, it is successfully tested on different systems and compared with several of the most recently published ED methods.
Introduction
With respect to the growing demand of renewable energy sources especially wind power, additional challenges have been introduced to maintain economical and reliable operation of modern power systems. In contrast to the various promising advantages such as reducing the dependence on fossil fuels, enhancing the independence and flexibility of large power grid; intermittent nature of wind and its uncertainty put it at a disadvantage of more complexity in analyzing power systems. An appropriate modeling of the wind’s variability is required to minimize the total operating cost in the ED model. The ED problem aims to cope with the minimum cost of power production in electrical power system analysis by assigning particular outputs to various available generators [1, 2]. Adoption and variation of high penetration wind power can affect economic dispatch of power system which is the outcome of not using fossil fuel.
Equality and inequality constraints associated to the Practical ED result in complexity and nonlinearity for the problem. The machines or associated auxiliary generating units or the network may impose mentioned characteristics to the issue which preserve reliability of the system during disturbances. For generating units, operating in Prohibited Operating Zones (POZs) may cause amplification of vibrations in their shaft bearing. That is the reason in practical application, generating units are avoided to be operated in these regions [3]. Therefore, several algorithms such as Integrated Artificial Intelligence (IAI) technique [4], Decomposition Technique (DT) [3], Genetic Algorithm (GA) [5], Particle Swarm Optimization (PSO) approach [6] and advantageous decision space approach [7] take POZs into account in the ED problem. Besides, ramp rate limits in actual operation of units are consequences of change restriction in the unit generation output [5]. Monitoring system operating reserve and its ability to respond to disturbances within the system is a major task to be sure about reliability of practical ED [8]. The system spinning reserve has a key role while there are uncertainties included the system such as uncertainty in generation. Considering proper reserve capacity results in overcoming unscheduled generator outages and major load forecast errors without load shedding. Furthermore some techniques such as decomposition technique [3], PSO with the sequential quadratic programming (SQP) [9], and Bender’s Decomposition (BD) [10] presented to deal with reserve constraints in ED problem.
In addition, massive new generating units with multi-valve steam turbines produce a ripple-like heat rate curve. In these units, a number of steam admission valves are opened sequentially to acquire increasing output from each unit [11]. Numerous methods such as Dynamic Programming (DP) [1], Improved GA (IGA) [12], Hybrid GA (HGA) [13], and Evolutionary Strategy Optimization (ESO) [14], Evolutionary Programming (EP) [15], Hybrid PSO (HPSO) [16], Modified PSO (MPSO) with a dynamic search space reduction strategy [17], GA approach [11, 18], CSA [19], a fast EP using the weighted mean of Gaussian and Cauchy mutations, called MFEP [19] and hybrid EP combined with SQP [20] have presumed valve-loading effect. As long as many generating units are provided with multiple fuel options, they should be organized to burn the most economic fuel [21]. Several heuristic algorithms such as Hierarchical Method (HM) [21], and Taguchi Method (TM) [22] have considered this subject.
By considering all above-mentioned issues, we must think about ramp rate limits, POZs, system spinning reserve, valve loading effects, and multiple fuel source options to solve a realistic ED problem. Aforesaid situation makes it hard to find the optimum solution. Thus, a variety of intelligent optimization methods and mathematical programming are utilized to solve the classic ED. Some of these algorithms can be named as follows: PSO approach [23], neural networks [24], Lagrange relaxation [25] and mixed integer programming [26]. While, a few recently published papers such as various Evolutionary Algorithms (EA) [27], improved GA with Multiplier Updating (IGA-MU) [28] and New PSO (NPSO) with Local Random Search (NPSO-LRS) [29] considered both valve loading effects and multi-fuel sources together to solve ED. Panigrahi et al. proposed a novel stochastic optimization approach using hybrid Bacterial Foraging (BF) technique by including ramp rate limits and POZs [30]. In order to find a solution for ED problem, a GA method called Real Coded GA (RCGA) is proposed in [31] while no uncertainties is considered in the model.
In [32], a new evolutionary algorithm called self-tuning Hybrid Differential Evolutionary (self-tuning HDE) was presented which take POZs and valve loading effects into account simultaneously.
As a matter of fact, increase in penetration of wind turbines causes troubles for the power system operator such as allocating system power among conventional generators and also among various available wind-powered generators. The stochastic nature of wind speed is the principal feature that determines the differences between wind-powered and conventional generators. In spite of the development in the combination of thermal and wind energy sources in ED modeling, it is still necessary to specify the stochastic nature of the wind speed. Recently, new researches have been conducted in the area of classical economic dispatch modeling which includes wind energy conversion system generator in [33]. In order to assess the marginal amount of reserve capacity needed for secure operation of the power system, a probabilistic technique has been represented. In this model, the author aims to reduce the total operating cost to minimum amount based on the cost function and distribution of the wind. In comparison with fossil fuels based generation, the operating cost of the wind power is much less expensive. With respect to the possibility of over/underestimation of wind power (due to wind variability), the factors for both under and over estimation of available wind power should also be included in the ED problem. The Weibull probability distribution function is employed to demonstrate variability of the wind speed i.e. the future wind speed and hence the power. The proposed method in [34] models the wind powers as inputs parameters while spinning reserve is taken into consideration. As a matter of fact, during daytime power load is heavy while output of wind farms is small. On the other side, this situation reverses during night time which leads to a dramatic change in output of wind farms in different periods of a day. In [35], a Monte Carlo Simulation (MCS) technique is utilized to produce a variety of scenarios to represent the volatility of wind power based on forecasting error. In this paper, two separated steps are considered for calculation of objective function and examination of the violation of constraints.
The main purpose of this paper is to incorporate a Probabilistic ED (PED) with wind-powered generators and solve the problem via numerical solutions. In this regard, the Unscented Transform (UT) [36] is employed instead of traditional approach based on MCS. To compare UT with MCS method, requiring fewer amounts of data causes development from the viewpoint of computational effort and time for TU. The UT approach resolves 2×m+1 deterministic ED to obtain the behavior of m random variable which is its major advantage. Nevertheless, both MCS and UT methods employ deterministic approach to find a solution for probabilistic problems by using only few first statistical moments of input random variables such as mean, variance, skewness, and kurtosis [37, 38].
As it is said in previous discussion, our paper focuses on a kind of sophisticated nonlinear optimization problem with equality and inequality constraints for more practical PED. The Social Spider Optimization (SSO) is a new optimization algorithm which finds a set of optimal solutions based on choosing three sub-modification methods. Each of these answers can be the best satisfying solution according to the system operator.
The basic outline of this paper can be categorized as follows:
In the first section a PED is introduced by incorporating uncertainties of wind generation as well as the load value in the realistic ED problem. Besides, Ramp rate limits, system spinning reserve, POZs, influence of valve loading and multiple fuel choices are assumed to have more practical solution for PED problem. A UT based procedure is used to handle the uncertainties which has the benefit of saving time and reducing the computational burden.
Here, a modification of Social Spider Optimization (MSSO) is suggested. The modified optimization method is able to provide a more diverse search in the solution space. Therefore, better optimum solutions with low computation burden can be assessed compared to other stochastic search techniques proposed in the previous researches. Finally, the usefulness of the proposed algorithm is validated by examination on different test systems and comparison with several ED techniques.
Problem formulation
The procedure for ED problem which incorporate wind turbines involves the allocation of generation among wind and thermal plants. So that the total production costs should be minimized while satisfying various constraints.
Objective function
The ultimate goal in ED problem is to reduce the total generation cost of the power system to minimum point during a specific time interval. Besides, there are various constraints which should be met. Therefore, we can formulate the optimization problem as below:
In the mentioned formulation, F
i
(P
i
) refers to the fuel cost of a thermal generating unit. It is supposed as a second order polynomial function of its generation P
i
as in (2).
Meanwhile, valve effects cause a nonlinear cost variation when the output of generating unit changes [11]. By adding this variation to the above cost function, we have:
It should be mentioned that . Now, final formulation by taking valve loading effect and multiple fuel choice into consideration can be summarized as follows:
In the classical models, only the generation limits and power balance constraint had been exerted to the problem. Other than that, overall active loss of the system P Loss was ignored or approximated by function of the unit power outputs and the transmission loss matrix coefficients called B matrix loss formula [29, 30–32]. However, in practical situations, scheduling is affected by complex and nonlinear characteristics of ED problem and many equality and inequality constraints. In this regard, this paper take the following constrains into account:
(1) Power balance constraints:
Regards to the classical approach, P
Loss
is ignored or approximated B matrix loss formula as below:
(2) Ramp rate limits: we assume the ramp rate limits based on the power outputs of units in the previous hour P0i.
(3) Generating capacity constraints: by employing aforesaid ramp rates, the upper and lower limit on generating capacity of the units will face a modification as follows:
(4) Prohibited Operating Zones: The following description depicts the feasible operating zones of the ith generating unit:
It is crystal clear that if we consider NP i prohibited zones for a generating unit, a convex set is unified from its NP i + 1 disjoint operating regions [3].
(5) System spinning reserves capacity: System spinning reserve by means of load regulation is one of the reasons that make units to operate in the POZs. Consequently, POZs strictly have direct influence on the flexibility of the unit to regulate system load. Accordingly, spinning reserve is assumed not to chip in units with POZs. Therefore, the spinning reserve constraint considering prohibited operating zones constraints are described as below [31]:
The output of wind generators is stochastic and depends on availability of the primary source, i.e., wind. Because of the fact that its output cannot be regulated, wind energy forecasting is needed in optimization procedure. Forecasting tools for wind generation are important issues which are beyond the purpose of this text. The alternation of wind and randomness of its speed can be mentioned as two major drawbacks of wind generators. Thus, the prediction of wind power develops the uncertainty largely. As a result, this paper has considered a mathematical approach for uncertainty analysis [39]. According to [40], the Weibull distribution function can model the wind speed at a certain location. Two parameters have been utilized named as: scale parameter, C, and shape parameter, k.
Weibull distribution plays the role of stochastic distribution of the wind speed. It is possible to derive probability of wind power distribution from the wind speed distribution. By using the power curve of the turbine and based on the predicted wind speed, the active power generation of each wind turbine can be easily calculated. The power curve of the turbine is specified by the manufacturer. The wind power production is restricted between the cut-in speed and the cut-out. A constant power output is gained from the interval of the rated speed and cut-out speed. In the following formulation the output power of a wind farm is described:
Due to random characteristic of the natural phenomena, most of the engineering problems are performed in a stochastic environment with related uncertainty [41, 42, 41, 42]. This situation develops the demand of a powerful stochastic structure to solve the problem with uncertainty. In [43], authors have proposed one of the most powerful tools which is applicable for modeling uncertainty in the nonlinear correlated transformations. Easy coding and simplicity are two profits of this technique. Moreover, another advantage of this approach can be named as: high ability of capturing uncertainty, low computational burden and high capability for modeling uncertainty in the correlated environment. Actually, in the UT technique, the process of approximating a probability distribution function (PDF) is quite simple in comparison with an arbitrary nonlinear function [43]. In order to preserve sufficient information about the input variable’s PDF, proper samples of input variables is generated. The results in [43] illustrate the high eligibility of UT technique over the other well-known methods. To better perceive forenamed algorithm, assume that we want to model the uncertainty of the nonlinear stochastic problem y = f(X). If the problem has n number of uncertain parameters, then the input vector X is a vector of the length n with the mean value μ and covariance P xx . While, y is the output vector and f is the nonlinear function. It can be realized that the symmetrical elements of matrix P xx are the variance of the uncertain variables. On the other hand, the non-symmetrical elements are the covariance among different uncertain parameters. For the sake of modeling the uncertainty, the UT method will solve a problem with n uncertain variables 2n+1 times. The following steps have been accomplished by the UT technique to attain the mean μ y and covariance matrix P yy of the output y:
In a symbolic definition, the kth row or column of matrix A is described as (A) k and w0 is the weight of the mean value μ.
It should be note that the summation of the weighting factors should obey the following expression:
Original Social Spider Optimization (SSO)
For the first time, in 2012, the cooperative behavior of social spiders in a colony was modeled in an optimization algorithm named SSO [44]. Female spider’s socialization differs from male insects. They attract or repulse other spiders regardless of their gender. Since the intensity of this interaction is determined by the spider weight and their distance, thus bigger spiders in the swarm are better ones. Generally, in this algorithm, male insects are classified in two groups of dominant and non-dominant according to their size. Dominant males are elected to mate with female insects and consequently ensure evolution and survival of the colony. Besides, the non-dominant males gather in the center of the male population. This labor prevents the search process from chasing abnormal and unusual individuals. The optimization procedure starts with N
F
and N
M
female and male individuals, respectively, which are generated randomly. The population size can be demonstrated by N
P
= N
F
+ N
M
. 65% – 90% of the population is formed by female spiders which show the female-based characteristic of population. Once the finesses are calculated, a weight (w
i
) is assigned to each spider. w
i
is calculated by using the best (f
b
) and worst (f
w
) values of fitness function asfollows:
The position adjustment of females which is influenced by their interactive is formulated as below:
The aforesaid equation determines both attractive and repulsive interactions of ith female spider by positive and negative signs, respectively. Attractive interaction will take place, providing a generated random number (α4) is less than attraction threshold (P attraction ). In the other condition, the repulsive interaction will be accomplished. In the above formulation, we make use of the closest spider (X c ), the best solution (X b ), and random item (α3) which is represented by the last term in (25). It is evident that Euclidian distance of ith individual to her closest spider (d ic ) and the best solution (d ib ) have an impact on the aforementioned movement. It was said that male spiders are grouped in dominant and non-dominant sets that helps in updating their position. In this regard, the male population is categorized based on weight values. In this arrangement, the middle individual is tagged as median member. So that, male members which weight above the median are classified into dominant set and others belong to non-dominant set. Thus, the updating process of dominant male’s position obeys the following description:
On the other side, since the non-dominant males are attracted to the weighted mean of the male population (M
w
), the formulation is modified as below:
In the end, the dominant males and female spiders will mate within a particular mating range (r
m
). This process in done based on Roulette Wheel Mechanism (RWM). The mating range is given below:
The generated spider can inherits its gender by replacing the solution with worst weight or fitness value and inherits its gender. Figure 1 depicts the flowchart of the original SSO.
Several privilege of SSO technique such as simple idea, generic application, high robustness, persuade us to use this algorithm. Similar to any other evolutionary algorithm such as TLBO [45, 46], FA [47] and CSA [48], the main disadvantage of the proposed MSSO algorithm is the stochastic nature of the algorithm. This is a general feature among the evolutionary algorithms. In this section, a novel self-adaptive modification method is implemented to increase the total search ability of the SSO. In addition, this improvement avoids the procedure to face a premature convergence. Three modification techniques have been considered for proposed self-adaptive modification as below:
Modification method 1: In the first modification method, Lévy flight is employed to enhance the diversity of the population. References [49] use a heavy-tailed probability distribution to describe the step-lengths of random walk in Lévy flight. The following mathematical formulation explains the Lévy flight:
The last solution is replaced by the new one if it has a better result.
Modification method 2: This modification method benefits crossover and mutation operators from GA. This manner increase the diversity of the spiders in the population. Therefore, three spiders S1, S2 and S3 are selected for each spider X
i
from the population. This process is performed randomly such that S1 ≠ S2 ≠ S3 ≠ i. afterward, based on the mutation operation from the genetic algorithm, two new test spiders are assessed as below:
Again, the best solution amongst XTest,1, XTest,2 and X i will replace X i .
Modification method 3: This modification method attempt to move the average of the population toward the best solution detected during the optimization. The evaluation the mean value of the population is operated column-wise M
p
. then, the following equation is utilized to perform the updating procedure of each solution in the population:
Prosperity of the proposed approach to solve proposed realistic ED and PED problem is examined by means of three test systems. We can attain the optimal ED only if each unit works with the most economic fuel to burn. The first example ran with ten generating units considering valve-point effects and multi-fuel options in the same time. On the other hand, the second test case consists of 15 dispatching units considering POZs for the generating units. The specifications of all 15 thermal units have been described in the respective reference. The requested load of the system is equal to 2630 MW. In order to assess power losses of the system, The B loss coefficients matrix was put into practice. In addition, the third test system consists of 40 units taking valve-point effects into account [19]. This problem is mostly probable to trap in many local minima, while the global minimum is hard to get. Based on previous discussion, the active power generation of each wind turbine can be easily computed by employing power curve of the turbine. The Table 1, mentioned below, submits the characteristics of power generation model.
The presented MSSO starts with a random initial population. For aforesaid test systems, the best, average, and worst solutions of the proposed MSSO are illustrated in Tables 2–4 respectively. In order to have better comparison, the results of several recently published ED solution methods (with similar trial runs) are also mentioned in these Tables. It should be noted that the results of the other methods reported in these tables have been directly repeated from their own references. Also, the convergence diagrams of algorithm are plotted in Figs. 2–4. From Figs. 2–4, it is seen that the proposed algorithm could converge in the first step. These figures show the fast convergence ability of the algorithm clearly.
It is evident from Tables 2–4, that the proposed MSSO reaches better results in the best, average, and worst solutions compared to the all other methods in all two ED test systems. The other subject that can be seen in all Tables is that the worst result of the proposed MSSO is better than the best result of all other methods. These comparisons display the efficiency of the presented MSSO as a proficient tool in solving the practical ED problem in different test systems. Besides, the fifth column of Tables 2 and 4 illustrate the effectiveness of the proposed MSSO on reducing the computational time and burden. Despite the fact that the algorithm does not improve in time consumption in comparison with the previous method, Table 2 shows better results. It should be noted that the mean CPU time in the previous referred methods is about 2.3261 minutes. However, the proposed MSSO has shown better performance in minimizing the cost function in a significant less time.
In order to prove the success of the proposed MSSO in solving PED, the uncertainty in wind generation is brought up in the test systems. In both systems three situations are simulated, 1) Turning a blind eye to wind generation, 2) Taking the wind generation into consideration as a deterministic procedure and 3) Performing the PED optimization by considering the randomness of wind generation using a Weibull distribution to present the wind speed. Since no wind-powered generator is involved in the PED problem of the first method, the results are similar to ED test systems completely. In the second method the wind generator with deterministic generation is and its output is dispatched like thermal units through optimization problem. In the end, the variation of the wind generation is added to the optimization problem. In this regard, the proposed MSSO attempts to minimize the generating cost of the thermal units by assigning economic outputs to them. For the first and second test systems, Tables 5 and 7 show the best, average and worst costs and Tables 6 and 8 give the detailed output of the generating units.
It can be seen that in comparison with the first method, generation costs for the 10-unit system rise in the third method. Tables 5 and 6 clarify this raise in the cost function. The first and second units from the 10-unit system were supposed to be wind turbines which leads to less power generation in the third method. This decrement in power generation caused by these two units forces other units to increase their output. Actually this labor is performed to satisfy the power balance between supply and demand. The same issue can be discussed for 15-unit test system as evident from Tables 7 and 8. Considering wind-power units (i.e. unit 13, 14 and 15) beside the uncertainty effects (due to stochastic framework) would reduce the total system costs. A thorough observation on the results gives beneficial evidence to prove the robustness and effectiveness of the proposed MSSO. It is crystal clear that noting the intermittency and randomness of the wind generation is a critical issue. The choice of the adequate reserve guarantees a successful dispatch of the units or better to say assures less risk and cost. In Tables 9 and 10, a more inclusive conclusion is presented for the third test system. Accordingly, we can diminish the cost of operation by considering the randomness of the wind speed. The wind turbines located in unit 27, 28 and 29 generate more power in the stochastic method thus forcing other units to decrease their output to preserve the power balance in the network. As a consequence, this reduction leads to a decrement in generation costs and pollution caused by thermal-units.
Conclusion
This paper proposes a novel stochastic PED model by including ramp rate limits, POZs, system spinning reserve, valve loading effects, and multiple fuel options under uncertainty of wind power generation. Other than that, a new MSSO is introduced to solve the proposed realistic ED and PED model which gives a better optimum solution by providing a more diverse search. In comparison with previous stochastic search techniques, the proposed MSSO saves time with lower computation burden for solving ED and PED problems. As it was shown, the proposed optimization technique successfully compromises between discovering the global minimum and diminishing computation time. Moreover, in order to handle the uncertainties imposed by the wind generation, UT seems appropriate. In the end, the results demonstrate the robustness and efficacy of the suggested algorithm.