Long-term load forecasting based on gravitational search algorithm

Abstract

This paper presents a novel Long-Term Load Forecasting (LTLF) technique based on the new heuristic method, namely Gravitational Search Algorithm (GSA). The objective of the suggested approach is establishing a more accurate LTLF model to minimize the average error of modeling. In order to estimate different fitting functions based on the proposed algorithm, two different case studies include Egyptian and Kuwaiti grids are selected. Also, the results are compared with a conventional approach, namely Least Squares (LS) method, and Particle Swarm Optimization (PSO) as a heuristic algorithm, to select the best LF model. Finally, based on the average and maximum errors arise from the estimations as a decision condition; the best function is selected for the LTLF problem.

Keywords

Energy forecasting demand forecasting long-term forecasting electricity regression gravitational search algorithm

1 Introduction

Load Forecasting (LF) is one of the main and basic processes in the power system planning and operation. It contains both geographical and electrical predictions. In fact, it predicts the electrical load magnitudes and locations for a specified period. Different classification based on the planning horizon durations is introduced. Reference [18] categorized the LF as following:

Short-Term Load Forecasting (STLF): refers to hour-by-hour predictions of 1 day up to 1 week,

Medium-Term Load Forecasting (MTLF): this approach results in daily forecast which, in general, focuses on the load peak of day for several weeks to several month (<1 year),

and Long-Term Load Forecasting (LTLF): this method predicts the load peak of month or the season for several years.

Power system operation in terms of economical and control is affected through the accuracy of LF. In this context, the total demand load depends on many factors such as class of customers, price forecast, time, weather conditions, population, economic indicators (such as Gross Domestic Product (GDP)), and so on (Fig. 1). In fact, it is affected by many individual load behaviors. In order to predict the load more accurately, different load forecasting models are introduced. In this regards, upward and downward prediction errors increase the operational cost [5]. Also, [5] by using a transformation technique, described a regression based daily peak load forecasting method. A similar work for STLF is done in [9]. A hybrid intelligent LF model to take into account both linearity and nonlinearity of load time series as a requirement of precise forecasting for STLF purpose is proposed in [16]. Also, it introduced a Modified Honey Bee Mating Optimization (MHBMO) algorithm to optimize the value of the support vector regression parameters. A review and classification of different models can be found in [17]. Reference [8] provided a comparison of large neural networks (networks with a large number of neurons and weights) with several classical methods. The classical methods ranged from naive forecasting approaches over smoothing filters and combination of them with linear regression. Furthermore, hybrids of smoothing filters and neural networks were considered. The task was to forecast the 24 hours load profile based on data from a local utility in Rio de Janeiro.

Reference [7] proposed a Genetic Algorithm (GA) with variable parameters to predict the electrical demand and GA parameters are tested together to found the best coefficients with minimum error. The Genetic Programming (GP) is used in [12] to predict the future load demand. A similar method based on the Particle Swarm Optimization (PSO) is introduced in [2]. In [1], the PSO approach is extended to LTLF. Reference [11] compared the survey of GA and GP methods, focusing on the solutions of the represented scheme of GA and GP deeply. Another approach to forecast the load demand based on the neuro-fuzzy method is suggested in [6] which used the past annual energy data to train an Adaptive Network based on the Fuzzy Interface System (ANFIS). In this approach, signal trend identification is performed by ANFIS while the power consumption patterns are captured by Artificial Neural Network (ANN). In [14], a combination of Firefly Algorithm (FA) and ANN to reach a more reliable and accurate LF model for STLF is employed in order to forecast the daily peak load of a part of a distribution system. A hybrid method based on the Self Adaptive Modified Bat Algorithm (SAMBA) and ANN is presented in [3] to estimate an accurate STLF model. Reference [13] established the LF model using a knowledge-based expert system, considering the various electrical and nonelectrical variables, obtaining a set of decision rules and storing in the knowledge base to obtain the LF model. A long-term load forecasting using ANN proposed in [4] in which the results are compared with regression approaches provided by EViews package. Also, real GDP and population as two independent variables are selected in [4].

Recently, a novel heuristic search algorithm based on the gravitational law and laws of motion has been proposed by Rashedi et al. in [15], namely Gravitational Search Algorithm (GSA). This new algorithm has been successfully applied to the various nonlinear functions and results demonstrated that it has high performance and is flexible enough to enhance exploitation and exploration abilities.

In this paper, a new algorithm based on the GSA to solve LTLF problem is proposed to establish various LF models (linear and quadratic functions and logistic model). In fact, the scope of this paper is focused on the application of GSA technique for the LTLF. Because of its characteristics, this new approach seems to be a good candidate to solve the real world LTLF problems and as long as authors know it has not been tested on this kind of problems before. Also, different test cases are selected to verify the accuracy and efficiency of the proposed method.

The main advantage of our work is due to its ability in establishing more accurate LTLF models with lower average error. It is clear that the LF study, if properly applied, holds a great saving potential for electric utility corporations. For example, a better estimation by a 1% reduction in forecasting error for a 10,000 MW utility can save up to $1.6 million annually [9]. It should be noted that the contribution in this area derives from the capability of the suggested algorithm in being robust i.e. always capable of finding a good quality solution without convergence problems and mostly yielding a better optimum which results in economical benefits which is our main performance indicator.

The rest of this work is organized as follows: In Section 2, problem formulation in terms of modeling LTLF and its error are formulated. In Section 3, the proposed GSA-based LTLF is illustrated and a brief review of two approaches which the obtained results are compared with them is presented. The LTLF results for two different case studies are presented in Section 4 comparing with PSO algorithm and Least Squares (LS) approach. And finally the conclusions are drawn in Section 5.

2 Problem formulation

2.1 Modeling LTLF

The first step to make a proper LF model is constructing it. There are four different basic methods for LTLF as follows [17]:

Trend analysis: It uses the past data to predict the electrical demand of the future. In this analysis, the curve fitting may be employed. The main drawback of this method is assuming the driving parameters remained unchanged during the study period in finding the demand of the target year.

End-use analysis: This method uses the total energy consumption (based on the estimations) of the appliances of the residential loads explicitly to forecast the energy consumption. Finally, some indirect approaches may be used to convert the forecasted energy to the electrical demand.

Econometric modeling: This method is widely used which is based on the establishing a mathematical relationship among the driving parameters (independent or control variables) and the load demand (dependent variable). Also, it can be applied to the different customer classes and the system as a whole.

Combined analysis: This method combines the econometric and end-use analysis to predict the load.

In this paper, we applied the econometric and logistic models to forecast the future load. Three functions considered in this work to predict the peak of load (annual peak against the year of operation) as follows:

Linear Curve Fitting (LCF) function: $Peak of Load = c_{0} + c_{1} year$ (1)

Second order Curve Fitting (SCF) function: $Peak of Load = c_{0} + c_{1} year + c_{2} {year}^{2}$ (2)

Logistic curve fitting function: $Peak of Load = \frac{c_{0}}{1 + e^{(c_{1} + c_{2} year)}}$ (3)

where year = 1, … q is the time (in year); c_i with i = 1, 2, 3 represents the coefficients of the function which must be determined. It should be noted that in this paper, the year of operation is selected as the independent variable and this can be changed to another one.

2.2 Error of modeling LTLF

Selecting a more accurately LF model is based on some critical decisions. In fact, the obtained results are evaluated based on these. The peak of load of kth year can be expressed as follows (regression model): $\begin{matrix} {Actual Peak of Load}_{k} = \\ {Peak of Load}_{k} + {residual}_{k} \end{matrix}$ (4) in which PeakofLoad_k can be obtained using Equation (1–3); residual_k is the residual load at year k∈ { 1, …, q }. The kth modeling error (in %) can be represented as $\begin{matrix} {Error}_{k} \\ = 100 \\ \times | \frac{{Actual Peak of Load}_{k} - {Peak of Load}_{k}}{{Actual Peak of Load}_{k}} | \\ = 100 \times | \frac{{residual}_{k}}{{Actual Peak of Load}_{k}} | \end{matrix}$ (5)

Analysis of error of modeling is formulated as follows:

Maximum modeling error (maximum absolute error in %) which can be expressed as $\max {{Error}_{1}, \dots, {Error}_{q}}$ (6)

Average modeling error (average absolute error in %) which can be calculated as $Avg . Error = \frac{\sum_{year = 1}^{q} {Error}_{year}}{q}$ (7)

3 GSA-based LTLF

The “gravitation” is the tendency of masses to accelerate toward each other. In fact, each mass (particle) in the universe attracts every other one. In other words, gravity is everywhere and these masses obey the following laws [15]:

Law of gravity: each mass attracts every other mass and the gravitational force between two masses is directly proportional to the product of their masses and inversely proportional to the distance between them.

Law of motion: the current velocity of any mass is equal to the sum of the fraction of its previous velocity and the variation in the velocity. Variation in the velocity or acceleration of any mass is equal to the force acted on the system.

Consider a system with N agents (masses or particles) in which the position of the ith agent can be represented by

$\begin{matrix} X_{i} & = & {[x_{i}^{1}, \dots, x_{i}^{d}, \dots, x_{i}^{n}]}^{T}; \\ i & = & 1, 2, \dots, N \end{matrix}$ (8)

where $x_{i}^{d}$ represents the ith agent’s position in dimension d. (.) ^T is the transposition of (.).

The way Newton’s gravitational force behaves is called “action at a distance”. This means gravity acts between separated masses without any delay and intermediary. In the Newton law of gravity, each mass attracts every other one with a “gravitational force” [10]. This force at a specific time t in dimension d which acting on ith mass from jth mass is directly proportional to the product of their masses and inversely proportional to the square of the distance between them [10] (in this paper, R_ij instead of $R_{ij}^{2}$ is used, because according to [15], R_ij provides better performances than $R_{ij}^{2}$ ) as follows:

$\begin{matrix} F_{ij}^{d} (t) \\ = G (G_{0}, t) \times \frac{M_{i} (t) \times M_{j} (t)}{R_{ij} (t) + ɛ} (x_{j}^{d} (t) - x_{i}^{d} (t)) \end{matrix}$ (9) where $\begin{matrix} R_{ij} (t) = X_{i} (t), X_{j} {(t)}_{2} \\ G (G_{0}, t) = G_{0} e^{- (\frac{α \times t}{{iteration}_{\max}})} \end{matrix}$ and where M_i (t) denotes the ith gravitational mass at time (age or iteration) t; R_ij (t) represents the Euclidian distance between ith and ith mass at time t; G (G₀, t) is the gravitational constant with initial value G₀ at time t; α and iteration_max denote a constant term and maximum iteration (total age of the universe), respectively; ɛ is a small constant.

It should be noted that due to the effect of decreasing gravity, the actual value of the G₀ depends on the actual age of the universe (denoted by t or iteration). A heavier mass means a more efficient agent and corresponds to better solution. This means that better agents have higher attractions and walk more slowly.

To give a stochastic characteristic to the suggested algorithm, it is supposed that the total force $F_{i}^{d} (t)$ that acts on the ith agent in the dth dimension be a randomly weighted sum of dth components of the forces exerted from other masses as follows: $F_{i}^{d} (t) = \sum_{\begin{matrix} j \in K_{best} \\ j \neq i \end{matrix}} {rand}_{j} F_{ij}^{d} (t)$ (10) where rand_j represents a random number in the interval [0, 1]. In the above equation, the number of agents with lapse of time is reduced by performing a good compromise between exploration and exploitation. The exploration is the ability of expanding search space, where the exploitation is the ability of finding the optima around a good solution. Hence, we propose only a set of agents with bigger mass apply their force to the other. In order to avoid trapping in a local optimum, the algorithm must use the exploration at beginning. It is because of the fact that in premier iterations, it explores the search space to find new solutions. By lapse of time, exploration must fade out and exploitation must fade in. So, the algorithm tunes itself in semi-optimal points [15]. For these reasons, only the K_best masses will attract the others. K_best is a function of time, with the initial value K₀ at the beginning and decreasing linearly with lapse of time and finally there will be just one agent applying force to the others [15].

By using the law of motion, the acceleration of the ith mass at time t, and in the dth direction is given as follows: $a_{i}^{d} (t) = \frac{F_{i}^{d} (t)}{M_{i} (t)}$ (11)

The above concepts are shown in Fig. 2.

The next velocity of an agent $v_{i}^{d} (t + 1)$ is considered as a fraction of its current velocity $v_{i}^{d} (t)$ added to its acceleration $a_{i}^{d} (t)$ . Also, the next position of an agent $x_{i}^{d} (t + 1)$ is a function of its current position $x_{i}^{d} (t)$ added to its velocity $v_{i}^{d} (t + 1)$ . Therefore, its position and its velocity could be calculated as follows:

$\begin{matrix} v_{i}^{d} (t + 1) & = & {rand}_{i} \times v_{i}^{d} (t) + a_{i}^{d} (t); \\ i & = & 1, 2, \dots, N \end{matrix}$ (12) $\begin{matrix} x_{i}^{d} (t + 1) & = & x_{i}^{d} (t) + v_{i}^{d} (t + 1); \\ i & = & 1, 2, \dots, N \end{matrix}$ (13) Each mass in each iteration (time t) must be updated as follows: $M_{i} (t) = \frac{m_{i} (t)}{\sum_{j = 1}^{N} m_{j} (t)}$ (14) where $m_{i} (t) = \frac{{fit}_{i} (t) - worst (t)}{best (t) - worst (t)}$ and where fit_i (t) represents the fitness value of the ith agent at time t. best (t) and worst (t) are defined as $\begin{matrix} forminimization \\ problem \end{matrix} {\begin{matrix} best (t) = min_{j \in {1, \dots, N}} {fit}_{j} (t) \\ worst (t) = max_{j \in {1, \dots, N}} {fit}_{j} (t) \end{matrix}$ (15) $\begin{matrix} formaximization \\ problem \end{matrix} {\begin{matrix} best (t) = max_{j \in {1, \dots, N}} {fit}_{j} (t) \\ worst (t) = min_{j \in {1, \dots, N}} {fit}_{j} (t) \end{matrix}$ (16)

The suggested GSA-based LTLF is illustrated in Fig. 3. This algorithm has the following steps:

Step 1. Initialize the parameters of GSA, including G₀, α, K₀, N, and iteration_max.

Step 2. Set the initial control variables c_i using random function.

Step 3. Evaluate the fitness function fit for each agent (mass). In fact, fitness function represents the objective one. In this paper, the following function is considered to be minimized by the proposed algorithm: $fit = Avg . Error (7)$ (17)

In fact, the main purpose of the algorithm isminimizing the average modeling error arises from estimated model. Hence, c_i as the main unknowns (control variables) in Equations (1–3) can be obtained.

Step 4. Update G (t), best (t), worst (t), and M_i (t) for each set of masses. Also, calculate the total gravitational force using Equation (10) for all agents. Then, determine the accelerating and velocity of them through Equations (11) and (12), respectively. Finally, the updated position of each mass can be calculated using Equation (13).

Step 5. If the age of algorithm is equal or less than the maximum iteration (i.e. t ≤ Iteration_max), then repeat Step 3-4. Otherwise, go to Step 6.

Step 6. Print the best results.

3.1 Main approaches for comparison purpose–a briefly review

3.1.1 Least squares method

This approach is a standard method to estimation a mathematical function when there are more equations than unknowns. A common LS method refers to Nonlinear LS (NLS). LS estimation minimizes the sum of squared residuals for finding the parameters values as follows: $\sum_{year = 1}^{q} {residual}_{year}^{2}$ (18)

There is no closed form solution for the parameter estimates; but, the estimates should be satisfied the first-order conditions with respect to the control parameters.

3.1.2 PSO Algorithm

PSO is a global optimization technique. In this algorithm, a number of particles form a “swarm” that fly or evolve throughout the feasible hyperspace to search the optimal solution (if exists) in a fruitful regions. In [1], PSO is used to establish different LF models. In this approach, each particle has two vectors (position and velocity) associated with it, in which, in each iteration, one particle with the best solution shares its position information with the rest of the swarm. In other words, this algorithm updates the population of particles by applying an operator according to the fitness information obtained from the environment so that the individuals of the population can be expected to move towards the better solution [15].

In order to fully understand how the algorithm works, it is interesting to compare the main differences between GSA and another well-known metaheuristic algorithm such as PSO [15]:

In both PSO and GSA, the optimization is obtai-ned by the agent’s movement in the search space; however the movement strategy is different.

In PSO the direction of an agent is calculated using only two best positions, i.e. pbest and gbest. But, in GSA, the direction of an agent is calculated based on the overall force obtained by all other masses.

In PSO, the updating procedure is performed without considering the quality of the solutions and the fitness values are not important in the updating procedure; differently, in GSA the force is proportional to the fitness value and so the masses see the search space around themselves in the area of influence of the force.

PSO uses a kind of memory for updating the velocity (due to pbest and gbest). Differently, GSA is memory-less and only the current position of the agents plays a major role in updating procedure.

4 Simulation and results

LF study if properly done, holds a great saving potential for electric utility corporations. For this purpose, an accurate LF model should be employed to predict the peak of load of the future years. The GSA parameters are selected as G₀ = 400, α = 1, K₀ = N = 200, and iteration_max = 500.

4.1 Case study A–Egyptian grid

In this case, the peak loads of Egyptian power system during for 17 years of operation (1977–1993) are used to estimate the parameters of linear, nonlinear, and logistic long-term forecasting models. The peak demands are illustrated in Fig. 4 [1].

The obtained coefficients of three mentioned functions by the LS approach, PSO method [1], and the proposed algorithm are shown in Table 1 (for LCF), Table 2 (for SCF), and Table 3 (for logistic function).

The load forecasting results obtained by the suggested approach for LCF, SCF, and logistic function are compared with those obtained by the PSO algorithm [1] and LS method in Figs. 5–7. Also, Table 4 illustrates the model errors in terms of maximum and average ones as Equations (6) and (7), respectively.

The results indicate that the introduced algorithm for three mentioned functions has lowest average error (3.80% , 3.33% , and 2.25% for LCF, and SCF, and logistic model, respectively); but the maximum error in comparison with PSO approach for LCF is higher. Hence, logistic model based on the GSA is more accurate for LTLF in the Egyptian network with maximum error 7.70% and average one 2.25% .

4.2 Case study B–Kuwaiti grid

Peak demands of Kuwaiti power network during 1992–2005 (14 operation years) are used to estimate the parameters of LCF, SCF, and logistic long-term forecasting models. The peak demands for Kuwaiti grid are shown in Fig. 8 [1].

The results in terms of coefficients of three mentioned functions are illustrated in Tables 5–7, respectively.

The load forecasting results obtained by the suggested approach for three presented models are compared with those obtained by the LS and PSO [1] methods in Figs. 9–11. Also, the model errors are shown in Table 8. Based on the results, it is clear that the proposed algorithm can estimate LCF, SCF, and logistic functions with lower average errors in which SCF model with maximum and average errors 3.80% and 1.33% , respectively, for GSA is the best choice for LTLF in the Kuwaiti network.

5 Conclusions

In this paper, a novel LTLF approach based on a heuristic and powerful algorithm, namely GSA is suggested to construct an accurate LF model. For this purpose, firstly, three widely used models, namely LCF, SCF, logistic functions are selected and then a forecasting method is suggested to find an LF model with lowest average error. Finally, the proposed technique is evaluated on two different case studies, namely Egyptian and Kuwaiti grids, and the obtained results are compared with those obtained by PSO and LS methods to introduce the best models based on the minimum maximum and average errors. The suggested algorithm indicated the capability of providing a good solution with higher benefits since it yields a better search of the global optimum to find a more accurate LTLF model. The simulation results show that the suggested algorithm is capable to solve different prediction problem and provide better estimation.

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