Scenario-based wind speed estimation using a new hybrid metaheuristic model: Particle swarm optimization and radial movement optimization

Abstract

This paper presents a new hybrid metaheuristic model in order to estimate wind speeds accurately. The study was started by the training process of artificial neural networks with some metaheuristic algorithms such as evolutionary strategy, genetic algorithm, ant colony optimization, probability-based incremental learning, particle swarm optimization, and radial movement optimization in the literature. The success of each model is recorded in graphs. In order to make the closest estimation and to increase the system stability, a new hybrid metaheuristic model was developed using particle swarm optimization and radial movement optimization, and the training process of artificial neural networks was performed with this new model. The data were obtained by real-time measurements from a 63-m-high wind measurement station built at the coordinates of UTM E 263254 and N 4173479, altitude 1313 m. Two different scenarios were created using actual data and applied to all models. It was observed that the error values in the designed new hybrid metaheuristic model were lower than those of the other models.

Keywords

Radial movement optimization particle swarm optimization artificial neural networks wind speed estimation hybrid metaheuristic algorithm

Introduction

Wind energy is an environment-friendly, unlimited, most economical type of energy which has a broad area of usage and is developing rapidly.^1,2 However, the stochastic and intermittent structure of wind is the main factor that makes it difficult to estimate the speed and power of wind.^3,4

The demand for electricity production by wind energy is accelerating day by day. This demand brings about an increase in the number of wind farms as well as some related problems. The main ones among these problems may be listed as grid reliability, power system quality, and interconnected linking operations.^2,5,6 Performing procedures of wind speed and wind power estimation is effective in solving such problems. In addition, the procedure of wind speed and wind power estimation is also highly important in terms of determining the locations of new wind farms to be built,⁷ facility maintenance and energy planning,⁸ energy transformation efficiency and reduction of surge risks,⁹ and optimal management of the benefits and risks of wind farms.¹⁰ Wind speed and power estimation studies are discussed under four groups in terms of time. These are immediate-run estimation, short-run estimation, medium-run estimation, and long-run estimation.^11–13

Wind speed and wind power estimation studies may also be categorized based on their estimation models as the persistence method,^13–15 physical method,¹⁶ statistical method,¹⁷ and hybrid method.¹⁸ The persistence method is the simplest method for estimating wind and is based on the assumption of a high correlation between the existent and future wind values. This method assumes that the wind speed value of any moment “t” will be the same as the value at a moment “t + Δt.”^13,15 In the physical method, wind speed estimations are based on physical parameters and meteorological projections. The statistical method involves wind power estimations to be produced using past wind data.^19,20 Table 1 shows the categorization of wind estimation in terms of time.

Table 1.

Categorization of wind estimation in terms of time.¹³

Period	Interval	Practices
Immediate run	From a few minutes up to 30 min	Electricity market balancing Real-time grid operations Regulatory operations
Short run	From 30 min up to 6 h	Economic load distribution planning Making reasonable decisions on load Operational security in the electricity market
Medium run	From 6 h up to 1 day	Unit commitment decisions Reserve requirement decisions Generator online/offline status decisions
Long run	From 1 day up to 1 week or longer	Maintenance planning Establishment management Optimum operational cost Feasibility for wind farm design

While the physical method is highly successful in long-run estimations, the statistical method is successful in short-run estimations. The physical method uses parameters like terrain, slope, pressure, and temperature to make estimations of wind speed and power, while the statistical method uses statistical models.^8,21,22 The hybrid method is a method that combines different methods that have unique abilities and characteristics. The purpose of such a method is to obtain the most suitable estimation performance by utilizing the advantages of each model that is included.¹⁸ Studies in the literature developed hybrid models which utilized two, three, and even four components. Some of these studies are listed in Table 2.

Table 2.

Hybrid models for wind speed estimation in the literature.

Hybrid models	Recording frequency	Number of records	Error models	References
GA + BP + EMD	10 min 1 h	700 500	MAPE, RMSE	Wang et al.²³
FEEMD + CS + WNN	10 min	2930	MAE, MAPE, MSE	Xu et al.²⁴
ANN + GFS	30 min	7856	MAE	Sanz et al.²⁵
W + SVM	1-2-3-4-5 h	80	MAE, MAPE, RMSE	Chen et al.²⁶
ARIMA + DENFIS	2 h 6 h 12 h	732 120 62	MAE, MAPE, RMSE	Ye et al.²⁷
HHT + ANN HHT + SVM HHT + RF HHT + BT	4 h 6 h 24 h	900 600 150	MAE, RMSE	Spiryaev et al.²⁸
AA + WLR	24 h	300	ARE	Guo et al.²⁹
EMD + MkRVR	24 h	200	MAPE	Fei³⁰
AR + GPR	1 h	1464	MAE, MAPE, RMSE	Zhang et al.³¹
EWT + CSA + LSSVM	30 min	100	MAE, MAPE, RMSE	Hu et al.³²
GA + APSO + WNN	10 min	3030	MAE, MAPE, MSE	Wang et al.³³
EEMD + SVM	1 month	72	MAE, MAPE	Hu et al.²⁰
EMD + FS + ANN/(DSF-ANN) EMD + FS + SVM/(DSF-SVM)	1 h	744	MAE, RMSE	Zhang et al.³⁴
ANN + MC	a few minutes	4200	MAPE	Pourmousavi Kani and Ardehali³⁵
ARIMA + ELM + SVM + LSSVM	15 min 30 min	576	MAPE	Wang and Hu³⁶
FEEMD + MLP FEEMD + ANFIS WP + MLP WP + ANFIS	30 min	700	MAE, MAPE, MSE	Liu et al.³⁷
SC + ESN + GA	15 min	5800	MAPE, MAE, nRMSE	Liu et al.³⁸
ARIMA + ANN	1 h	48	ME, MAE, MSE	Cadenas and Rivera³⁹
EMD + ANN	1 h	500	MAE, MSE, MAPE	Liu et al.⁴⁰
ARIMA + KF + PSO	24 h	120	ARE, MAE, RMSE	Su et al.⁴¹
WD + ABC + RVM	24 h	365	MAPE	Fei and He⁴²
WP + BFGS WP + ARIMA + BFGS W + BFGS	30 min	700	MAE, MAPE, MSE	Liu et al.⁴³
WT + SVM + GA	30 min	1440	MAE, MAPE, RMSE	Liu et al.⁴⁴

ABC: artificial bee colony algorithm; ANFIS: adaptive neuro-fuzzy inference system; ANNs: artificial neural networks; ARIMA: auto-regressive integrated moving average; BP: backpropagation; ELM: extreme learning machine; EMD: empirical mode decomposition; ESN: echo state networks; FEEMD: fast ensemble empirical mode decomposition; FIS: fuzzy inference system, GA: genetic algorithm; GS: grid search; KF: Kalman filter; MC: Markov chain; PSO: particle swarm optimization; SC: spectral clustering; SVM: support vector machine; WD: wavelet decomposition; WT: wavelet transform; DENFIS: dynamic evolving neural-fuzzy inference system; GPR: Gaussian process regression; EWT: empirical wavelet transform; CSA: coupled simulated annealing; LSSVM: least-squares support vector machine; APSO: adaptive particle swarm optimization; WP: wavelet packet; MLP: multilayer perceptron; RVM: relevance vector machine; BFGS: Broyden–Fletcher–Goldfarb–Shanno; MAPE: mean absolute percentage error; MSE: mean square error; RMSE: root mean square error; MAE: mean absolute error; nRMSE: normalized root mean square error; WNN: Wavelet Neural Network; CS: Cuckoo Search; ME: Mean Error; GFS: Global Forecasting System; HHT: Hilbert–Huang Transform; RF: Radial Function; BT: Boosting Trees; AA: Apriori Algorithm; WLR: Weighted Local-Region; DSF: Decomposition Selection Forecasting; MkRVR: Multiple-Kernel Relevance Vector Regression; ARE: Absolute Relative Error.

In this study, training studies of artificial neural networks (ANNs) were considered as a problem and it was aimed to develop a model to be used in the ANN training process which is accurate and fast and works stably. With this aim, the study was started using some metaheuristic algorithms in the literature in order to be used in the ANN training process. The success of the developed hybrid metaheuristic models of ANNs trained with evolutionary strategy (ES), ANNs trained with genetic algorithm (GA), ANNs trained with ant colony optimization (ACO), ANNs trained with probability-based incremental learning (PBIL), ANNs trained with particle swarm optimization (PSO), and ANNs trained with radial movement optimization (RMO) was recorded and plotted. In order to make the closest estimation and to increase the system stability, a new hybrid metaheuristic model was developed using PSO and RMO, and the training process of ANNs was performed with this new hybrid metaheuristic model. Two scenarios were created and all the models were applied. The data were used in scenarios obtained from a wind measurement station built in a 63-m-high university campus. The success of the new designed hybrid metaheuristic model (ANNs trained with PRO + RMO) was compared with that of the other hybrid models.

Data processing

As shown in Figure 1, the wind measurement station was mounted at the coordinates of UTM E 263254 and N 4173479 and an altitude of 1313 m by steel guy ropes.

Figure 1.

Geographical location of the wind measurement station.

The three-dimensional (3D) depiction of the wind measurement station is shown in Figure 2.

Figure 2.

3D depiction of the wind measurement station.

The measurement mast consisted of 21 modules of 3 m height and a total height of 63 m. The devices that were installed on the wind measurement station are given in Table 3.

Table 3.

Devices on the wind measurement station.

Devices	Height (m)
Anemometers	61
	30
Wind aspect sensors	59
	28
Temperature sensor	4
Humidity sensor	4
Pressure sensor	3.5
Data logger	3.5

Installation works of the wind measurement station are shown in Figure 3.

Figure 3.

Installation works of the wind measurement station.

Statistical values of the data set (August 2014) are given in Table 4.

Table 4.

Statistical values of the data set (August 2014).

Parameters	Average	Median	Standard deviation	Mode	Max	Min	Skewness	Kurtosis	Variance	Number of samples
Wind speed 61 m (m/s)	4.592	4.435	0.991	5.165	8.797	2.432	0.939	2.604	0.981	4464
Wind speed 31 m (m/s)	4.313	4.153	0.927	4.258	8.431	2.343	1.129	3.283	0.859	4464
Wind aspect 59 m (°)	161.689	156.717	56.931	149.179	282.038	41.013	0.092	−0.729	3241.174	4464
Wind aspect 29 m (°)	175.019	168.025	59.581	142.464	299.036	29.847	0.035	−0.734	3549.893	4464
Temperature (°C)	23.101	23.396	2.356	21.087	28.409	17.217	−0.259	−0.436	5.551	4464
Humidity (%)	46.638	47.642	13.329	61.670	76.673	26.791	0.292	−1.089	177.654	4464
Pressure (mbar)	865.763	866.348	2.354	867.423	870.448	861.358	−0.136	−0.640	5.544	4464

Design of a new hybrid metaheuristic model

Metaheuristic algorithms are one step higher than heuristic algorithms, and they can select and apply heuristic algorithms based on the solution of the problem. They have some advantages of minimizing the complexity of calculations, success in nonlinear problems, capabilities of recording previous findings in local searches, and possibilities of producing effective results within a short time.^45–47 The algorithms that constituted the model are presented in sections “RMO” and “PSO.”

RMO

RMO is a swarm-based, fast, simple, and effective metaheuristic optimization model which uses the spherical boundaries of a vector in the search space to find the optimal solution and was designed for the spherical optimization of complex and nonlinear optimization problems.⁴⁶ In this algorithm, iteration cannot be transferred among the position and speed of the particles. Particles move from an updated point in each iteration, and therefore there is only a need for memory that has fewer calculations. The presence of a global best vector in the updating process prevents the algorithm from getting stuck in a local optimum.⁴⁸

The RMO algorithm is initiated by starting the particles in the search space that indicates the solutions of the problem. The positions of the particles are shown with a nop (number of particles) × nod (number of dimensions) matrix that is named as X_ij. Nop is normally equal to the number of variables that would be optimized, but it can be selected by the user on demand.^48,49

The step after obtaining the center point (cp) once is the dispersion of the particles from cp along the radius. This moves the particles from V_ij-vector-based cp along the radius as straight lines. The V_ij vector is a random nop × nop vector that is obtained by the equation below. The coefficient k in the equation should be a carefully selected integer^45,47–49

V_{ij} = rand (0, 1) \times V_{\max (j)} \to {\begin{matrix} V_{\max (j)} = \frac{X_{\max (j)} - X_{\min (j)}}{k} \\ i = 1, 2, \dots, nop \\ j = 1, 2, \dots, nod \end{matrix}

(1)

As shown in equation (2), the speed vector has a weight of inertia (W) that is associated with and determines the convergence speed of the algorithm to an extent. This weight has a value that decreases along the generation iteration. Equation (2) shows the relationship between the weight of inertia and generation iterations^45,47–49

W_{k} = W_{\max} - (\frac{W_{\max} - W_{\min}}{Generatio n_{\max}}) \times Generatio n_{k}

(2)

As the speed vector $V_{i, j}^{k}$ is dependent on W_k, the values of W_max and W_min determine the effects of speed vectors on motion. These values can be selected by the user. The next step is to use the objective function to assess the suitability of all particles after dispersion^45,47–49

V_{i, j}^{k} = W_{k} \times rand (0, 1) \times V_{\max (j)}

(3)

The particle that contains the optimal value is taken as the radial best (Rbest) vector. The accuracy value associated with the position of this particle represents the Rbest particle. The positions of the Gbest and Rbest particles are used along with the updating vector (up) to update the optimal center point position as shown in equations (4) and (5). C1 and C2 are the coefficients that need to be adjusted before running the algorithm^47–49

c p^{k + 1} = c p^{k} + up

(4)

up = C 1 \times (Gbest - c p^{k}) + C 2 \times (Rbest - c p^{k})

(5)

Vectorial depiction of updating cp along the updating vector (up) is shown in Figure 4.

Figure 4.

Vectorial depiction of updating cp along the updating vector (up).^47,48

After the center point cp is updated, dispersion of the particles starts from the new cp. In the next generation, the Gbest value in the previous generation is updated by comparison to the Rbest value that is obtained during the dispersion of the particles in the new generation iteration. If Rbest provides a better solution than Gbest, their positions should be switched. The process continues until Gbest reaches a certain value or the number of generations reaches a defined maximum value.^47,48 Figure 5 shows the sprinkling of the particles along the radii where V_max is the radius of the sphere and updating the cp by up vector.

Figure 5.

The sprinkling of the particles along the radii where V_max is the radius of the sphere and updating the cp by up vector.^47–49

PSO

PSO is a two-dimensional simulation of the collective behaviors of swarms of birds and fish in cases such as looking for food. It is a metaheuristic algorithm. It is efficient and effective in terms of calculation and easy in terms of comprehension and implementation.^50,51

PSO depicts each bird or fish as a particle and communities of birds or fish as swarms. In the algorithm, first, the initial positions, speeds, and population are randomly created. Accuracy values are calculated by determining the initial values for each particle in the population. The position value of the particle that has the best (minimum value) among the accuracy values is selected as Gbest. The other particles update their positions and speeds based on this value. The cycle continues until the finalization step.⁵²

In PSO, $X_{i}^{k}$ represents the position vector. The position vector in the problem space that consists of j particles is expressed as $X_{i}^{k} = [x_{i 1}^{k}, x_{i 2}^{k}, \dots, x_{ij}^{k}]$ . $V_{i}^{k}$ is the speed vector and the speed vector for a particle is shown as $V_{i}^{k} = [v_{i 1}^{k}, v_{i 2}^{k}, \dots, v_{ij}^{k}]$ . Local best neighbors are the position vectors that have the local best values, and they consist of elements equal to the number of particles in each iteration. They are shown as $Pbes t_{i} = [p_{i 1}, p_{i 2}, \dots, p_{ij}]$ . Global best neighbors are the position values that have the best values in each iteration and are expressed as $G_{best} = [g_{1}, g_{2}, \dots, g_{j}]$ . The expression c1 is the effect of the best results obtained by the particle on the next iteration. The expression c2 is the effect of the best results obtained by the swarm on the next iteration. The expression w is the weight of inertia and it controls the effects of the previous speed vector on the current speed vector. In the case that this component which takes value in the range of 0–1 has a high value, the particle performs a search in the global environment, speeds up the solution, and may lead to overlooking the target area. In the case of low values, the particle performs a search in the local environment and slows down the solution. The speeds and positions of the particles can be updated by equations (6) and (7)^50–53

\begin{matrix} v_{i}^{k + 1} = v_{i}^{k} + c_{1} \cdot {rand}_{1}^{k} ({pbest}_{i}^{k} - x_{i}^{k}) \\ + c_{2} \cdot {rand}_{2}^{k} (gbes t^{k} - x_{i}^{k}) \end{matrix}

(6)

x_{i}^{k + 1} = x_{i}^{k} + v_{i}^{k + 1}

(7)

Figure 6 shows the vectorial depiction of the motion of a particle k.

Figure 6.

Vectorial depiction of the motion of a particle k.⁷

A new hybrid metaheuristic model

The main aim here is to design a new hybrid metaheuristic model that is different to those in the literature, has high accuracy rates, is fast, and works stably. Thus, the model was developed using PSO and RMO, and the training process of ANNs was performed with this new model. With this training process, it is aimed to optimize the linear and nonlinear components of the data sets and large random fluctuations. The flowchart of the new model is shown in Figure 7.

Figure 7.

Flowchart of the new hybrid metaheuristic model.

This new hybrid metaheuristic model was designed to make these two different features in a hybrid algorithm by taking advantage of PSO and RMO’s distinctive individual features and to increase the system stability. Thus, PSO was used thanks to it’s features such as storage of coordinates and speeds of particles, ability to take into account the best available availability of particles, in determining the next movements, and it has benefited from its best past coordinates and its ability to consider the experiences of its most successful neighbor. Furthermore, RMO was used thanks to its excellent features such as the ability to search around the target point with an intensely focused feature, need for low memory, fast operation, and the ability to continue searching without getting pinched by the local optimum.

The data set was divided into three sets of data: 70% training, 15% verification, and 15% test data, while the model code is shown in Figure 8.

Figure 8.

Dividing the data set into three parts.

Interface for new hybrid metaheuristic model (ANNs trained with PSO + RMO)

A user interface was designed using MATLAB App Designer software. Thus, a new and an alternative hybrid metaheuristic estimation model that is highly accurate, fast working, practical, and easy to use was supported by an interface. The characteristics of the designed interface are given below:

Allows selection of the file to be processed;

Displays information about the data set being processed;

Provides opportunity to use seven different estimation models (ANNs trained with ES, ANNs trained with GA, ANNs trained with ACO, ANNs trained with PBIL, ANNs trained with PSO, ANNs trained with RMO, and ANNs trained with PSO + RMO);

Allows drawing real-time plots of the actual and estimated values;

Can calculate six different error values such as normalized root mean square error (nRMSE), root mean square error (RMSE), mean square error (MSE), mean absolute percentage error (MAPE), peak signal-to-noise ratio (PSNR), and R and print them on the screen;

Allows easy modification of the W_max, W_min, c1, and c2 parameters for PSO + RMO, the W, C1, C2, and C3 parameters for RMO, and the W_max, W_min, c1, and c2 parameters for PSO;

Allows changing the number of iterations and number of particles (nop);

Allows plotting the Gbest values;

Can bring up real-time information about the stage of the process on the screen;

Can reset the command history with the “clear” button.

The interface designed using MATLAB App Designer is shown in Figure 9. The process starts by selecting the file to be run in the interface screen. Then the “create neural network” button is clicked on to create a neural network. The parameters in the bottom right corner of the screen are changed for a desired model (ANNs trained with ES, ANNs trained with GA, ANNs trained with ACO, ANNs trained with PBIL, ANNs trained with PSO, ANNs trained with RMO, ANNs trained with PSO + RMO) manually, that is, the parameter values that are considered to be the most suitable for a selected model are typed in. At this stage, it is also possible to change the number of iterations. Then the button for the desired model is clicked on. The progress of the process can be followed on the progress information part. After the process is completed, the actual and estimated values will be displayed on the screen along with the Gbest value. The error values (nRMSE, RMSE, MSE, MAPE, PSNR, R) are calculated one by one and shown on the error analysis part in the upper right corner of the screen.

Figure 9.

A user interface designed by MATLAB App Designer.

Furthermore, it was seen that the ES, GA, ACO, and PBIL metaheuristic algorithms have been applied in various fields to solve a large number of problems and the results have been presented.^54–56 These metaheuristic algorithms were used in this paper, thanks to excellent contributions of the literature.

Performance results

This section examines the performance of the designed hybrid metaheuristic estimation model for wind speed estimations. For this aim, the data sets obtained from the wind measurement station were converted into two different scenarios, and the performance results of the hybrid metaheuristic estimation model are presented. Figure 10 which shows the scenario implementations:

Figure 10.

Scenario implementations.

Scenario 1. In this scenario, wind speed estimations were made according to the data on temperature, humidity, and pressure. The data used were recorded in January 2014 (recording frequency: 6 h, recording time: 10 days) and June 2014 (recording frequency: 24 h, recording time: 30 days).

Scenario 2. In this scenario, wind speed estimations were made according to the data on wind aspect, temperature, and humidity. The data used were recorded in August 2014 (recording frequency: 24 h, recording time: 15 days). The input and output variables used in Scenario 1 and Scenario 2 are shown in Table 5.

Table 5.

Input and output variables used in Scenario 1 and Scenario 2.

Variables and sensor heights used in estimation (input variables)				Estimated variables and sensor heights (output variables)
Wind aspect	Temperature	Humidity	Pressure	Wind speed
59 m	4 m	4 m	3.5 m	61 m

Figure 11 shows the actual and estimated plots of wind speed of Scenario 1—January. It includes the success of wind speed estimation of the hybrid models ANNs trained with ES, ANNs trained with PBIL, ANNs trained with GA, ANNs trained with PSO, ANNs trained with ACO, ANNs trained with RMO, and ANNs trained with PSO + RMO.

Figure 11.

Actual and estimated plots of wind speed of Scenario 1—January: (a) ANNs trained with ES, (b) ANNs trained with PBIL, (c) ANNs trained with GA, (d) ANNs trained with PSO, (e) ANNs trained with ACO, (f) ANNs trained with RMO, and (g) ANNs trained with PSO + RMO (new hybrid metaheuristic model).

Figure 12 shows the actual and estimated plots of wind speed of Scenario 1—June. It includes the success of wind speed estimation of the hybrid models ANNs trained with ES, ANNs trained with PBIL, ANNs trained with GA, ANNs trained with PSO, ANNs trained with ACO, ANNs trained with RMO, and ANNs trained with PSO + RMO.

Figure 12.

Actual and estimated plots of wind speed of Scenario 2—June: (a) ANNs trained with ES, (b) ANNs trained with PBIL, (c) ANNs trained with GA. (d) ANNs trained with PSO, (e) ANNs trained with ACO, (f) ANNs trained with RMO, and (g) ANNs trained with PSO + RMO (new hybrid metaheuristic model).

Figure 13 shows the actual and estimated plots of wind speed of Scenario 2—August. It includes the success of wind speed estimation of the hybrid models ANNs trained with ES, ANNs trained with PBIL, ANNs trained with GA, ANNs trained with PSO, ANNs trained with ACO, ANNs trained with RMO, and ANNs trained with PSO + RMO.

Figure 13.

Actual and estimated plots of wind speed of Scenario 2—August: (a) ANNs trained with ES, (b) ANNs trained with PBIL, (c) ANNs trained with GA, (d) ANNs trained with PSO, (e) ANNs trained with ACO, (f) ANNs trained with RMO, and (g) ANNs trained with PSO + RMO (new hybrid metaheuristic model).

The best nRMSE, RMSE, MSE, MAPE, PSNR, and R values for the test error (50 runs) of Scenario 1—January are shown in Table 6. The min, average, max, and standard deviation values for Scenario 1—January test error (MAPE, 50 runs) are given in Table 7.

Table 6.

The best nRMSE, RMSE, MSE, MAPE, PSNR, and R values for the test error (50 runs) of Scenario 1—January.

Test error (50 runs) of Scenario 1 (recording frequency: 6 h, recording time: 10 days, January)						ANNs trained with ACO						ANNs trained with ES						ANNs trained with PBIL
						nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R
						0.174	0.069	0.004	1.012	71.229	0.999	0.613	0.122	0.015	1.677	66.355	0.999	0.626	0.125	0.015	1.843	66.17	0.999
ANNs trained with GA						ANNs trained with PSO						ANNs trained with RMO						ANNs trained with PSO + RMO
nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R
0.296	0.059	0.003	1.000	72.657	0.999	0.110	0.088	0.007	1.044	69.210	0.999	0.291	0.233	0.054	2.755	60.77	0.998	0.049	0.039	0.001	0.528	76.18	1.000

nRMSE: normalized root mean square error; RMSE: root mean square error; MSE: mean square error; MAPE: mean absolute percentage error; PSNR: peak signal-to-noise ratio; ANN: artificial neural network; ACO: ant colony optimization; ES: evolutionary strategy; PBIL: probability-based incremental learning; GA: genetic algorithm; PSO: particle swarm optimization; RMO: radial movement optimization.

Table 7.

The min, average, max, and standard deviation (SD) values for Scenario 1—January test error (MAPE, 50 runs).

Test error (MAPE, 50 runs) of Scenario 1 (recording frequency: 6 h, recording time: 10 days, January)				ANNs trained with ACO				ANNs trained with ES				ANNs trained with PBIL
				Min	Avg	Max	SD	Min	Avg	Max	SD	Min	Avg	Max	SD
				1.012	4.881	9.929	2.655	1.677	5.962	10.033	2.540	1.843	4.107	8.616	2.109
ANNs trained with GA				ANNs trained with PSO				ANNs trained with RMO				ANNs trained with PSO + RMO
Min	Avg	Max	SD	Min	Avg	Max	SD	Min	Avg	Max	SD	Min	Avg	Max	SD
1.000	4.751	9.016	2.491	1.044	1.387	3.734	0.642	2.755	6.827	10.07	1.957	0.528	0.752	2.142	0.413

MAPE: mean absolute percentage error; ANN: artificial neural network; ACO: ant colony optimization; ES: evolutionary strategy; PBIL: probability-based incremental learning; GA: genetic algorithm; PSO: particle swarm optimization; RMO: radial movement optimization.

The best nRMSE, RMSE, MSE, MAPE, PSNR, and R values for the test error (50 runs) of Scenario 1—June are given in Table 8. The min, average, max, and standard deviation values for Scenario 1—June test error (MAPE, 50 runs) are given Table 9.

Table 8.

The best nRMSE, RMSE, MSE, MAPE, PSNR, and R values for the test error (50 runs) of Scenario 1—June.

Test error (50 runs) of Scenario 1 (recording frequency: 24 h, recording time: 30 days, June)						ANNs trained with ACO						ANNs trained with ES						ANNs trained with PBIL
						nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R
						0.285	0.403	0.162	6.584	56.020	0.993	0.244	0.244	0.118	5.224	57.377	0.995	0.293	0.413	0.171	6.318	55.793	0.993
ANNs trained with GA						ANNs trained with PSO						ANNs trained with RMO						ANNs trained with PSO + RMO
nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R
0.207	0.292	0.085	4.405	58.806	0.996	0.116	0.177	0.031	2.520	63.130	0.998	0.147	0.226	0.051	3.683	61.050	0.997	0.083	0.127	0.016	2.144	66.020	0.999

Table 9.

The min, average, max, and standard deviation (SD) values for Scenario 1—June test error (MAPE, 50 runs).

Test error (MAPE, 50 runs) of Scenario 1 (recording frequency: 24 h, recording time: 30 days, June)				ANNs trained with ACO				ANNs trained with ES				ANNs trained with PBIL
				Min	Avg	Max	SD	Min	Avg	Max	SD	Min	Avg	Max	SD
				6.584	9.176	10.588	1.171	5.224	9.623	12.540	1.244	6.318	9.284	13.127	1.441
ANNs trained with GA				ANNs trained with PSO				ANNs trained with RMO				ANNs trained with PSO + RMO
Min	Avg	Max	SD	Min	Avg	Max	SD	Min	Avg	Max	SD	Min	Avg	Max	SD
4.405	8.741	11.148	1.563	2.520	3.856	7.516	1.234	3.683	17.974	27.320	7.288	2.144	2.158	2.372	0.055

The best nRMSE, RMSE, MSE, MAPE, PSNR, and R values for the test error (50 runs) of Scenario 2—August are given in Table 10. The min, average, max, and standard deviation values for Scenario 2—August test error (MAPE, 50 runs) are given in Table 11.

Table 10.

The best nRMSE, RMSE, MSE, MAPE, PSNR, R results for the test error (50 runs) of Scenario 2—August.

Test error (50 runs) of Scenario 2 (recording frequency: 24 h, recording time: 15 days, August)						ANNs trained with ACO						ANNs trained with ES						ANNs trained with PBIL
						nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R
						0.371	0.122	0.015	2.092	66.351	0.999	0.323	0.106	0.011	1.849	67.565	0.999	0.306	0.101	0.010	1.929	68.023	0.999
ANNs trained with GA						ANNs trained with PSO						ANNs trained with RMO						ANNs trained with PSO + RMO
nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R	nRMSE	RMSE	MSE	MAPE	PSNR	R
0.269	0.089	0.007	1.589	69.139	0.999	0.090	0.056	0.003	1.102	73.140	0.999	0.321	0.199	0.039	3.255	62.150	0.997	0.100	0.062	0.003	0.922	72.250	0.999

Table 11.

The min, average, max, and standard deviation (SD) values for Scenario 2—August test error (MAPE, 50 runs).

Test error (MAPE, 50 runs) of Scenario 2 (recording frequency: 24 h, recording time: 15 days, August)				ANNs trained with ACO				ANNs trained with ES				ANNs trained with PBIL
				Min	Avg	Max	SD	Min	Avg	Max	SD	Min	Avg	Max	SD
				2.093	5.044	7.585	1.629	1.849	5.786	7.559	1.498	1.929	4.837	7.237	1.531
ANNs trained with GA				ANNs trained with PSO				ANNs trained with RMO				ANNs trained with PSO + RMO
Min	Avg	Max	SD	Min	Avg	Max	SD	Min	Avg	Max	SD	Min	Avg	Max	SD
1.589	4.542	7.572	1.718	1.102	2.031	5.977	1.311	3.255	10.490	18.530	4.808	0.922	1.222	2.063	0.210

Conclusion

Wind may show momentary changes due to its nature, and it causes highly difficult and complicated estimations. Wind speed and wind power estimation studies done as accurately as possible are highly important in terms of determining the locations for wind farms, shaping energy unit costs, efficient and effective energy investments, and grid reliability.

In this study, a new hybrid metaheuristic (ANNs trained with PSO + RMO) estimation model was designed using the PSO and RMO models. It means that the training process of ANNs was performed using PSO and RMO. The designed new hybrid metaheuristic model was compared to each of the other hybrid models (ANNs trained with ES, ANNs trained with GA, ANNs trained with ACO, ANNs trained with PBIL, ANNs trained with PSO, and ANNs trained with RMO). An interface was designed using MATLAB App Designer (R2017b) and the whole model was visualized.

The performance of ANNs trained with ES, ANNs trained with GA, ANNs trained with ACO, ANNs trained with PBIL, ANNs trained with PSO, ANNs trained with RMO, and ANNs trained with the PSO + RMO combination were observed separately by two different scenarios (Scenario 1 and Scenario 2). According to the analysis of Scenario 1 (January and June) given in Tables 6 and 8, the nRMSE, RMSE, MSE, MAPE, PSNR, and R error values (test error, 50 runs) for the new hybrid metaheuristic model (ANNs trained with PSO + RMO) were much lower than those of the other hybrid models. Likewise, according to the analysis of Scenario 2 (August) given in Table 10, the nRMSE, RMSE, MSE, MAPE, PSNR, and R error values (test error, 50 runs) for the new hybrid metaheuristic model (ANNs trained with PSO + RMO) were much lower than those of the other hybrid models. It shows the success of the new designed hybrid metaheuristic estimation model.

The average MAPE values for Scenario 1 and Scenario 2 (test error, 50 runs) are shown in Figure 14. A newly designed hybrid metaheuristic model (ANNs trained with PSO + RMO) has achieved the most successful results among all the hybrid models. These are the lowest errors among the all models, 0.752, 1.222, and 2.158 for Scenario 1 (January), Scenario 2 (August), and Scenario 1 (June), respectively.

Figure 14.

The average MAPE values for Scenario 1 and Scenario 2 (test error, 50 runs).

As a result of the analyses, it was observed that ANNs trained with the PSO + RMO model showed a highly accurate and reliable performance in the solution of nonlinear problems such as wind speed estimation. Thus, a new hybrid metaheuristic model (ANNs trained with PSO + RMO) which has high accuracy, is fast, and works stably has become a contribution to the literature with this study.

Footnotes

Acknowledgements

The authors thank the project coordinator Prof. Dr Serdar Salman who provided insight and expertise that greatly assisted the research. Furthermore, special thanks to Drs Rasoul Rahmani and İsmail Kırbaş for their excellent contribution to this study.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This research was supported by West Mediterranean Development Agency (BAKA, Project No. TR61/13/DFD/036) and Mehmet Akif Ersoy University Scientific Research Projects Commission (Project No. 0212-Gudumlu-13).

ORCID iD

Alper Kerem

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