Probabilistic optimal planning in active distribution networks considering non‐linear loads based on data clustering method
2021; Institution of Engineering and Technology; Volume: 16; Issue: 4 Linguagem: Inglês
10.1049/gtd2.12320
ISSN1751-8695
AutoresHasan Ebrahimi, Saeed Rezaeian‐Marjani, Sadjad Galvani, Vahid Talavat,
Tópico(s)Microgrid Control and Optimization
ResumoIET Generation, Transmission & DistributionVolume 16, Issue 4 p. 686-702 ORIGINAL RESEARCH PAPEROpen Access Probabilistic optimal planning in active distribution networks considering non-linear loads based on data clustering method Hasan Ebrahimi, Hasan Ebrahimi Department of Power Engineering, Faculty of Electrical and Computer Engineering, Urmia University, Urmia, IranSearch for more papers by this authorSaeed Rezaeian-Marjani, Saeed Rezaeian-Marjani Department of Power Engineering, Faculty of Electrical and Computer Engineering, Urmia University, Urmia, IranSearch for more papers by this authorSadjad Galvani, Corresponding Author Sadjad Galvani s.galvani@urmia.ac.ir Department of Power Engineering, Faculty of Electrical and Computer Engineering, Urmia University, Urmia, Iran Correspondence Sadjad Galvani, Department of Power Engineering, Faculty of Electrical and Computer Engineering, Urmia University, Urmia, Iran. Email: s.galvani@urmia.ac.irSearch for more papers by this authorVahid Talavat, Vahid Talavat Department of Power Engineering, Faculty of Electrical and Computer Engineering, Urmia University, Urmia, IranSearch for more papers by this author Hasan Ebrahimi, Hasan Ebrahimi Department of Power Engineering, Faculty of Electrical and Computer Engineering, Urmia University, Urmia, IranSearch for more papers by this authorSaeed Rezaeian-Marjani, Saeed Rezaeian-Marjani Department of Power Engineering, Faculty of Electrical and Computer Engineering, Urmia University, Urmia, IranSearch for more papers by this authorSadjad Galvani, Corresponding Author Sadjad Galvani s.galvani@urmia.ac.ir Department of Power Engineering, Faculty of Electrical and Computer Engineering, Urmia University, Urmia, Iran Correspondence Sadjad Galvani, Department of Power Engineering, Faculty of Electrical and Computer Engineering, Urmia University, Urmia, Iran. Email: s.galvani@urmia.ac.irSearch for more papers by this authorVahid Talavat, Vahid Talavat Department of Power Engineering, Faculty of Electrical and Computer Engineering, Urmia University, Urmia, IranSearch for more papers by this author First published: 16 October 2021 https://doi.org/10.1049/gtd2.12320Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Renewable energies have a significant portion in supplying energy demands in modern distribution networks. Due to the wide use of power electronic devices, these networks may have power quality problems. The unpredictable nature of renewable energies, besides the effect of non-linear loads brings out serious planning and operating challenges for distribution systems. Basicly, harmonic distortion is a severe problem for both electric efficiency and power energy customers. This study proposes an optimal scheduling strategy for wind turbine's integrated distribution networks with non-linear loads using a multi-objective individualized instruction mechanism teaching-learning-based optimization algorithm and the best solution is selected via the TOPSIS technique. In the proposed strategy, energy storage systems are optimally scheduled besides wind turbines, and reactive power compensators. Also, to use the distribution network more efficiently, an optimal network reconfiguration is applied. The wind turbine's output and load demands have probabilistic nature. The proposed scheme reduces the total harmonic distortion as well as total costs. The efficacy of the proposed management scheme is investigated using the IEEE standard 33 bus distribution network. Also, the performance of the multi-objective individualized instruction mechanism teaching-learning-based optimization algorithm is compared with the multi-objective particle swarm optimization algorithm. 1 INTRODUCTION 1.1 Motivation The uses of non-linear loads connected to distribution networks such as static converters of power electronic devices, arc discharge devices, saturated magnetic devices, and electronic appliances have been significantly increased. Non-linear loads change the sinusoidal shape of the alternating current, resulting in harmonic distortion in the distribution networks. Harmonic distortion may reduce system performance, system equipment failure, and error in protection and measurement devices. This harmonic distortion also increases losses and heating in numerous electromagnetic devices [1, 2]. Besides the mentioned power quality problem, distribution networks experience some other problems such as weak grid structure and low reliability due to their radial configuration as they only one power source for a group of customers. To overcome the mentioned challenges, distribution networks are restructured from traditional systems to active systems by deploying DGs and ESSs. DGs and ESSs bring environmental advantages as well as technical advantages. Also, distribution systems reconfiguration besides reactive power compensators helps distribution networks to operate more efficiently. However, these active distribution networks encounter more uncertainty due to more utilization of renewable energy sources. The uncertainty of loads, besides the uncertainty of renewable energies, has brought a significant challenge to the distribution network planning and operating [3, 45]. Reconfiguration of distribution systems, DG, ESS, and reactive power planning are the primary tools for optimizing the performance of distribution systems. The economic considerations in distribution systems operating and planning, besides satisfying operational requirements, become the purpose of the active distribution networks planning and operating [6]. Besides economic objectives, other technical objectives should be considered in distribution systems operation, and planning and a proper trade-off should be taken between various objectives. In this regard, classic mathematical-based optimization algorithms have some deficiencies encountering such a non-linear optimization problem, and the application of efficient optimization algorithms is necessary for conducting this type of optimization problem [7]. The uncertainty of stochastic input variables should be considered to get suitable solutions. The uncertainty management methods and the probabilistic assessment of distribution systems have an essential role in effective decision-making for planning and operating. Application of methods with acceptable accuracy and reasonable calculation time for online applications is necessary [8]. This study presents a practical planning and scheduling program to determine the optimal location and size of reactive power compensators, DG unit, and ESS unit (charge and discharge rate). Also, the optimal configuration for the distribution system is obtained. The objective functions are the minimization of harmonic distortion and total cost. These objective functions are optimized in a multi-objective environment using the INM_TLBO method. Also, to make the results more realistic, the uncertainty of the wind speed, and load demands have been considered and an efficient data clustering strategy based on the k-means and hierarchical approachs are used. 1.2 Literature review In [6], a development scheme of active distribution networks is introduced, and DG installation, network reconfiguration, and construction of new feeders are considered. ESS and reactive power compensation devices are not employed in this study. Reference [9] investigated the method of enhancing power supply capacity through coordinated control of network, load, and source in active distribution network. Knowing the importance of active distribution network uncertainties, the purpose of reference [10] is a detailed review of uncertainty modelling methods in distribution networks. In these two references, uncertain factors of distribution networks such as DGs’ output and demand response are considered. However, these studies do not pay attention to harmonic distortion minimization. Also, ESS programming has not been considered in these studies. ESS is paid into attention in [11-13]. A probabilistic active distribution network management scheme in the presence of ESS and DG is studied in [11]. A two-stage optimization scheme is offered for DG management with ESS integration in [12]. In [13], an active distribution network that includes renewable DGs is expanded by obtaining optimal capacity, location, and power rating of the ESSs. However, these references consider the stochastic nature of the active distribution network, but they do not consider the reactive power compensation, reconfiguration, and THD relaxation. References [14-17] paid attention to reactive power compensation in distribution networks. The authors in [14] are investigates applying a capacitor-less distribution static synchronous compensator for reactive power compensation and harmonic relaxation. The backward/forward sweep-based harmonic load flow is expanded to the quick and exact solution for capacitor compensated distribution networks by the authors [15]. In [16] the authors are focused on a problem associated with the capacitor effects in power systems considering harmonic distortion. Also, reference [17] studies the voltage and current harmonics injected by large-scale renewable DG resources in distribution networks. However, reference [14] does not consider cost optimization. Also, the uncertain variables are not investigated. In paper [15], the authors are focused on the network reconfiguration in harmonic polluted distribution network while they are does not minimize the costs. Also, they don't optimize the site and size of the DG and ESS units. The main goal of [16] is to analyse the behaviour of a power system in the presence of harmonic distortion when capacitors are installed. Despite this investigation, the authors ignored the DG, ESS, uncertain variables, costs, and network reconfiguration. Reference [17] does not consider the cost objective function. Also, reactive compensation and network reconfiguration are not considered, too. Optimization problems with multi-objective functions have been widely studied in various engineering fields. The dominance concept and Pareto fronts are essential in conducting these algorithms into final solutions. However, multi-objective problems may be multimodal, non-linear, multi-distributed, making the actual Pareto front very complex and sometimes inefficient. For instance, In the standard edition of NSGA-II, the Pareto data is utilized to choose, elicit steps, and compare, but the dominance correlation between different results is not fully considered in the chosen step. In the INM-TLBO algorithm, an individualized instruction mechanism is unified with the TLBO and the non-dominated classification, to overcome the mentioned issue [18, 19]. 1.3 Paper contribution In this study, four different plans including, DG utilizing, network reconfiguration, scheduling of ESS charge/discharge, and reactive power compensation, are presented. In the network reconfiguration, the status of tie switches is determined. In DG utilizing, the hourly contribution of dispatchable DGs is optimized. The status of charge/discharge of ESSs is hourly determined in the scheduling of ESS charge/discharge, and the reactive power compensation is determined in the reactive power compensation scheme. The minimization of the total cost including, DG cost, charge/discharge of ESSs, and losses, besides the minimization of THD, are objectives of these plans. The optimization is conducted by the multi-objective INM-TLBO algorithm. As a critical contribution of this paper, the INM-TLBO is used in these optimization problems for the first time. Other unique characteristics of this paper are as follows. There are some non-linear loads in the network that make the study more applicable and much closer to actual conditions. The uncertainty of loads and renewable energy sources are considered, as uncertainties of active distribution networks may ultimately affect the operating/planning decisions. The technique for order of preference by similarity to ideal solution (TOPSIS) is used for selecting the final solution among Pareto solutions obtained by the multi-objective INM-TLBO. The performance of four different plans is comprehensively compared. A comparison is conducted for comparing the performance of multi-objective INM-TLBO algorithm with the MOPSO algorithm. Plus, two k-means and hierarchical clustering techniques are utilized and the results are compared. 1.4 Paper organization In continuation, Section 3 presents the mathematical model for the optimization problem. In Section 4 non-linear loads and harmonic load flow are discussed. The probabilistic analysis method is presented in Section 5. Sections 6, and 7 introduce the INM-TLBO algorithm and TOPSIS method. Study results are presented in Section 8, and the last section is the conclusion of this paper. 2 PROBLEM FORMULATION The proposed scheduling program is presented in this section. First, the objective functions for assigning the optimal locations and sizes of DG, capacitors, ESS, and tie switches number is introduced, and then operational constraints are presented. 2.1 Objective function The total cost related to equipment installation and the operation of the network, besides total harmonic distortion, are considered as objective functions. 2.1.1 Total cost Minimization of the installation of equipment cost, besides operational cost is the first objective function, which is formulated as Equation (1). C o p e = ∑ y 1 1 + r y − 1 ∑ t P y . t C y . t p + P y . t l o s s C y . t p + ∑ d C d . y E S S + ∑ g f g D G ( P g . y D G ) \begin{eqnarray} && {\rm{\;}}{C^{ope}} = \mathop \sum \limits_y \frac{1}{{{{\left( {1 + r} \right)}^{y - 1}}}}\;\nonumber\\[10pt] && \left( {\mathop \sum \limits_t \left( {{P_{y.t}}C_{y.t}^p + P_{y.t}^{loss}C_{y.t}^p} \right) + \mathop \sum \limits_d C_{d.y}^{ESS} + \mathop \sum \limits_g f_g^{DG}(P_{g.y}^{DG})} \right)\nonumber\\\end{eqnarray} (1) C o p e ${C^{ope}}$ includes (1) the purchasing energy cost P y . t C y . t p ${P_{y.t}}C_{y.t}^p$ , (2) power losses cost P y . t l o s s C y . t p $P_{y.t}^{loss}C_{y.t}^p$ , (3) ESS's annual costs C d . y E S S $C_{d.y}^{ESS}$ , and 4) DG's annual costs f g D G P g . y D G $f_g^{DG}P_{g.y}^{DG}$ . In Equation (1), the term 1 ( 1 + r ) y − 1 $\frac{1}{{{{( {1 + r} )}^{y - 1}}}}$ is added for converting the future value of the cost at the end of the yth year to the present value. 2.1.2 Total harmonic distortion THD minimization is the other objective function, which is defined in Equation (2). According to IEEE standards, THD is described as the divide of the square root of the harmonic elements to the square root quantity of the base value, which is uttered as the percent of the base value. T H D v = ∑ t 1 t ∑ i 1 N V 2 . t . i 2 + V 3 . t . i 2 + ⋯ V h . t . i 2 V 1 . i \begin{equation}{\rm{\;}}TH{D_v} = \mathop \sum \limits_t \frac{1}{t}\left( {\mathop \sum \limits_i \frac{1}{N}\left( {\frac{{\sqrt {V_{2.t.i}^2 + V_{3.t.i}^2 + \cdots V_{h.t.i}^2} }}{{{V_{1.i}}}}} \right)} \right)\end{equation} (2) 2.2 Constraints 2.2.1 Branches capacity constraint Branches currents should be bounded to their allowable rating against excessive currents: I b . y . t ≤ I b m a x b = 1 , 2 , … , N b \begin{equation} \def\eqcellsep{&}\begin{array}{l} \left| {{I_{b.y.t}}} \right| \le I_b^{max}\\[10pt] b\; = \;1,2,\; \ldots ,{N_b} \end{array} \end{equation} (3) 2.2.2 Buses voltage constraint To preserve the bus voltages in the permissible range, Equation (4) restriction should be satisfied: V i m i n ≤ V i . y . t ≤ V i m a x i = 1 , 2 , … , N \begin{equation} \def\eqcellsep{&}\begin{array}{l} V_i^{min} \le {V_{i.y.t}} \le V_i^{max}\\[10pt] i\; = \;1,2,\; \ldots ,N \end{array} \end{equation} (4) 2.2.3 Active management constraints The DG and capacitors limitations are considered: P g . m i n D G ≤ P g D G ≤ P g . m a x D G \begin{equation}P_{g.min}^{DG} \le P_g^{DG} \le P_{g.max}^{DG}\end{equation} (5) Q m . m i n C ≤ Q m C ≤ Q m . m a x C \begin{equation}Q_{m.min}^C \le Q_m^C \le Q_{m.max}^C\end{equation} (6)where Equation (5) is the DG dispatch constraints, and Equation (6) is the reactive power compensator regulating limit. Also, the DG and the reactive power compensator's location constraint is equal to i = 1 , 2 , ⋯ , N $i\; = \;1,2,\; \cdots ,N$ . 2.2.4 Network reconfiguration constraints The network radiality is an essential constraint in the distribution network reconfiguration. Usually, the distribution network designs are poor ring networks, so that distribution network functions are radial to retain integrity in the protection of distribution networks and other aspects. Thus, this factor should be maintained during the network reconfiguration procedure. The radiality of the distribution network in the reconfiguration procedure is retained by satisfying the graph rules [20]. In graph theory, a connected graph without loops is known as a tree. Therefore, the radial structure of a distribution system is compared with a tree. A tree is a subgraph with ( n − 1 ) $( {n - 1} )$ lines. Consequently, the structure of a distribution system with N $N$ buses and N b ${N_b}$ branches is radial if it satisfies the constraint in Equation (1) and the constraint in Equation (2): Constraint 1: the obtained reconfiguration solution must have ( n − 1 ) $( {n - 1} )$ branches. Constraint 2: All the buses (loads) must be connected. Note that only constraint 1 does not guarantee the radiality of a distribution system; so, constraint 2 must be satisfied too [21]. 2.2.5 ESS operation constraints ESS constraints cab be represented by Equations (7)–(10). E d . m i n E S S ≤ E d . y . t E S S ≤ E d E S S \begin{equation}E_{d.min}^{ESS} \le E_{d.y.t}^{ESS} \le E_d^{ESS}\end{equation} (7) E d . y . t E S S = E d . y . t − 1 E S S + Δ t · P d . y . t E S S · η \begin{equation}{\rm{\;}}E_{d.y.t}^{ESS} = E_{d.y.t - 1}^{ESS}{\rm{\;}} + \Delta t \cdot P_{d.y.t}^{ESS} \cdot \eta \end{equation} (8) E d . y . 0 E S S = E d . y . i n i E S S \begin{equation}{\rm{\;}}E_{d.y.0}^{ESS} = E_{d.y.ini}^{ESS}\end{equation} (9) − R D d E S S ≤ P d . y . t E S S − P d . y . t − 1 E S S ≤ R U d E S S \begin{equation} - RD_d^{ESS} \le P_{d.y.t}^{ESS} - P_{d.y.t - 1}^{ESS} \le RU_d^{ESS}\end{equation} (10) Constraint in Equation (7) is the maximum and minimum energy value of ESS. Equations (8) and (9) are energy storage constraints. Also, Equation (10) is the ramp-up and ramp-down constraints of the ESS power [22]. Furthermore, the ESS location constraint is equal to i = 1 , 2 , ⋯ , N $i\; = \;1,2,\; \cdots ,N$ . 2.2.6 THD constraint The constraint on T H D v $TH{D_v}$ can be represented by Equation (11). T H D v ≤ T H D v m a x \begin{equation}TH{D_v} \le THD_v^{max}\end{equation} (11) To retain T H D v $TH{D_v}$ in the permissible range determined by the network manager, Equation (11) constraint should be satisfied. Some of these are inequality constraints and some of them are equality constraints. They are generally represented by Equations (19) and (20) in Subsection 4.2. 3 NON-LINEAR LOADS AND HARMONIC LOAD FLOW The radial distribution system diagram is illustrated in Figure 1. w i ${w_i}$ is assumed as the non-linear loads coefficient in the i t h ${i^{th}}$ bus, and subsequently, ( 1 − w i ) $( {1 - {w_i}} )$ is the linear loads coefficient. Note that both linear and non-linear loads at bus i have the same power factor at base frequency. Furthermore, the magnitude of the generated harmonic current is calculated by the following equations. I i 1 = P i − j Q i V i 1 ∗ \begin{equation}I_i^1 = \frac{{{P_i} - j{Q_i}}}{{V_i^{1*}}}\end{equation} (12) I i h = w i I i 1 h h = 3 , 5 , … H \begin{equation}I_i^h = {w_i}\;\frac{{I_i^1}}{h}{\rm{\;}}h\; = \;3,5, \ldots H\end{equation} (13) FIGURE 1Open in figure viewerPowerPoint The radial distribution network diagram In these equations, I i 1 $I_i^1$ is the current of the i th ${i^{{\rm{th}}}}$ bus at the base frequency and I i h $I_i^h$ is the injected harmonic currents by the non-linear portion of the load at h $h$ th frequency. P i ${P_i}$ and Q i ${Q_i}$ are the active and reactive power at the i th ${i^{{\rm{th}}}}$ bus. V i 1 ∗ $V_i^{1*}$ is represents the complex conjugate of base voltage at the i th ${i^{{\rm{th}}}}$ bus. H $H$ is the upper harmonic level. Further, harmonic load flows depend on system component modelling at higher frequencies. In the following, system modelling at fundamental and harmonic frequencies is described. 3.1 Modelling at the base frequency In classic papers, the load is simulated as a fixed P–Q load model. The simple fixed impedance model is precise than the fixed P–Q model [23]. In this study, loads are defined as a fixed admittance. The fixed load admittances are possible in series and parallel representation. Thus, the load admittances in parallel representation are calculated by the following equation: Y l o a d i = P i − j Q i V i 1 2 \begin{equation}{\rm{\;}}Yloa{d_i} = \frac{{{P_i} - j{Q_i}}}{{{{\left| {V_i^1} \right|}^2}}}\end{equation} (14) V i 1 $V_i^1$ is obtained from the base level. 3.2 Modelling at harmonic frequencies Accurate models are possible for distribution lines and shunt capacitors at harmonic frequencies. If proximity and skin effects are ignored, the feeder segment and parallel capacitors admittance and described by the following equations: Y s e i j h = 1 / R s e i j + j · h · X s e i j \begin{equation}{\rm{\;}}Yse_{ij}^h = 1/\left( {Rs{e_{ij}} + j \cdot h \cdot Xs{e_{ij}}} \right)\end{equation} (15) y c i h = h · y c i 1 \begin{equation}{\rm{\;}}yc_i^h = \;h \cdot yc_i^1\end{equation} (16) R s e i j $Rs{e_{ij}}$ and X s e i j $Xs{e_{ij}}$ are the series impedance between the i th ${i^{{\rm{th}}}}$ and j th ${j^{{\rm{th}}}}$ buses. y c i 1 $yc_i^1$ the admittance of the capacitor size Q C ${Q_C}$ at base frequency. For linear loads, it is suggested in [24] to apply a generalized model, which is included a resistance with an inductance in a parallel model (respective of the active and reactive powers at base frequency). The load admittance at the h th ${h^{{\rm{th}}}}$ load level is determined by: Y l o a d i h = 1 − w i P i − j Q i / h V i 1 2 \begin{equation}Yload_i^h = \;\left( {1 - {w_i}} \right)\frac{{{P_i} - j{Q_i}/h}}{{{{\left| {V_i^1} \right|}^2}}}\end{equation} (17) Y l o a d i h $Yload_i^h$ is the load admittance in the h th ${h^{{\rm{th}}}}$ harmonic level. Non-linear loads are represented by simple ideal current sources; Thus, the complex load can be modelled as an admittance (the linear part) in shunt with a current source (the non-linear part). The network model at h th ${h^{{\rm{th}}}}$ harmonic frequency is represented in Figure 2. The h th ${h^{{\rm{th}}}}$ harmonic portion of the voltage is calculated using Equation (18). V 1 h V 2 h ⋮ V n h = Y B U S − 1 I 1 h I 2 h ⋮ I n h \begin{equation}{\rm{\;}}\left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {V_1^h}\\[6pt] {V_2^h} \end{array} }\\[6pt] \vdots \\[6pt] {V_n^h} \end{array} } \right] = {\left[ {{Y_{BUS}}} \right]^{ - 1}}\;\left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {I_1^h}\\[6pt] {I_2^h} \end{array} }\\[6pt] \vdots \\[6pt] {I_n^h} \end{array} } \right]\end{equation} (18) Y B U S ${Y_{BUS}}$ is the overall network admittance matrix. FIGURE 2Open in figure viewerPowerPoint Equivalent circuit of distribution network at h harmonic frequency The overall voltage at the h th ${h^{{\rm{th}}}}$ load level can be obtained by the following equation. V i = ∑ h = 1 H V i h 2 \begin{equation}{\rm{\;}}\left| {{V_i}} \right| = \sqrt {\mathop \sum \limits_{h = 1}^H {{\left| {V_i^h} \right|}^2}} \end{equation} (19) Another important factor that describes the voltage waveform is the THD explained by Equation (20) [25]. T H D = V 2 2 + V 3 2 + ⋯ V h 2 V 1 × 100 \begin{equation}THD\; = \frac{{\sqrt {V_2^2 + V_3^2 + \cdots V_h^2} }}{{\left| {{V_1}} \right|}}\; \times 100\end{equation} (20) 4 PROBABILISTIC ANALYSIS 4.1 Uncertainty modelling One of the most essential steps of probabilistic evaluation is to propose a proper statistical model for uncertain variables. In this study, wind speed, and power demands are modelled as uncertain elements, which are used as follows: 4.1.1 Power demands modelling Usually, the uncertain load models follow the normal distribution function. The following equation defines the probability density function of the normal distribution [26]. f x = 1 σ x 2 π . e − x − E x 2 2 σ x 2 \begin{equation}f{\rm{\;}}\left( x \right) = \frac{1}{{\left( {{{\sigma}}\left[ {\rm{x}} \right]} \right)\sqrt {2\pi } }}{\rm{\;}}.{e^{ - \frac{{{{\left( {x - {\rm{E}}\left[ x \right]} \right)}^2}}}{{2{{\sigma}}{{\left[ {\rm{x}} \right]}^2}}}}}\end{equation} (21) 4.1.2 Relevant WTs generation and wind speed modelling The Weibull distribution is commonly used for the probabilistic behaviour of the wind speed. Equation (22) expresses the probability density function of the Weibull distribution [27]. f v = A B . v B A − 1 . e − v B A v ≥ 0 0 v < 0 \begin{equation} f{\rm{\;}}\left( v \right) = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {\frac{A}{B}.{{\left( {\frac{v}{B}} \right)}^{A - 1}}.{e^{ - {{\left( {\frac{v}{B}} \right)}^A}{\rm{\;}}}}\;v \ge 0}\\[6pt] {0\;v < 0} \end{array} \right.\end{equation} (22) The power generation of the WT can be calculated according to wind speed Equation (23). P w v = 0 v ≤ v i n c o r v ≥ v o u t c v − v i n c v r a t e d − v i n c P r w v i n c ≤ v ≤ v r a t e d P r w v r a t e d ≤ v ≤ v o u t c \begin{equation} {P^w}{\rm{\;}}\left( v \right) = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{2}{c}@{}} 0&{v \le v_{in}^c{\rm{\;}}or{\rm{\;}}v \ge v_{out}^c}\\[6pt] {\frac{{v - v_{in}^c}}{{{v_{rated}} - v_{in}^c}}P_r^w}&{v_{in}^c \le v \le {v_{rated}}}\\[6pt] {P_r^w}&{{v_{rated}} \le v \le v_{out}^c} \end{array} \right.\end{equation} (23) 4.2 Data clustering methods 4.2.1 K-means data clustering The k-means method follows a simple way of sorting the data collection through a fixed number of clusters. k-means is presented by Mac Queen in 1967 [28]. The interval of every cluster member to its centre is minimized [29]. Briefly, the following steps are the algorithm steps [30]: The number of clusters ( K $K$ ) is specified. Randomly, K centres or agents initialized a k , k = 1 , 2 , … , K ${a_k},\;k\; = \;1,2, \ldots ,\;K$ . Data is assigned to the clusters based on the minimum distance with K agents Equation (24): i f D d n , a k < D d n , a l ⇒ d n ∈ G k n = 1.2 . … . N , k = 1.2 . … . K , l = 1.2 . … . K , ( l ≠ k ) \begin{equation} \def\eqcellsep{&}\begin{array}{l} if\;D\left( {{d_n},{a_k}} \right) < D\left( {{d_n},{a_l}} \right) \Rightarrow {d_n} \in {G_k}\\[6pt] n\; = \;1.2. \ldots .N,\\[6pt] k\; = \;1.2. \ldots .K,\\[6pt] l\; = \;1.2. \ldots .K,\\[6pt] (l \ne k) \end{array} \end{equation} (24) N $N$ and d n ${d_n}$ are the numbers of data, and the n th ${n^{{\rm{th}}}}$ data. a k ${a_k}$ , a l ${a_l}$ , are the k th ${k^{{\rm{th}}}}$ , the l th ${l^{{\rm{th}}}}$ centers and, the G k ${G_k}$ is the k th ${k^{{\rm{th}}}}$ group or cluster. D ( d n , a k ) $D( {{d_n},{a_k}} )$ is distance between d n ${d_n}$ and a k ${a_k}$ . D ( d n , a l ) $D( {{d_n},{a_l}} )$ is distance between d n ${d_n}$ and a l ${a_l}$ . The distance that is employed to assess the distinction between data can be calculated by Euclidean distance, Manhattan or city block distance, Pearson correlation coefficient, Kendall's (tau) distance, or other methods [31]. Euclidean distance between two d i ${d_i}$ and d j ${d_j}$ is represented as Equation (25). D d i , d j = ∑ m = 1 M d i m − d j m 2 2 \begin{equation}D\;\left( {{d_i
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