Interest and Applicability of Meta-Heuristic Algorithms in the Electrical Parameter Identification of Multiphase Machines

energies Article Interest and Applicability of Meta-Heuristic Algorithms in the Electrical Parameter Identification of Multiphase Machines † Daniel Gutierrez-Reina 1, Federico Barrero 2,* , Jose Riveros 3, Ignacio Gonzalez-Prieto 4, Sergio L. Toral 2and Mario J. Duran 5 1Department of Engineering, Loyola University Andalusia, 41014 Seville, Spain; dgutierr[email protected] 2Electronic Engineering Department, University of Seville, 41092 Sevilla, Spain; st[email protected] 3Faculty of Engineering, University of Talca, Curicó3340000, Chile; [email protected] 4Thermal and Electrical Engineering Department, University of Huelva, 21007 Huelva, Spain; [email protected] 5Department of Electrical Engineering, University of Malaga, 29071 Malaga, Spain; [email protected] *Correspondence: fbarrer[email protected]; Tel.: +34-954481304 † This paper is an extended version of our paper published in Riveros, J.A.; Reina, D.G.; Barrero, F.; Toral, S.L.; Durán, M.J. Five-Phase Induction Machine Parameter Identification using PSO and Standstill Techniques. In Proceedings of the IECON 2015—41st Annual Conference of the IEEE Industrial Electronics Society, Yokohama, Japan, 9–12 November 2015. Received: 26 November 2018; Accepted: 16 January 2019; Published: 19 January 2019   Abstract: Multiphase machines are complex multi-variable electro-mechanical systems that are receiving special attention from industry due to their better fault tolerance and power-per-phase splitting characteristics compared with conventional three-phase machines. Their utility and interest are restricted to the definition of high-performance controllers, which strongly depends on the knowledge of the electrical parameters used in the multiphase machine model. This work presents the proof-of-concept of a new method based on particle swarm optimization and standstill time-domain tests. This proposed method is tested to estimate the electrical parameters of a five-phase induction machine. A reduction of the estimation error higher than 2.5% is obtained compared with gradient-based approaches. Keywords: multiphase drives; off-line identification methods; meta-heuristic algorithms 1. Introduction Electromechanical systems such as multiphase variable speed drives have attracted the interest of the scientific community in recent times. They have been found as an attractive alternative to three-phase drives in particular industrial applications [ 1 ], where the electrical stresses on the machine and power electronic components as well as the harmonic content must be reduced and/or an inherent fault-tolerant capability is required. The interest in recent research works aims to exploit the inherent characteristics of multiphase drives, improving the overall reliability and performance of the system in order to favor their industrial applicability. However, their higher number of phases, in comparison with three-phase drives, results in more complex controllers due to higher number of freedom degrees. Most of the control techniques that have been proposed for multiphase drives are an extension of conventional three-phase control structures, aiming for a high speed/torque performance of the drive in healthy and faulty situations, and giving particular attention to multiphase machines of five and six phases [ 1 , 2 ]. Then, field oriented control (FOC) techniques, direct torque controllers (DTCs) or model-based predictive control (MPC) methods have been successfully used in multiphase drives, Energies 2019,12, 314; doi:10.3390/en12020314 www.mdpi.com/journal/energies Energies 2019,12, 314 2 of 15 where an accurate knowledge of the electrical parameters of the machine is required to yield the highest performance behavior of a system [ 1 , 2 ]. Note however that multiphase drives can be considered like an emerging technology, where most existing units have been built by rewinding conventional three-phase machines and reshaping the distribution of the stator slots [ 3 , 4 ]. The resulting machine is neither the most optimal nor its parameters correspond with those of the original three-phase drive. Therefore, methods and algorithms for the estimation of the rewound machine’s parameters are required to get adjustable speed multiphase drives with appropriate control performances. While the research on the identification of the electrical parameters of conventional three-phase drives is a mature field, this is not the case in the multiphase drives’ area [ 1 ]. Many off-line and on-line methods have been proposed to obtain the electrical parameters of three-phase machines, where standstill identification techniques can be highlighted for being accurate and easy to apply in commercial variable frequency drives [ 5 , 6 ]. Standstill methods are off-line identification tools based on injecting dc or ac electrical signals using the power converter of the drive, normally a Voltage Source Inverter (VSI), which does not produce a rotating field and keeps the electrical machine stopped. Then, the identification procedure is applied to fit the real response with the simplified machine model, where adaptive filters, recursive least-squares (RLS)-based algorithms, or maximum likelihood methods have been used for this purpose [ 7 ]. The extension of these methods for the multiphase case is barely found in the scientific literature. The standstill methods have been successfully applied for the identification of the electrical parameters of a symmetrical 5-phase induction machine with distributed windings in [ 8 , 9 ]. In [ 8 , 9 ], the stator and rotor resistors, the mutual inductance and the stator and rotor leakage inductances of the machine modelling are estimated using the non-torque capability of particular harmonic components that are injected in the estimation process. A RLS procedure was applied to fit the real response with the machine model, also complemented with sinusoidal excitation methods to tune and adjust the estimated parameters. The obtained results offer however bad accuracy and high deviation in some trials (up to 50% for certain cases in the estimation of the magnetizing inductance) because it is based on gradient-following-based algorithms that cannot properly fit the non-linear performance of a real machine. The algorithm proposed in [ 8 , 9 ] shows also a high dependency on the established forgetting factors, requiring an initial value for the estimated parameters close to the optimum result to find the global minimum solution. In this work, the method in [ 8 , 9 ] is extended to find an identification scheme that avoids the aforementioned drawbacks and propagation errors, adding the ability of detecting constructive asymmetries in the machine if desired. Meta-heuristic algorithms may represent an interesting alternative in this field [ 10 ]. These methods can offer a suitable guided search even in non-differentiable or nonlinear spaces, where conventional gradient-based methods are usually unsuccessful [ 11 ] because they get stuck in local minima. Among the available meta-heuristic optimization techniques, the particle swarm optimization (PSO) algorithm [ 12 , 13 ] is an interesting tool for solving optimization complex engineering problems [ 14 , 15 ]. It is based on the metaphor of social interaction during the movement into a multidimensional space and it has been widely applied for solving power systems optimization problems [ 14 ]. In this paper, the PSO optimization technique is proposed to minimize the mean square error (MSE) in two operation subspaces, namely α – β and x–y, between the responses of the simulated and real systems in standstill configuration for a multi-variable electro-mechanical system like a five-phase induction machine. To the authors’ knowledge, this is the first study that applies a bio-inspired algorithm like the PSO for the estimation of electrical parameters in electro-mechanical systems. The main idea is that the simulated model reaches the same responses of the real system, as the estimated parameters get closer to the real ones guided by the PSO algorithm. Therefore, the main contributions of this paper are: • The analysis of the utility of PSO algorithms in an application-oriented case like the estimation of the electrical parameters of a five-phase induction machine. • The comparison of the proposed PSO estimation technique with gradient-following-based algorithms [16]. The proposed technique clearly outperforms the gradient-based technique. Energies 2019,12, 314 3 of 15 This paper continues as follows: Section 2overviews the five-phase induction drives, which is the multi-variable electro-mechanical system used as case example and the PSO algorithm. Section 3 analyses the proposed estimation procedure that combines standstill tests and the PSO algorithm. Section 4provides the estimated electrical parameters achieved by the proposed method and the validation of the obtained parameters in a real test. Finally, conclusions are given in Section 5. 2. Background: Five-Phase Induction Machines and PSO Algorithm This section is divided into two parts. First, an introduction of the five-phase induction machine used in the paper is presented. Second, the PSO algorithm used and its configuration parameters are described in details. 2.1. Five-Phase Induction Machines The case under study is a symmetrical five-phase induction machine, where the stator windings are equally displaced ( ϑ = 2 π /5) and sinusoidally distributed along the stator. The multiphase drive is power-supplied using a two-level VSI, as can be observed in Figure 1. Energies 2018, 11, x FOR PEER REVIEW 3 of 15 This paper continues as follows: Section 2 overviews the five-phase induction drives, which is the multi-variable electro-mechanical system used as case example and the PSO algorithm. Section 3 analyses the proposed estimation procedure that combines standstill tests and the PSO algorithm. Section 4 provides the estimated electrical parameters achieved by the proposed method and the validation of the obtained parameters in a real test. Finally, conclusions are given in Section 5. 2. Background: Five-phase Induction Machines and PSO Algorithm This section is divided into two parts. First, an introduction of the five-phase induction machine used in the paper is presented. Second, the PSO algorithm used and its configuration parameters are described in details. 2.1. Five-phase Induction Machines The case under study is a symmetrical five-phase induction machine, where the stator windings are equally displaced ( ϑ = 2π/5) and sinusoidally distributed along the stator. The multiphase drive is power-supplied using a two-level VSI, as can be observed in Figure 1. abcde -a -b -c -d -e +a +b +c +d +e V dc i sa i sb i sc i sd i se v sa v se v sd v sc v sb r ω Figure 1. General scheme of the system under study. The model of the system is more complex than the one obtained for a three-phase case due to the higher number of phases. However, the general theory of electrical machines is also applied to obtain the model of the system and the following assumptions are taken into account to obtain a set of continuous-time phase voltage equilibrium equations: machine windings are identical and equally distributed around the stator, magnetic field saturation and eddy currents are not considered, nonlinearity in relation with temperature or frequency changes are not considered, and the machine air gap is assumed to be uniform and of constant density without any variation due to rotor eccentricities or machine slots. These equations can be simplified to avoid the dependence of the rotor position of certain parameter matrices using the Clarke transformation, which is used by the vector space decomposition theory to determine two orthogonal planes completely decoupled from each other (called α-β and x-y), plus an axis that contains the homopolar component (z-component). The obtained equations are detailed in (1)–(7), where the electrical parameters to be estimated are shown (the stator and rotor resistances, Rs and Rr, respectively, the mutual inductance represented by Lm, and the stator and rotor leakage inductances, Lls and Llr, respectively). It is interesting to mention that the fundamental supply component plus harmonics of the order 10n ± 1 (n = 0,1,2,3,...) are within the α–β subspace, which is the torque-producing plane. The rest of harmonic components are into the non-torque producing planes, including the x–y subspace, where supply harmonics of the order 10n ± 3 (n = 0,1,2,3,...) are considered, and the z-axis that contains harmonic components of the order 5n, with n = 1,2,3,... and only exists if the neutral point is not isolated. This is not our case because isolated neutral point is assumed and (7) is no longer required because isz = 0. Therefore, 32 (25) switching states and 30 active, and 2 zero voltage vectors can be generated in the α-β and x-y subspaces. Figure 2 identifies all available voltage vectors that can be applied to the multiphase machine, identified by using the decimal number corresponding to the binary code of the switching state Sa, Sb, Sc, Sd, Se, Figure 1. General scheme of the system under study. The model of the system is more complex than the one obtained for a three-phase case due to the higher number of phases. However, the general theory of electrical machines is also applied to obtain the model of the system and the following assumptions are taken into account to obtain a set of continuous-time phase voltage equilibrium equations: machine windings are identical and equally distributed around the stator, magnetic field saturation and eddy currents are not considered, non-linearity in relation with temperature or frequency changes are not considered, and the machine air gap is assumed to be uniform and of constant density without any variation due to rotor eccentricities or machine slots. These equations can be simplified to avoid the dependence of the rotor position of certain parameter matrices using the Clarke transformation, which is used by the vector space decomposition theory to determine two orthogonal planes completely decoupled from each other (called α - β and x-y), plus an axis that contains the homopolar component (z-component). The obtained equations are detailed in (1)–(7), where the electrical parameters to be estimated are shown (the stator and rotor resistances, R s and R r , respectively, the mutual inductance represented by L m , and the stator and rotor leakage inductances, L ls and L lr , respectively). It is interesting to mention that the fundamental supply component plus harmonics of the order 10n ± 1 (n= 0,1,2,3,...) are within the α – β subspace, which is the torque-producing plane. The rest of harmonic components are into the non-torque producing planes, including the x–y subspace, where supply harmonics of the order 10n±3 (n= 0,1,2,3,...) are considered, and the z-axis that contains harmonic components of the Energies 2019,12, 314 4 of 15 order 5n, with n= 1,2,3,... and only exists if the neutral point is not isolated. This is not our case because isolated neutral point is assumed and (7) is no longer required because i sz = 0. Therefore, 32 (2 5 ) switching states and 30 active, and 2 zero voltage vectors can be generated in the α - β and x-y subspaces. Figure 2identifies all available voltage vectors that can be applied to the multiphase machine, identified by using the decimal number corresponding to the binary code of the switching state S a ,S b ,S c ,S d ,S e , being S a and S e the most and least significant bits, respectively. The modelling of the machine is finally complemented with a differential equation that describes the rotor movement depending on the electrical and load torques. Since this study focuses on the estimation of the electrical parameters of the machine, the movement equation is omitted here for simplicity (more details on the modelling of system can be found in [1–4]): vsα=Rs+Lsd dt isα+Lmdirα dt (1) vsβ=Rs+Lsd dt isβ+Lm dirβ dt (2) 0=Rr+Lrd dt irα+Lmdisα dt +ωrLrirβ+ωrLmisβ(3) 0=Rr+Lrd dt irβ+Lm disβ dt −ωrLrirα−ωrLmisα(4) vsx =Rs+Lls d dt isx (5) vsy =Rs+Lls d dt isy (6) vsz =Rs+Lls d dt isz (7) Energies 2018, 11, x FOR PEER REVIEW 4 of 15 being Sa and Se the most and least significant bits, respectively. The modelling of the machine is finally complemented with a differential equation that describes the rotor movement depending on the electrical and load torques. Since this study focuses on the estimation of the electrical parameters of the machine, the movement equation is omitted here for simplicity (more details on the modelling of system can be found in [1–4]): 𝑣 =𝑅𝑠+𝐿𝑠𝑑 𝑑𝑡𝑖𝑠𝛼 +𝐿𝑚𝑑𝑖𝑟𝛼 𝑑𝑡 (1) 𝑣 =𝑅𝑠+𝐿𝑠𝑑 𝑑𝑡𝑖𝑠𝛽 +𝐿𝑚𝑑𝑖𝑟𝛽 𝑑𝑡 (2) 0=𝑅+𝐿𝑑 𝑑𝑡𝑖 +𝐿𝑑𝑖 𝑑𝑡 +𝜔𝐿𝑖 +𝜔𝐿𝑖 (3) 0=𝑅+𝐿𝑑 𝑑𝑡𝑖 +𝐿𝑑𝑖 𝑑𝑡 −𝜔𝐿𝑖 −𝜔𝐿𝑖 (4) 𝑣 =𝑅+𝐿 𝑑 𝑑𝑡𝑖 (5) 𝑣 =𝑅+𝐿 𝑑 𝑑𝑡𝑖 (6) 𝑣 =𝑅+𝐿 𝑑 𝑑𝑡𝑖 (7) 31 1 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 y 0 14 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 β 0αx Figure 2. Generated voltage vectors in the α-β and x-y planes. Note that the same switching state produces two different vectors in every plane. 2.2. PSO Algorithm PSO is a meta-heuristic population-based technique. It is inspired by the social behavior of bird flocking and fish schooling; therefore, it is based on the swarm intelligence concept [17]. PSO refers to artificial intelligence systems where the collective behavior of unsophisticated agents that interact locally with their environment creates coherent global functional patterns [12,15]. In general, PSO algorithm uses a population of particles that fly throughout the problem hyperspace [18]. All the particles have fitness values that are evaluated by the fitness function to be optimized and have velocities vectors, which determine the movement of the particles in the search space. These velocities are stochastically adjusted throughout the execution of the algorithm according to the historical best position for the particle itself and the neighborhood (other neighbor particles) [12,15]. Therefore, the particles or candidate solutions fly throughout the problem search space attracted by the positions of the best particles found during the execution of the algorithm. PSO-based methods have been used in a wide range of engineering areas to solve complex continuous optimization problems, such as product design and manufacturing [19], automotive industry [20], structural design [21], and computer networks [22], among others [23,24]. Figure 2. Generated voltage vectors in the α - β and x-yplanes. Note that the same switching state produces two different vectors in every plane. 2.2. PSO Algorithm PSO is a meta-heuristic population-based technique. It is inspired by the social behavior of bird flocking and fish schooling; therefore, it is based on the swarm intelligence concept [ 17 ]. PSO refers to artificial intelligence systems where the collective behavior of unsophisticated agents that interact locally with their environment creates coherent global functional patterns [ 12 , 15 ]. In general, Energies 2019,12, 314 5 of 15 PSO algorithm uses a population of particles that fly throughout the problem hyperspace [ 18 ]. All the particles have fitness values that are evaluated by the fitness function to be optimized and have velocities vectors, which determine the movement of the particles in the search space. These velocities are stochastically adjusted throughout the execution of the algorithm according to the historical best position for the particle itself and the neighborhood (other neighbor particles) [ 12 , 15 ]. Therefore, the particles or candidate solutions fly throughout the problem search space attracted by the positions of the best particles found during the execution of the algorithm. PSO-based methods have been used in a wide range of engineering areas to solve complex continuous optimization problems, such as product design and manufacturing [ 19 ], automotive industry [ 20 ], structural design [ 21 ], and computer networks [22], among others [23,24]. Mathematically, the PSO algorithm is formulated as follows. First, a set of Pparticles (population) is randomly initialized. Note that the position of each particle is a possible solution for the estimation algorithm and it is represented by a d-dimensional vector in the problem space xi=(xi1,xi2, . . . , x) , being i= 1, 2, . . . , P and s∈R . Thus, each particle is randomly placed in the d-dimensional space as a candidate solution and its performance is evaluated using a predefined fitness function. The velocity of the ith particle vi=vi1 , vi2 , . . . , vid , veR , is defined as the change of its position. Depending on the number of objectives considered by the fitness function, the PSO algorithms can be classified as single and multi-objective algorithms [25]. The information available for each particle is based on its own experience and the knowledge of the performance of other particles in its neighborhood. Therefore, each particle adjusts its trajectory based on its own previous best local position and the previous best global position attained by any particle of the swarm, namely p id and p gd . The velocities and positions of particles are updated using Equations (8) and (9), respectively: vid(t+1)=wvid(t)+c1rand1(pid −xid(t)) +c2rand2pgd −xid(t)(8) xid(t+1)=xid(t)+vid(t)(9) where tis the iteration counter, wis the inertia weigh, c 1 and c 2 are the acceleration coefficients, and rand 1 and rand 2 are two random numbers uniformly distributed in the interval [0, 1]. The inertia weight controls the impact of previous velocities on the current velocity and it is used to control the convergence of the PSO [ 12 ]. To reduce this weight over the iterations allowing the algorithm to exploit some specific areas, wis updated according to the following equation: w=wmax −wmax −wmin itermax iter (10) where w max and w min are the maximum and minimum values that the inertia weight can take, iter the current iteration of the algorithm and iter max the maximum number of iterations. The acceleration coefficients c 1 and c 2 control how far a particle moves in a single iteration. The velocity update in Equation (8) has three major components. The first one is the inertia, which models the tendency of the particle to continue in the same direction that it has been travelling. The second component is usually referred as memory and it is the linear attraction towards the best position ever found by the given particle p id scaled by a random weight c1rand1 . The last component, usually referred as cooperation or social knowledge, is the linear attraction towards the best position found by any particle p gd , scaled by another random weight c2rand2. Energies 2019,12, 314 6 of 15 Algorithm 1 Objective function f(x),xi=(xi1,xi2, . . . , x) Initialize locations xiand velocity vi,i=1, 2, . . . , P Find pgd from min{f(x1), . . . , f(xP)}at (t = 0) While (criterion) For loop over all Pparticles and all ddimensions Generate new velocity vid(t+1)using (8) Calculate new locations xid(t+1)using (9) Evaluate objective function at new locations xid(t+1) Find pid for each particle xid End for Find the current pgd Update t = t + 1 End while Output the final results xid and pgd with p id the PSO algorithm tries to force exploitation around local optimums, while with p gd the algorithm explores new areas of the search space. Both features are the main tools for the PSO algorithm to achieve satisfactory results in complex optimization problems like the one presented in this work. Algorithm 1 represents the original implementation of the PSO algorithm used in this work. Furthermore, in this work, each individual will represent the set of electrical parameters to be estimated using the PSO algorithm, whose result will be proven to converge to an optimal solution. 3. Suggested Estimation Procedure The proposal presented in this work utilizes both the standstill technique and the PSO procedure that have been particularized to the system under study, which is a symmetrical five-phase induction machine with distributed windings fed by a two-level VSI. In order to have a better understanding of the estimation procedure, this section will detail the standstill scheme, where an insight into how the electrical parameters are estimated is provided. Then, the application of the search engine based on the PSO method to obtain the final estimation is described. 3.1. Standstill Procedure in Five-Phase Induction Drives The basis of standstill identification schemes is that the machine model can be simplified when the rotor speed is zero ( ωr = 0). This can be obtained with an appropriate stator winding arrangement that avoids the generation of electrical torque. Several stator winding arrangements can be chosen, generating different stator current components. Table 1summarizes two winding arrangements proposed in [ 9 ] for the identification of the electrical parameters in the α - β (first row) and x-y (second row) subspaces. The first one maximizes the α –axis component with respect to the x-axis component (winding connection 1), while the remaining components are zero. This arrangement allows two identification processes in the α - β subspace for the estimation of the rotor parameters (R r , L lr ) and the magnetizing inductance (L m ). The second one maximizes the x-axis component with respect to the α -axis component (winding connection 2), generating null components in the rest. Then, this second arrangement allows one identification process in the x-y subspace to estimate the stator resistance (R s ) and stator leakage inductance (L ls ) parameters. The resulting discrete dynamics models, which will be used in the identification process, are obtained as follows. Energies 2019,12, 314 7 of 15 Table 1. Two available windings’ arrangements in a five-phase induction machine for the single-phase standstill estimation procedure. Winding Connections Voltage Vectors Decoupled Voltage Components Energies 2018, 11, x FOR PEER REVIEW 7 of 15 𝐼(𝑧) 𝑉(𝑧)=𝑍󰇫1−𝑒  𝑠1+𝑠𝜏 𝐾𝑠󰇬=𝜏+(𝑇+𝜏)𝑧 𝐾(1−𝑧) (15) where T s is the sampling period. This model, also called “full-order transfer function model in the α-β subspace”, provides information of current response in the α-β plane. In essence, the same model has been so far used in the identification process of three-phase machines using standstill techniques whose parameters are identified using this transfer function. The model in the x-y subspace is now studied. The continuous-time transfer function that describes the x-axis current response is obtained after creating a stator voltage using the winding arrangement shown in the second row in Table 1: Table 1. Two available windings’ arrangements in a five-phase induction machine for the singlephase standstill estimation procedure. Winding Connections Voltage Vectors Decoupled Voltage Components 𝑣  =1.1708⋅𝑉  𝑣  =−0.1708⋅𝑉  𝑣  =0 𝑣  =𝑣  =0 𝑣  =−0.1708⋅𝑉  𝑣  =1.1708⋅𝑉  𝑣  =0 𝑣  =𝑣  =0 𝐼(𝑠) 𝑉(𝑠)=1 (𝑅+𝑠𝐿)=1 𝑅(1+𝑠𝜏) (16) where τ ls =L ls /R s . This model in the x-y subspace can be referred as the “stator leakage inductance model” because it contributes to the estimation allowing the identification of the L ls parameter. The input voltage in the x–axis depends on V dc, as it is detailed in Table 1 (fourth column). Notice that the obtained stator voltage is not null in α-β plane. Therefore, certain disturbance to the identification process is generated as in the previous case. The model of the current response is then discretized using a zeroorder hold as follows: 𝐼(𝑧) 𝑉(𝑧)=𝑍󰇫1−𝑒  𝑠1 𝑅(1+𝑠𝜏)󰇬= 1−𝑒  /  𝑧 𝑅(1−𝑒  /  𝑧) (17) The stator leakage inductance model provides additional information, compared with the threephase case, about the identification of R s and L ls parameters, and will be used for this purpose. 3.2. Search Engine for the Estimation Process Using PSO The main idea of using the PSO algorithm in this complex application is to converge towards a good solution of the estimated electrical parameters of a five-phase induction machine. Each particle is composed of a set of electrical parameters like an unknown vector x = [R s R r L m L ls L lr ] to be accurately estimated. The fitness function used to evaluate the quality of every particle in the population is the mean squared error (MSE) between the outputs given by the real system (the multiphase induction machine, 𝑦 and 𝑦 in Equation (18) and the outputs given by a modelled system (using Matlab and named 𝑦 and 𝑦). Both systems (the real machine and the Matlab-based model) are governed using stator voltages in the standstill configuration to generate first a response in the α-β subspace and then in the x-y plane. The full-order model is avoided to guarantee that the estimation of the α-β Energies 2018, 11, x FOR PEER REVIEW 7 of 15 𝐼(𝑧) 𝑉(𝑧)=𝑍󰇫1−𝑒  𝑠1+𝑠𝜏 𝐾𝑠󰇬=𝜏+(𝑇+𝜏)𝑧 𝐾(1−𝑧) (15) where T s is the sampling period. This model, also called “full-order transfer function model in the α-β subspace”, provides information of current response in the α-β plane. In essence, the same model has been so far used in the identification process of three-phase machines using standstill techniques whose parameters are identified using this transfer function. The model in the x-y subspace is now studied. The continuous-time transfer function that describes the x-axis current response is obtained after creating a stator voltage using the winding arrangement shown in the second row in Table 1: Table 1. Two available windings’ arrangements in a five-phase induction machine for the singlephase standstill estimation procedure. Winding Connections Voltage Vectors Decoupled Voltage Components 𝑣  =1.1708⋅𝑉  𝑣  =−0.1708⋅𝑉  𝑣  =0 𝑣  =𝑣  =0 𝑣  =−0.1708⋅𝑉  𝑣  =1.1708⋅𝑉  𝑣  =0 𝑣  =𝑣  =0 𝐼(𝑠) 𝑉(𝑠)=1 (𝑅+𝑠𝐿)=1 𝑅(1+𝑠𝜏) (16) where τ ls =L ls /R s . This model in the x-y subspace can be referred as the “stator leakage inductance model” because it contributes to the estimation allowing the identification of the L ls parameter. The input voltage in the x–axis depends on V dc, as it is detailed in Table 1 (fourth column). Notice that the obtained stator voltage is not null in α-β plane. Therefore, certain disturbance to the identification process is generated as in the previous case. The model of the current response is then discretized using a zeroorder hold as follows: 𝐼(𝑧) 𝑉(𝑧)=𝑍󰇫1−𝑒  𝑠1 𝑅(1+𝑠𝜏)󰇬= 1−𝑒  /  𝑧 𝑅(1−𝑒  /  𝑧) (17) The stator leakage inductance model provides additional information, compared with the threephase case, about the identification of R s and L ls parameters, and will be used for this purpose. 3.2. Search Engine for the Estimation Process Using PSO The main idea of using the PSO algorithm in this complex application is to converge towards a good solution of the estimated electrical parameters of a five-phase induction machine. Each particle is composed of a set of electrical parameters like an unknown vector x = [R s R r L m L ls L lr ] to be accurately estimated. The fitness function used to evaluate the quality of every particle in the population is the mean squared error (MSE) between the outputs given by the real system (the multiphase induction machine, 𝑦 and 𝑦 in Equation (18) and the outputs given by a modelled system (using Matlab and named 𝑦 and 𝑦). Both systems (the real machine and the Matlab-based model) are governed using stator voltages in the standstill configuration to generate first a response in the α-β subspace and then in the x-y plane. The full-order model is avoided to guarantee that the estimation of the α-β vsα=1.1708 ·Vdc vsx =−0.1708 ·Vdc vsz =0 vsβ=vsy =0 Energies 2018, 11, x FOR PEER REVIEW 7 of 15 𝐼(𝑧) 𝑉(𝑧)=𝑍󰇫1−𝑒  𝑠1+𝑠𝜏 𝐾𝑠󰇬=𝜏+(𝑇+𝜏)𝑧 𝐾(1−𝑧) (15) where T s is the sampling period. This model, also called “full-order transfer function model in the α-β subspace”, provides information of current response in the α-β plane. In essence, the same model has been so far used in the identification process of three-phase machines using standstill techniques whose parameters are identified using this transfer function. The model in the x-y subspace is now studied. The continuous-time transfer function that describes the x-axis current response is obtained after creating a stator voltage using the winding arrangement shown in the second row in Table 1: Table 1. Two available windings’ arrangements in a five-phase induction machine for the singlephase standstill estimation procedure. Winding Connections Voltage Vectors Decoupled Voltage Components 𝑣  =1.1708⋅𝑉  𝑣  =−0.1708⋅𝑉  𝑣  =0 𝑣  =𝑣  =0 𝑣  =−0.1708⋅𝑉  𝑣  =1.1708⋅𝑉  𝑣  =0 𝑣  =𝑣  =0 𝐼(𝑠) 𝑉(𝑠)=1 (𝑅+𝑠𝐿)=1 𝑅(1+𝑠𝜏) (16) where τ ls =L ls /R s . This model in the x-y subspace can be referred as the “stator leakage inductance model” because it contributes to the estimation allowing the identification of the L ls parameter. The input voltage in the x–axis depends on V dc, as it is detailed in Table 1 (fourth column). Notice that the obtained stator voltage is not null in α-β plane. Therefore, certain disturbance to the identification process is generated as in the previous case. The model of the current response is then discretized using a zeroorder hold as follows: 𝐼(𝑧) 𝑉(𝑧)=𝑍󰇫1−𝑒  𝑠1 𝑅(1+𝑠𝜏)󰇬= 1−𝑒  /  𝑧 𝑅(1−𝑒  /  𝑧) (17) The stator leakage inductance model provides additional information, compared with the threephase case, about the identification of R s and L ls parameters, and will be used for this purpose. 3.2. Search Engine for the Estimation Process Using PSO The main idea of using the PSO algorithm in this complex application is to converge towards a good solution of the estimated electrical parameters of a five-phase induction machine. Each particle is composed of a set of electrical parameters like an unknown vector x = [R s R r L m L ls L lr ] to be accurately estimated. The fitness function used to evaluate the quality of every particle in the population is the mean squared error (MSE) between the outputs given by the real system (the multiphase induction machine, 𝑦 and 𝑦 in Equation (18) and the outputs given by a modelled system (using Matlab and named 𝑦 and 𝑦). Both systems (the real machine and the Matlab-based model) are governed using stator voltages in the standstill configuration to generate first a response in the α-β subspace and then in the x-y plane. The full-order model is avoided to guarantee that the estimation of the α-β Energies 2018, 11, x FOR PEER REVIEW 7 of 15 𝐼(𝑧) 𝑉(𝑧)=𝑍󰇫1−𝑒  𝑠1+𝑠𝜏 𝐾𝑠󰇬=𝜏+(𝑇+𝜏)𝑧 𝐾(1−𝑧) (15) where T s is the sampling period. This model, also called “full-order transfer function model in the α-β subspace”, provides information of current response in the α-β plane. In essence, the same model has been so far used in the identification process of three-phase machines using standstill techniques whose parameters are identified using this transfer function. The model in the x-y subspace is now studied. The continuous-time transfer function that describes the x-axis current response is obtained after creating a stator voltage using the winding arrangement shown in the second row in Table 1: Table 1. Two available windings’ arrangements in a five-phase induction machine for the singlephase standstill estimation procedure. Winding Connections Voltage Vectors Decoupled Voltage Components 𝑣  =1.1708⋅𝑉  𝑣  =−0.1708⋅𝑉  𝑣  =0 𝑣  =𝑣  =0 𝑣  =−0.1708⋅𝑉  𝑣  =1.1708⋅𝑉  𝑣  =0 𝑣  =𝑣  =0 𝐼(𝑠) 𝑉(𝑠)=1 (𝑅+𝑠𝐿)=1 𝑅(1+𝑠𝜏) (16) where τ ls =L ls /R s . This model in the x-y subspace can be referred as the “stator leakage inductance model” because it contributes to the estimation allowing the identification of the L ls parameter. The input voltage in the x–axis depends on V dc, as it is detailed in Table 1 (fourth column). Notice that the obtained stator voltage is not null in α-β plane. Therefore, certain disturbance to the identification process is generated as in the previous case. The model of the current response is then discretized using a zeroorder hold as follows: 𝐼(𝑧) 𝑉(𝑧)=𝑍󰇫1−𝑒  𝑠1 𝑅(1+𝑠𝜏)󰇬= 1−𝑒  /  𝑧 𝑅(1−𝑒  /  𝑧) (17) The stator leakage inductance model provides additional information, compared with the threephase case, about the identification of R s and L ls parameters, and will be used for this purpose. 3.2. Search Engine for the Estimation Process Using PSO The main idea of using the PSO algorithm in this complex application is to converge towards a good solution of the estimated electrical parameters of a five-phase induction machine. Each particle is composed of a set of electrical parameters like an unknown vector x = [R s R r L m L ls L lr ] to be accurately estimated. The fitness function used to evaluate the quality of every particle in the population is the mean squared error (MSE) between the outputs given by the real system (the multiphase induction machine, 𝑦 and 𝑦 in Equation (18) and the outputs given by a modelled system (using Matlab and named 𝑦 and 𝑦). Both systems (the real machine and the Matlab-based model) are governed using stator voltages in the standstill configuration to generate first a response in the α-β subspace and then in the x-y plane. The full-order model is avoided to guarantee that the estimation of the α-β vsα=−0.1708 ·Vdc vsx =1.1708 ·Vdc vsz =0 vsβ=vsy =0 The first identification model focuses on the α - β plane and it is shown in the upper row of Table 1. Then, a winding arrangement is chosen to minimize stator voltage in the x-y subspace and reduce any interference between orthogonal frames. However, notice that the obtained stator voltage is not null in the x-y plane, so certain disturbance in the identification process is generated. The stator and rotor current responses in the α–axis can be described by the following equations: Vsα(s)=(Rs+sLs)Isα(s)+sLmIrα(s)0=(Rr+sLr)Irα(s)+sLmIsα(s)(11) The transfer function that models the current response in the α–βsubspace is as follows: Vsα(s)=(Rs+sσLs)Isα(s)+sKT 1+sτr Isα(s)(12) where KT=Lm2/Lr,τr=Lr/Rrand σLs=Ls–KT. The continuous-time transfer function that describes the α –axis stator current response can be simplified using the term V sr (s) detailed in (13), as it is shown in (14), and discretized using a zero-order holder as it is stated in (15): Vsr(s)=Vsα(s)−(Rs+sσLs)Isα(s)(13) Isα(s) Vsr(s)=(1+sτr) KTs(14) Isα(z) Vsr(z)=Z1−e−sTs s·1+sτr KTs=τr+(Ts+τr)z−1 KT(1−z−1)(15) where Tsis the sampling period. This model, also called “full-order transfer function model in the α - β subspace”, provides information of current response in the α - β plane. In essence, the same model has been so far used in the identification process of three-phase machines using standstill techniques whose parameters are identified using this transfer function. The model in the x-y subspace is now studied. The continuous-time transfer function that describes the x-axis current response is obtained after creating a stator voltage using the winding arrangement shown in the second row in Table 1: Isx(s) Vsx(s)=1 (Rs+sLls)=1 Rs(1+sτls)(16) where τls =Lls/Rs. Energies 2019,12, 314 8 of 15 This model in the x-y subspace can be referred as the “stator leakage inductance model” because it contributes to the estimation allowing the identification of the L ls parameter. The input voltage in the x–axis depends on V dc, as it is detailed in Table 1(fourth column). Notice that the obtained stator voltage is not null in α - β plane. Therefore, certain disturbance to the identification process is generated as in the previous case. The model of the current response is then discretized using a zero-order hold as follows: Isx(z) Vsx(z)=Z1−e−sTs s·1 Rs(1+sτls)=1−e−Ts/τls z−1 Rs1−e−Ts/τls z−1(17) The stator leakage inductance model provides additional information, compared with the three-phase case, about the identification of Rsand Lls parameters, and will be used for this purpose. 3.2. Search Engine for the Estimation Process Using PSO The main idea of using the PSO algorithm in this complex application is to converge towards a good solution of the estimated electrical parameters of a five-phase induction machine. Each particle is composed of a set of electrical parameters like an unknown vector x = [R s R r L m L ls L lr ]to be accurately estimated. The fitness function used to evaluate the quality of every particle in the population is the mean squared error (MSE) between the outputs given by the real system (the multiphase induction machine, yα and yx in Equation (18) and the outputs given by a modelled system (using Matlab and named ˆ yα and ˆ yx ). Both systems (the real machine and the Matlab-based model) are governed using stator voltages in the standstill configuration to generate first a response in the α - β subspace and then in the x-y plane. The full-order model is avoided to guarantee that the estimation of the α - β parameters (involved in the main control magnitudes of the electrical drive such as the electrical torque and the stator flux production) is made without having any interference of the x-y plane, which is related to the electrical losses in a machine with distributed winding. For this reason, the same weights have been considered for both subspaces α - β and x-y. Consequently, the proposed fitness function gfor this study is defined as follows: g=pMSEx2+MSEα2 MSEα2=1 Nα Nα ∑ k=1 kyα(k)−ˆ yα(k)k2(18) MSEx2=1 Nx Nx ∑ k=1 kyx(k)−ˆ yx(k)k2 where the MSE α and MSE x values are the mean squared errors computed for the response in the α - and x-axis, respectively, and N α and N x regulate the desired accuracy in the estimation of the α - β and x-yparameters (in this case, the same accuracy has been selected). The complexity of the estimation procedure comes from adjusting simultaneously the two regression models of α - β and x-yplanes based on the response of the multiphase machine in standstill arrangements to known input signals. On the one hand, the regression model of α - β plane that allows the estimation of R r ,L lr and L m parameters. On the other hand, the x-yplane that enables the estimation of R s and L ls . Notice that the five electrical parameters have continuous values ranging from the intervals included in Table 3 (see Section 4.1 for more details). Therefore, the complexity of the optimization problem consists in finding the most optimal values that reduce the error among the simulated response of electrical machine model with the electrical parameters as inputs and the real response obtained from experiments. 4. Experimental Assessment The performance of the proposal is analyzed using an experimental test bench based on a symmetrical five-phase induction machine with distributed windings. The multiphase machine was Energies 2019,12, 314 9 of 15 built from a commercial three-phase induction machine that has been rewound and reassembled. Then, the proposed estimation technique is applied to obtain the unidentified vector x= [R s R r L m L ls L lr ] that represents the electrical parameters of the multi-phase machine. Figure 3shows an scheme of the experimental test bench, where pictures of electronic equipment are included. The VSI-based multiphase power converter is built from two commercial three-phase modules from Semikron (SKS21F) that are linked to a unique DC of up to 300 V. The controller is based on a well-known digital signal processor from Texas Instruments (12500 TI Boulevard, Dallas, TX, USA) and Technosoft (Avenue des Alpes 20, 2000 Neuchâtel, Switzerland), the TMS320LF28335 and the MSK28335 board, respectively. Sensing some electrical variables (stator currents and voltages) is a major requirement in the estimation strategy, which it is done using two different sensors from LEM (Chemin des Aulx 8, P.O. Box 35, 1228 Plan-les-Ouates, Switzerland), the LA-55P and LV-25P devices. It is important to highlight that the voltage electrical signals obtained from the sensors are filtered using analog low-pass filters with a cut-off frequency of 1.5 kHz. It is also interesting to remark that the windings of the multiphase machine must be rearranged to avoid torque generation and to assure the standstill behavior. This is done following the connection scheme shown in the first row of Table 1. Energies 2018, 11, x FOR PEER REVIEW 9 of 15 Figure 3. Scheme of the experimental test bench. 4.1. Identification of the Electrical Parameters of the System Two different stator voltages are applied using the proposed standstill tests in α-β and x-y subspaces and the current responses of the system are recorded. The winding connection shown in the first row of Table 1 is initially used and a step voltage from –20 to 20 V is applied to the machine, Figure 4a, to obtain the current response in the α–axis shown in Figure 4b. This voltage excites the electromagnetic circuit and rotor time constants at standstill in the α-axis, as it is detailed in Equation (15). The winding connection shown in the second row of Table 1 is then used and a three-level signal (–60, 0 and 60 V) with a fundamental frequency of 25 Hz is applied to the stator, see Figure 4c. This stator voltage excites the stator electromagnetic circuit detailed in Equation (17), producing the stator current in the x–axis shown in Figure 4d. The obtained stator current responses y are then compared with the modelled responses ŷ, evaluated with Equations (15) and (17) in order to compute the fitness function g (18) of each individual in the PSO algorithm. (a) (b) 00.5 11.5 2 -30 -20 -10 0 10 20 30 Time (s) Voltage (V) 1.1 1.15 1.2 1.25 1.3 1.35 0.8 1 1.3 1.6 1.8 Time (s) α -current (A) Figure 3. Scheme of the experimental test bench. 4.1. Identification of the Electrical Parameters of the System Two different stator voltages are applied using the proposed standstill tests in α - β and x-y subspaces and the current responses of the system are recorded. The winding connection shown in the first row of Table 1is initially used and a step voltage from –20 to 20 V is applied to the machine, Figure 4a, to obtain the current response in the α –axis shown in Figure 4b. This voltage excites the electromagnetic circuit and rotor time constants at standstill in the α -axis, as it is detailed in Equation (15). The winding connection shown in the second row of Table 1is then used and a three-level signal (–60, 0 and 60 V) with a fundamental frequency of 25 Hz is applied to the stator, see Figure 4c. This stator voltage excites the stator electromagnetic circuit detailed in Equation (17), producing the stator current in the x–axis shown in Figure 4d. The obtained stator current responses y are then compared with the modelled responses ˆ y, evaluated with Equations (15) and (17) in order to compute the fitness function g (18) of each individual in the PSO algorithm.