Real-time estimation and tracking of parameters in permanent magnet synchronous motor using a modified two-stage particle swarm optimization algorithm Elham Mohammadalipour Tofighi Department of Power Engineering Faculty of Electrical and Computer Engineering University of Tabriz, Tabriz, Iran
[email protected]
Amin Mahdizadeh Student member, IEEE
[email protected]
Abstract— Reliable controller design for permanent magnet synchronous motor requires knowledge of the motor parameters to build an accurate model of the system. In real-time operation, various environmental and internal factors affect the parameters value which consequently influence the performance of the controller over time. The former studies used the extracted data in off-line mode of operation to estimate the parameters. In this study, the particle swarm optimization algorithm is implemented to estimate the parameters of PMSM (stator resistance, inductances, and the rotor permanent magnet (PM) flux linkage) during motor operation. The effects of the temperature rise on PMSM parameters are investigated and the stator resistance change is successfully tracked with the proposed Two-Stage Single-Flock Particle Swarm Optimization method in steady-state operation condition. The simulation is based on a proper model of the PMSM, including electromagnetic and mechanical elements. The parameters are calculated and minimized via a normalized root mean square error. Keywords— Permanent magnet synchronous motor (PMSM), particle swarm optimization (PSO), on-line parameter estimation, electrical motor modeling
I.
INTRODUCTION
High efficiency, high torque-to-inertia ratio, sharp dynamic response, simple control and modeling, maintenancefree operation, and structural compactness [1] are some of the very favorable features of the permanent magnet synchronous motors. These qualities make PMSM a very reliable choice in variable speed high performance motor-drive applications [2]. Advanced controllers are required to drive these motors. The design of such controllers depends on the knowledge of PMSM parameters’ quality and sharpness and their variation with temperature and frequency changes during operation [3]. Any mismatch between the actual motor parameter values (e.g., stator resistance, inductance and the PM flux linkage) and those used in controller body will lead to deterioration in drive and consequently the motor performance. Hence, these
Mohammad Reza Feyzi Department of Power Engineering Faculty of Electrical and Computer Engineering University of Tabriz, Tabriz, Iran
[email protected]
parameters must be identified as accurate as possible. Different analytical [4,5] and experimental methods such as IEEE Std-112 with blocked-rotor and no-load test [6] and DC current decay test [7] have been developed and implemented in PMSM parameter estimation. Adaptive identification algorithms can also be implemented to estimate the electrical parameters of a PMSM based on output identification errors [8]. The above mentioned approaches use mathematical and engineering assumptions to simplify the nonlinearities so that they are adoptable to linear system and control theories. However, in high performance applications, the linear system and control theories are inadequate and cannot meet the necessary precision demanded by the whole process. With advancement in DSPs’ design and application in recent years, optimization techniques have been widely used in parameter identification of electrical machines addressing the problem of model and parameter identification for nonlinear systems. Genetic algorithm (GA) [9] and particle swarm optimization (PSO) have received special attention due to their simple nature in implementation and adjustment in comparison with other evolutionary algorithms. PSO itself has different versions such as diversity-guided particle swarm optimization (DGPSO) [10], particle swarm optimization with a constriction factor [11], particle swarm optimization with a time-varying inertia weight [12], dynamic particle swarm optimization (Dynamic PSO) with time-varying acceleration coefficients of the cognitive and social components [13] and a stretching particle swarm optimization (SPSO) [14] which differ mainly in concept of how to regulate the search and optimization evolution process. These methods are processed in offline mode and optimistically consider the original system to be under no change pressure during operation which is not the actual case in most applications. The online PSO has been applied in [15] for tuning the PI controller and parameter optimization of electrical motors. In [16] the stator resistance and the disturbed load torque are identified for a variable frequency PMSM drive. Although
PSO is a robust stochastic optimizationn algorithm, the convergence speed for complex systems may cause long running times [17] which in high frequenccy applications of motor-drive control is not acceptable. In [18], the admissible bounds of parameters are minimized using experimental tests which guarantee the accuracy enhancemeent, reduction of search space and convergence time decreasse of overall PSO procedure. In this paper, a two-stage sinngle-flock particle swarm optimization technique is introducedd. It estimates the permanent magnet synchronous motor parameters p during online operation mode by taking into accounnt the temperature rise effects on stator resistance variation. The remainder of this paper is organizzed as follow: the PMSM model and the problem of parameter estimation are demonstrated and formulated in section II. The standard particle swarm optimization algorithm and thhe developed twostage single-flock PSO are presented in secttion III. In section IV, the evaluation and simulation results are presented and the conclusion is summed up in section V respecctively. II.
SYSTEM MODELING
To design a high performance controlller for permanent magnet synchronous motor, a detailed andd precise model is essential. For this purpose, the influence of several properties of geometry (symmetry), material specificcations (nonlinear material effects) and the environment coonditions must be taken into account during the design process. The best choice then will be the finite element methods to uttilize these aspects of PMSM together. However, by using the t feedback and closed-loop control design, impact of many of these phenomena are suppressed and simpler moddels are applicable. The main assumptions in building a simple model m are [19]: • The stator winding is distributed sinnusoidally around the periphery of the air gap. (iinduced EMF is sinusoidal) • The effect of stator slots on the rotor angle a dependence of the inductances is neglected. • Eddy currents and hysteresis losses are negligible. • Saturation is neglected. • There is no cage on the rotor Among a variety of models presented in literature, l the twoaxis dq-model is the most popular model implemented in variable speed PMSM drive control applicattions. With Park’s dq-transformation, the three-phase stationarry variables of the PMSM are converted onto the rotating refereence frame and the transformation angle has the same value as that of the rotor’s electrical position angle θ r [20]. Fig. 1 represents the coordinate Park’s transformation model of thhe PMSM. According to the structure of the motor and a location of the permanent magnet on rotor, there are two types of PMSMs, i.e., the surface-mounted permanent maggnet synchronous motors (SPMSMs) and the interior peermanent magnet synchronous motors (IPMSMs). The diffeerence between a SPMSM and an IPMSM arises from the fact fa that there is a saliency on the rotor of the IPMSM type annd therefore the d-
Fig. 1, Park’s transfformation for PMSM
axis inductance is larger thaan the q-axis inductance. This difference between the d-axis and a q-axis inductances is due to either the asymmetric structuure of the PMSM or the flux induced magnetic saturation caaused by the permanent magnet characteristics [21]. A. Permanent magnet sunchroonous motor model A three-phase surface-m mounted permanent magnet synchronous motor (SPMSM M) based on a synchronously rotating dq-reference frame has h been implemented in this study. The dynamics of PM MSM are described by two subsystems, i.e. the electrical subsystem and the mechanical subsystem which both are preesented by nonlinear equations [22,23]. The parameters of thhe typical PMSM used in this paper are listed in Table I. TABLE I.
P PMSM PARAMETERS
υdr , υqr
dq-componentts of the stator voltage in synchronouslyy rotating rotor reference frame
idr , iqr
dq- componennts of the stator current in synchronouslyy rotating rotor reference frame
ψ mag Rs
Ld , Lq
permanent maagnet flux linkage stator resistancce per phase dq-componentts of the motor inductance
ωr
rotor electric angular a speed
Te
electromechannical torque developed by the machine
TL
Applied load torque t
J
moment of ineertia of the rotor
B
the viscous friiction coefficient
np
number of pairs of poles
The electrical equations are:
didr 1 r = (υd − Rs idr + ωr Lq iqr ) , dt Ld diqr dt
=
1 r (υq − Rs iqr − ωr Ld idr − ωrψ mag ) . Lq
(1)
(2)
Fig. 2 illustrates the equivalent steady-state electrical circuit of PMSM. The mechanical subsystem is composed of the rotor and its bearings. Motion components synthesize the mechanical model of the PMSM as:
⎤ d ωr ω 1⎡ = n p ⎢Te − B r − TL ⎥ , dt J ⎣⎢ np ⎦⎥
(3)
The electromechanical torque developed by the machine is calculated via: Te =
3 n p ⎡iqrψ mag + ( Ld − Lq )idr iqr ⎤⎦ . 2 ⎣
(4)
Equations (1) to (4) demonstrate the dynamic model of the PMSM; a highly coupled nonlinear 2-input 3-output system in dq-reference frame. The parameters idr , iqr and ωr are considered as state variables and υdr and υqr as control signals (inputs). The parameters of the PMSM, particularly the resistance and inductance vary significantly with operation conditions. In this paper only the temperature rise effects on stator resistance is studied. A temperature rise causes a nonlinear variation of resistance which can be typically modeled by a first-order step response: Rst (t ) = Rs , ref + (1 − e Δt T ) ,
(5)
where Δt represent the time duration for the temperature rise. T is the time constant and Rs , ref is the stator resistance value in the normal operation conditions before the temperature rise affects the motor windings’ characteristics. Based on above dynamic equations, the space-state model of the PMSM will be shown as:
x = F ( x, Θ, u ) ,
(6)
y = G ( x, Θ ) ,
(7)
where x represents the states vector. The initial condition is x(0)=x0. Θ is the permanent magnet synchronous machine parameters vector and u represents the inputs vector, and y is the measurable outputs vector: x = ⎡⎣i
r q
Θ = ⎡⎣ Rs
r d
i Lq
u = ⎡⎣υqr
Fig. 2, PMSM steady-state equivalent electrical circuit
ωr ⎤⎦ ,
(8)
Ld ψ mag ⎤⎦ ,
(9)
υ dr ⎤⎦ ,
(10)
y = ⎡⎣iqsr
idsr ⎤⎦ .
(11)
B. Parameter estimation problem formulation In real systems, the parameters vector Θ is unknown and the objective is to determine Θ via a comparison between the time dependent responses of the given system with an appropriate parameterized model using a performance function. The space-state representation of this model system is presented with, ˆ ,u , xˆ = F xˆ, Θ
(
)
(12)
ˆ , yˆ = G xˆ, Θ
(
(13)
)
where xˆ is the states vector of the model and initial condition ˆ is the PMSM model parameters vector and yˆ is xˆ(0) = x0 . Θ is the model’s measurable outputs vector respectively;
xˆ = ⎡⎣iˆqr ˆ = ⎡ Rˆ Θ ⎣ s
iˆdr Lˆq
yˆ = ⎡⎣iˆqsr
ωˆ r ⎤⎦ , Lˆd ψˆ mag ⎤⎦ ,
iˆdsr ⎤⎦ .
(14) (15) (16)
ˆ and applying the same Considering a known value for Θ input signal u to the main system and to the model of the system, yˆ is achieved by means of an objective function. This objective function is an output error comparison tool comparing y and yˆ then minimizing the output towards the optimum result. The fitness function for the parameter estimation problem is written as:
( )
t
ˆ = ( y ( t ) − yˆ ( t ) )2 dt. F Θ ∫
(17)
0
ˆ and is zero (minimum) The fitness F is a function of Θ ˆ whenever Θ = Θ . The optimization problem will be described as:
( )
ˆ , arg min F Θ
(18)
i represents the i-th particle of the swarm, 1≤ N ≤ n,
which will be subjected to: ˆ Θ
min
ˆ
