Improved Parameter Estimation Algorithms for Induction Motors

Transcription

1 Improved Parameter Estimation Algorithms for Induction Motors Julius Susanto A thesis submitted for the degree of Masters by Coursework (Electrical Utility Engineering) Department of Electrical and Computer Engineering November 2013

2 Abstract The performance of Newton-Raphson, Levenberg-Marquardt, Damped Newton- Raphson and genetic algorithms are investigated for the estimation of induction motor equivalent circuit parameters from commonly available manufacturer data. A new hybrid algorithm is then proposed that combines the advantages of both descent and natural optimisation algorithms. Through computer simulation, the hybrid algorithm is shown to significantly outperform the conventional algorithms in terms of convergence and squared error rates. All of the algorithms are tested on a large data set of 6,380 IEC (50Hz) and NEMA (60Hz) motors. Keywords: parameter estimation, induction motor, double cage ii

3 Acknowledgements I would like to thank my project supervisor Prof. Syed Islam for his patient support, ideas and encouragement throughout the project. I d also like to acknowledge my colleagues at DIgSILENT GmbH for their support and helping me to find this problem in the first place. Special thanks goes to my managers Manuel Castillo and Flavio Fernández for their encouragement, Oscar Muñoz and Goce Jauleski for assisting me trawl through the code and Nicholai Wilson for the banter. iii

4 Contents Abstract Acknowledgements ii iii 1 Introduction Rationale Thesis Outline Induction Motor Equivalent Circuits Single Cage Model Double Cage Model Higher Order Models Core and Mechanical Losses Calculating Torque and Current from the Equivalent Circuit Torque-Speed and Current-Speed Curves Parameter Estimation Problem Problem Formulation Ignoring Core Losses Single Cage Model (Ignoring Core Losses) Double Cage Model (Ignoring Core Losses) Problem Formulation Considering Core Losses Single Cage Model (with Core Losses) Double Cage Model (with Core Losses) Classes of Parameter Estimation Algorithms Descent Algorithms Requirement for Linear Restrictions Newton-Raphson Algorithm Parameter Constraints Adaptive Step Size Initial Conditions Convergence Criteria Levenberg-Marquardt Algorithm Choice of Damping Parameter iv

5 4.4 Damped Newton-Raphson Algorithm Comparison of Descent Algorithms Single Cage Model Double Cage Model Selection of Linear Restrictions Approaches for Selecting Linear Restrictions Computer Simulation Selection of Initial Conditions Methods for Calculating Initial Conditions Sets of Initial Conditions Performance of Initial Conditions Incorporation of Initial Conditions into Descent Algorithms Conclusions about Descent Algorithms Natural Optimisation Algorithms Genetic Algorithm Application of GA to Motor Parameter Estimation Computer Simulation Other Natural Optimisation Algorithms Conclusions about Natural Optimisation Algorithms Hybrid Algorithms Motivation for Hybrid Algorithms Proposed Hybrid Algorithm Computer Simulation Comparative Analysis of Algorithms Comparison of Algorithm Performance Convergence and Error Tolerance Algorithm Performance and Motor Rated Power Comparison of Algorithm Computation Time Conclusions and Future Work Conclusions Contributions Future Work References 78 A Motor Data Set 83 B MATLAB Source Code 84 B.1 Common Auxiliary Functions B.1.1 calc pqt B.1.2 get torque v

6 B.2 Descent Algorithms B.2.1 nr solver B.2.2 lm solver (Error Term Adjustment) B.2.3 lm solver2 (Gain Ratio Adjustment) B.2.4 dnr solver B.3 Genetic Algorithm B.3.1 ga solver B.4 Hybrid Algorithms B.4.1 hybrid nr B.4.2 hybrid lm B.4.3 hybrid dnr vi

7 List of Tables 4.1 Comparison of descent algorithms for the single cage model with fixed restrictions k x = 1 and k r = Comparison of descent algorithms for the double cage model with fixed restrictions k x = 1 and k r = Conventional Newton-Raphson algorithm results for double cage model using diffrent methods for selecting linear restrictions Squared error performance of different initial conditions for IEC motors Squared error performance of different initial conditions for NEMA motors Conventional Newton-Raphson algorithm results for double cage model with revised initial conditions Conventional Newton-Raphson algorithm results for double cage model with revised initial estimates for X s and R c (with k x = 1, k r = 0.5) Range of initial parameter estimates Standard deviations for mutation noise Default settings for genetic algorithm Results of the genetic algorithm for the double cage model Range of initial parameter estimates Default settings for hybrid algorithm Simulation results for baseline NR and hybrid algorithms Summary of simulation results for the double cage model Breakdown of motor data sets by motor rated power Algorithm performance broken down by rated power (IEC motors) 72 vii

8 7.4 Algorithm performance broken down by rated power (NEMA motors) Average algorithm solution time viii

9 List of Figures 2.1 General induction motor equivalent circuit Basic single cage model equivalent circuit (5 parameters) Basic double cage model equivalent circuit (7 parameters) Two minimum parameter double cage model equivalent circuits (6 parameters) Example of a higher order model (triple cage) Motor torque-speed curve Motor current-speed curve Flowchart for conventional NR algorithm Simplified motor equivalent circuit at locked rotor Approximate breakdown of stator current Flowchart for genetic algorithm Error rates vs maximum number of generations Flowchart for hybrid algorithm (with natural selection of R s and X r2 ) Convergence rate versus error tolerance plot for IEC motors Convergence rate versus error tolerance plot for IEC motors Visual depiction of error tolerance - Torque-speed curve of 75kW motor Visual depiction of error tolerance - Current-speed curve of 75kW motor ix

10 CHAPTER 1 Introduction 1.1 Rationale The three-phase induction motor is arguably the workhorse of modern industry, found in almost all industrial settings from manufacturing to mining. Equivalent circuit parameters of induction machines are essential for time-domain simulations where the dynamic interactions between the machine(s) and the power system need to be analysed, for example: Motor starting and re-acceleration Bus transfer studies Changes in motor loading Motor behaviour during faults Dynamic voltage stability However, motor manufacturers do not t to provide the equivalent circuit parameters for their machines. This is a problem because the parameters are generally motor specific and typical values found in the literature are often not of sufficient accuracy. Moreover, power system studies involving motors 1

11 1. Introduction 2 are normally performed during the design stages of projects, where the motors themselves have not yet been ordered and on-site testing is not possible. It is therefore desirable to estimate motor equivalent circuit parameters from the data that manufacturers make available in their catalogues, data sheets and technical brochures (i.e. performance parameters such as breakdown torque, locked rotor torque, full-load power factor, full-load efficiency, etc). A number of parameter estimation techniques have been proposed in the literature (for example, see [1], [2], [3], [4] and [5]). The de facto approach that has emerged, and which has been adopted by the majority of commercial software packages, has been to use an algorithm based on a Newton-Raphson approach. However, it has been observed from experience that the Newton-Raphson based algorithms can have poor convergence and error performance. Therefore, parameter estimation algorithms with improved performance would be preferred. In this project, the performance of a number of motor parameter estimation algorithms based on readily available manufacturer data is investigated, and a new hybrid algorithm that exhibits improved performance is proposed. 1.2 Thesis Outline The structure of this thesis is as follows: Chapters 2 and 3 provide background on induction motor equivalent circuits and the nature of the parameter estimation problem based on manufacturer data. Chapter 4 begins the tour of parameter estimation approaches with descent algorithms, which are algorithms based on variations of Newton s method. Three descent algorithms are investigated - the commonly used Newton-Raphson and Levenberg-Marquardt algorithms and the damped Newton-Raphson algorithm. Chapter 4 also contains a brief investigation on the selection of initial parameters.

12 1. Introduction 3 Chapter 5 looks at natural optimisation approaches to solving the parameter estimation problem. In particular, the genetic algorithm is explored and simulated on the motor data set. In Chapter 6, a hybrid descent and natural optimisation approach is proposed and investigated. Three variations of the hybrid algorithm are implemented (NR- GA, LM-GA and DNR-GA) and simulated on the motor data set. The results are then compared with the conventional NR algorithm. Chapter 7 provides a comparative analysis of the algorithms explored in Chapters 4, 5 and 6 in terms of algorithm performance, convergence criteria and computation times. Chapter 8 concludes the thesis by offering a generic workflow for solving motor parameter estimation problems. The contributions of this project and avenues for future work are also discussed.

13 CHAPTER 2 Induction Motor Equivalent Circuits An induction machine can be viewed as a generalised transformer with an air gap and a rotating short-circuited secondary winding. The equivalent circuit of an induction motor is therefore similar to that of a transformer. The key difference is in the rotor equivalent circuit, where the voltages and currents are proportional to the slip frequency. This is commonly represented in the equivalent circuit by a variable (slip-depent) rotor resistance (i.e. Rr s ). For balanced steady-state analysis, it is acceptable to use a per-phase equivalent circuit. Analysis is also made simpler by working with per-unit values, where the scaling factors required to calculate polyphase quantities (such as polyphase power) are not required. The general induction motor per-phase equivalent circuit with all parameters referred to the stator is shown in Figure 2.1. The parameters of this equivalent circuit are as follows: R s is the stator resistance (pu) X s is the stator leakage reactance (pu) X m is the magnetising reactance (pu) R c is the core loss component (pu) 4

14 2. Induction Motor Equivalent Circuits 5 Figure 2.1: General induction motor equivalent circuit X r is the rotor leakage reactance (pu) R r is the rotor resistance (pu) The motor equivalent circuit in Figure 2.1 shows that the parameters vary with frequency / slip, current (i.e. saturation effects) and temperature, for the following reasons: AC resistances for the copper windings are temperature depent [3] Inductances will vary due to eddy currents and saturation of teeth and core [2] Rotor resistance is frequency (slip) depent due to eddy currents in deep rotor bars or double cage rotors [6] Impedances are affected by the skin effect at different frequencies [6] For power system studies, it is desirable to use a motor equivalent circuit with constant parameters that are valid over the full range of motor speeds (i.e. from 0 to 1 pu). It is important to note that these parameters will not likely correspond to the real motor parameters, since as noted above, the real parameters are slip, current and temperature depent. However, the set of constant equivalent

15 2. Induction Motor Equivalent Circuits 6 circuit parameters will match the motor s performance characteristics over the full speed range (e.g. torque-slip and current-slip curves, power factor, efficiency, etc). While others have attempted to apply motor models with variable parameters (for example, Haque in [7]), there does not appear to be significant gains to be had by this approach and it only makes the model more complicated and less manageable for standard power systems analysis programs. Therefore, only constant parameter models are considered in this project. In the following subsections, a number of constant parameter equivalent circuit models are presented. 2.1 Single Cage Model The single cage model is simply the general equivalent circuit in Figure 2.1 with constant parameters (and excluding core losses). The single cage model is normally suitable to represent the performance characteristics of wound-rotor motors. Figure 2.2: Basic single cage model equivalent circuit (5 parameters)

16 2. Induction Motor Equivalent Circuits Double Cage Model To account for the effects of double-cage rotors or deep bar rotors (e.g. most squirrel-cage machines), a second rotor branch is added to the equivalent circuit of the single cage model. Figure 2.3: Basic double cage model equivalent circuit (7 parameters) In the equivalent circuit, the inner cage leakage reactance X r1 is always higher than the outer cage leakage reactance X r2, but the outer cage impedance is typically higher than the inner cage impedance on starting. These conditions can be resolved by including the following two inequality constraints in the model [5]: X r1 > X r2 R r2 > R r1 Corcoles et al [8] showed that since the double cage model (without core losses) has 6 model invariant functions (MIVs), then any model with greater than 6 parameters (such as the 7 parameter double cage model in Figure 2.3) can be reduced to a 6 parameter model without any loss of information. There are 5 so-called minimum parameter models for the double cage model, two of which are shown below in Figure 2.4.

17 2. Induction Motor Equivalent Circuits 8 Figure 2.4: Two minimum parameter double cage model equivalent circuits (6 parameters) 2.3 Higher Order Models Additional rotor branch circuits and other impedances (e.g. reactances between rotor branches) can be added to imbue the model with additional degrees of freedom. As will be shown later, adding more parameters to the model can be problematic for estimation purposes, as the system of equations becomes underdetermined and some parameters need to be constrained in order to solve the system, e.g. by linear restrictions. As discussed earlier, Corcoles et al [8] found that some higher order models can be algebraically reduced to lower order models without any loss of information. In other words, some parameters are redundant and their addition can turn a system with a univocally identifiable solution into a system with an infinite number of solutions (i.e. with parameters that need to be constrained).

18 2. Induction Motor Equivalent Circuits 9 Figure 2.5: Example of a higher order model (triple cage) 2.4 Core and Mechanical Losses In the equivalent circuit models described thus far, the shunt magnetising branch is represented only by a magnetising reactance (X m ) and the core losses are neglected. In a practical motor, there will also be eddy currents in the core laminations that manifest themselves as heat losses. These core (or iron) losses can be modelled as a shunt resistance as in the general motor model in Figure 2.1. Furthermore, there are mechanical losses due to friction on the rotor bearings, and while it isn t completely appropriate to model mechanical losses in an electrical circuit, these frictional losses can also be approximated by a shunt resistance. The reason for modelling these losses in the motor equivalent circuit is so that motor efficiencies can be properly estimated (this is described in more detail in section 3.2). The core and mechanical losses are lumped together as a single shunt resistance R c. To simplify further, the shunt resistance can be placed at the input of the equivalent circuit rather than parallel to the magnetising branch [9].

19 2. Induction Motor Equivalent Circuits Calculating Torque and Current from the Equivalent Circuit The electrical torque developed in an induction machine is proportional to the square of rotor current, i.e. where T is the electrical torque developed (N-m) p is the number of motor poles q is the number of stator phases f is the nominal frequency (Hz) R r is the equivalent rotor resistance (Ω) s is the motor slip (pu) I r is the rotor current (A) T = pq R r 4πf s I2 r (2.1) By using per-unit values, the constant terms can be eliminated and the equation above reduces to: T = R r s I2 r (2.2) where all the quantities in this equation are in per-unit values. It can be sees that for any given motor equivalent circuit, standard circuit analysis can be used to calculate the rotor current and therefore electrical torque. By way of example, the torque in the double cage model (without core losses) introduced earlier in section 2.2 will be calculated. Recasting the impedances as admittances:

20 2. Induction Motor Equivalent Circuits 11 1 Y s = R s + jx s (2.3) Y m = 1 jx m (2.4) 1 Y r1 = R r1 (2.5) + jx s r1 1 Y r2 = R r2 (2.6) + jx s r2 Applying Kirchoff s law, the voltage U 1 at the magnetising branch is: (U n U 1 ) Y s = U 1 (Y m + Y r1 + Y r2 ) (2.7) U n Y s = U 1 (Y s + Y m + Y r1 + Y r2 ) (2.8) U 1 = U n Y s Y s + Y m + Y r1 + Y r2 (2.9) The per-unit stator current I s is therefore: I s = (U n U 1 ) Y s (2.10) The per-unit rotor currents in each cage I r1 and I r2 are: I r1 = U 1 Y r1 (2.11) I r2 = U 1 Y r2 (2.12) Finally, the per-unit electrical torque developed in the motor is: T = R r1 s I2 r1 + R r2 s I2 r2 (2.13)

21 2. Induction Motor Equivalent Circuits 12 A similar kind of analysis can be done for other motor equivalent circuit models to calculate the electrical torque and current of the machine. 2.6 Torque-Speed and Current-Speed Curves Based on the torque and stator current equations developed in the previous section, torque-speed and current-speed curves can be constructed from the equivalent circuit for the full range of motor speeds (i.e. from standtill to synchronous speed). Examples of motor torque-speed and current-speed curves are shown in Figure 2.6 and Figure 2.7 respectively. Figure 2.6: Motor torque-speed curve

22 2. Induction Motor Equivalent Circuits 13 Figure 2.7: Motor current-speed curve

23 CHAPTER 3 Parameter Estimation Problem The characteristics of an induction motor are normally provided by manufacturers in the form of a standard set of performance parameters, with the following parameters being the most common: Nominal voltage, U n (V) Nominal frequency, f (Hz) Rated asynchronous speed, n fl (rpm) Rated (stator) current, I s,fl (A) Rated mechanical power, P m,fl (kw) Rated torque, T n (Nm) Full load power factor, cos φ fl (pu) Full load efficiency, η fl (pu) Breakdown torque, T b /T n (normalised) Locked rotor torque, T lr /T n (normalised) Locked rotor current, I lr /I s,fl ) (pu) 14

24 3. Parameter Estimation Problem 15 From previous sections, we know that a set of equivalent circuit parameters can yield specific torque-speed and current-speed curves. So given a set of performance parameters that contain features on the torque-speed and current-speed curves (e.g. breakdown torque, locked-rotor current, etc), is it possible to determine the corresponding equivalent circuit parameters that yield these features? This is the crux of the parameter estimation problem and can be posed as follows - How can the motor performance parameters be converted into equivalent circuit parameters?. While all of the performance parameters in the above set can be used in an estimation procedure, there are actually only six indpent magnitudes that can be formed from them: P m,fl, Q fl, T b, T lr, I lr and η fl [5]. These indepent magnitudes will thus form the basis of the problem formulation, where the indepent magnitudes calculated from the equivalent circuit are matched with the performance parameters supplied by the manufacturer. The basic double cage model in section 2.2 is used to illustrate how these six indepent magnitudes can be calculated from the equivalent circuit model. Stator and rotor currents at slip s can be readily calculated from the equivalent circuit as shown in section 2.5. Quantities for per-unit active power P, reactive power Q and power factor cos φ at slip s can be calculated as follows: S(s) = U n I s (s) (3.1) P (s) = T (s)(1 s) (3.2) Q(s) = I{S(s)} (3.3) cos φ(s) = R{S(s)} S(s) (3.4)

25 3. Parameter Estimation Problem 16 Nominal speed n s and full load slip s f is calculated as follows: n s = 120f p (3.5) s f = 1 n fl n s (3.6) where p is the number of motor poles f is the nominal frequency (Hz) n fl is the asynchronous speed at full load (rpm) Calculating the slip at maximum torque s max is found by solving the equation: dt ds = 0 (3.7) (Under the condition that the second derivative d2 ds 2 < 0) In the double cage model, the solution to this equation is not trivial and it is more convenient to use an estimate, e.g. based on an interval search between s = 0 and s = Problem Formulation Ignoring Core Losses Single Cage Model (Ignoring Core Losses) In the single cage model, the locked rotor torque T lr and locked rotor current I lr are not used because the single cage model does not have enough degrees of freedom to capture both the starting and breakdown torque characteristics without introducing significant errors [10]. As a result, it is more commonplace to only consider the breakdown torque T b in the single cage model and simply ignore the torque and current characteristics at locked rotor. For wound-rotor motors,

26 3. Parameter Estimation Problem 17 this yields sufficiently accurate results (i.e. in terms of the resulting torque-speed curve). However, a single-cage model is unable to accurately model the torquespeed characteristics of squirrel cage motors, especially those with deep bars, and thus a double cage model should be used for these types of motors. Without taking into account core losses, the full load motor efficiency η fl also cannot be used (see section 3.2 for more details). Therefore, there are only three indepent parameters that can be used in the problem formulation: P m,fl, Q fl and T b. These indepent parameters can be used to formulate the parameter estimation in terms of a non-linear least squares problem, with a set of non-linear equations of the form F(x) = 0: f 1 (x) = P m,fl P (s f ) = 0 (3.8) f 2 (x) = sin φ Q(s f ) = 0 (3.9) f 3 (x) = T b T (s max ) = 0 (3.10) (3.11) where F = (f 1, f 2, f 3 ) and x = (R s, X s, X m, R r, X r ) are the equivalent circuit parameters of the single cage model Double Cage Model (Ignoring Core Losses) In the double cage model, the locked rotor torque T lr and locked rotor current I lr are included as indepent parameters. As in the single cage model, the full load motor efficiency η fl cannot be used without taking into account core losses. Therefore, there are five indepent parameters and the following non-linear

27 3. Parameter Estimation Problem 18 least squares problem: f 1 (x) = P m,fl P (s f ) = 0 (3.12) f 2 (x) = sin φ Q(s f ) = 0 (3.13) f 3 (x) = T b T (s max ) = 0 (3.14) f 4 (x) = T lr T (s = 1) = 0 (3.15) f 5 (x) = I lr I(s = 1) = 0 (3.16) (3.17) where F = (f 1, f 2, f 3, f 4, f 5 ) and x = (R s, X s, X m, R r1, X r1, R r2, X r2 ) are the equivalent circuit parameters of the double cage model 3.2 Problem Formulation Considering Core Losses It was previously noted that without taking into account the core (and mechanical) losses, the motor full load efficiency η fl cannot be used as an indepent parameter in the problem formulation. This is because efficiency is calculated based on the ratio of output mechanical power to input electrical power. If the heat losses through the core and rotor frictional losses are not taken into account, then the equivalent circuit is not suitable to accurately estimate motor efficiency [7]. It follows that attempting to use the motor full load efficiency in the estimation of the equivalent circuit without a core loss component would cause errors in the parameter estimates (e.g. the stator resistance would be overestimated). When core losses are included in the model, then the motor full load efficiency η fl can also be used as an indepent parameter. The problem formulations are

28 3. Parameter Estimation Problem 19 restated below for the single cage and double cage models with core losses taken into account. Note that in this project, motor models including core losses are always used Single Cage Model (with Core Losses) The non-linear least squares problem for the single cage model with core losses is as follows: f 1 (x) = P m,fl P (s f ) = 0 (3.18) f 2 (x) = sin φ Q(s f ) = 0 (3.19) f 3 (x) = T b T (s max ) = 0 (3.20) f 4 (x) = ηfl η(s f ) = 0 (3.21) where F = (f 1, f 2, f 3, f 4 ) and x = (R s, X s, X m, R r, X r, R c ) are the equivalent circuit parameters of the single cage model (with core losses) Double Cage Model (with Core Losses) The non-linear least squares problem for the double cage model with core losses is as follows:

29 3. Parameter Estimation Problem 20 f 1 (x) = P m,fl P (s f ) = 0 (3.22) f 2 (x) = sin φ Q(s f ) = 0 (3.23) f 3 (x) = T b T (s max ) = 0 (3.24) f 4 (x) = T lr T (s = 1) = 0 (3.25) f 5 (x) = I lr I(s = 1) = 0 (3.26) f 6 (x) = ηfl η(s f ) = 0 (3.27) where F = (f 1, f 2, f 3, f 4, f 5, f 6 ) and x = (R s, X s, X m, R r1, X r1, R r2, X r2, R c ) are the equivalent circuit parameters of the double cage model (with core losses) 3.3 Classes of Parameter Estimation Algorithms The parameter estimation problems formulated in the preceding sections can be solved by a variety of non-linear least squares solver algorithms. As with all nonlinear least squares problems, closed form solutions are generally not available and iterative algorithms are used to converge on a solution by minimising error residuals. Motor parameter estimation algorithms generally fall under two broad classes: 1. Descent Methods: are the class of algorithms based on variations of Newton s method for convergence to a solution, e.g. Newton-Raphson, Levenberg-Marquardt, etc 2. Natural Optimisation Methods: are the class of algorithms based on processes found in nature where successive randomised trials are filtered for

30 3. Parameter Estimation Problem 21 fitness at each iteration, e.g. genetic algorithm, particle swarm optimisation, ant colony optimisation, simulated annealing, etc

31 CHAPTER 4 Descent Algorithms 4.1 Requirement for Linear Restrictions It can be seen from the problem formulations in Chapter 3 that in each case, the number of paramters to be estimated (i.e. unknown variables) exceeds the number of simultaneous equations. In other words, the systems of equations are all underdetermined. Therefore, in order to make the systems exactly determined and solvable with descent algorithms, we must either: 1. Fix two parameters a priori (i.e. parameters are known ) 2. Impose two constraints on the problem formulations, e.g. linear restrictions In this project, the use of linear restrictions was found to be superior to fixed parameters. Therefore, the baseline descent algorithms will include two linear restrictions by default. It was shown in [5] that the stator resistance R s was the least sensitive parameter in the equivalent circuit, i.e. variations in the value of R s had the least significant effect on the resulting torque-speed and current-speed curves. Therefore, R s can be subject to a linear restriction by linking it to the rotor resistance, leading to the first linear restriction: 22

32 4. Descent Algorithms 23 R s = k r R r (for the single cage model) R s = k r R r1 (for the double cage model) Where k r is a constant linear restriction Moreover, it is assumed that the rotor reactance is linearly related to the stator reactance, leading to the second linear restriction. X r = k x X s (for the single cage model) X r2 = k x X s (for the double cage model) Where k x is a constant linear restriction An investigation into the selection of linear restrictions is discussed in Section Newton-Raphson Algorithm Of the class of descent methods used to solve non-linear least squares problems, the Newton-Raphson (NR) algorithm is probably the most straightforward. The NR algorithm is an iterative method where each iteration is calculated as follows: x k+1 = x k h n J 1 F(x k ) (4.1) where x k+1 is the solution at the (k + 1)th iteration, x k is the solution at the kth iteration, h n is the step-size coefficient (more on this later) and J is the Jacobian matrix evaluated with the parameters at the kth iteration, x k. The Jacobian matrix J has the general form: J = f 1 f 1 x 6 x f 6 x 1... f 6 x 6 (4.2)

33 4. Descent Algorithms 24 For systems where it is impractical to compute the exact partial derivatives analytically, a numerical approximation may be used with finite difference equations: f i f i(x + δ j h) f i (x) x j h (4.3) where δ j is vector of zeros with a single non-zero value of 1 at the j-th element and h is a constant with a very small absolute value (e.g ). A modified form of the NR algorithm proposed in [9] for the double cage model is shown in Figure 4.1. This algorithm was selected because of its completeness, numerical accuracy and robustness compared to previously proposed methods (for example, in [1], [2] and [3]). Furthermore, the algorithm can be applied using commonly available manufacturer data, whereas other algorithms require more detailed data that may not be readily available (for example, the full torquespeed curve in [4]). Two other features of the algorithm that aid its robustness are worth highlighting: Parameter Constraints The inequality constraints of the double cage model (X r1 > X r2 and R r2 > R r1 ) can be implicitly included into the formulation by a simple change of variables [5]:

34 4. Descent Algorithms 25 Figure 4.1: Flowchart for conventional NR algorithm

35 4. Descent Algorithms 26 x 1 = R r1 x 2 = R r2 R r1 x 3 = X m x 4 = X s x 5 = X r1 k x X s x 6 = R c Furthermore, only the absolute values of the parameter estimates are used to ensure that no negative parameters are estimated Adaptive Step Size The step size h n in equation 4.1 is a scaling term that determines how far the algorithm should go along the descent direction J 1 F(x k ). Choosing a step size that is too large risks the algorithm not converging. On the other hand, choosing a step size that is too small can cause the algorithm to converge too slowly. An adaptive step size can avoid both these problems by starting with a high step size and only reducing it if the algorithm does not converge (refer to the flowchart in Figure 4.1) Initial Conditions For the base case NR algorithm, the initial parameter estimates are selected as follows [5]:

36 4. Descent Algorithms 27 R r1 = U ns f P m,fl X m = U n Q fl X s = 0.05X m R s = k r R r1 R r2 = 5R r1 X r1 = 1.2X s X r2 = k x X s R c = 10 An investigation into the selection of different initial conditions is discussed later in Section Convergence Criteria The default convergence criteria in this project is a squared error value of The algorithms will stop when the squared error is below this value. 4.3 Levenberg-Marquardt Algorithm The Levenberg-Marquardt (LM) algorithm, sometimes called the damped leastsquares algorithm, is another popular technique for solving least-saures problems [11] [12]. In the LM algorithm, each iteration is calculated as follows: x k+1 = x k [ J T J + λ diag(j T J) ] 1 J T F(x k ) (4.4) where x k+1 is the solution at the (k + 1)th iteration, x k is the solution at

37 4. Descent Algorithms 28 the kth iteration, λ is the damping parameter (more on this later) and J is the Jacobian matrix evaluated with the parameters at the kth iteration, x k (as described previously in Equation 4.2). Parameter constraints as implemented in the Newton-Raphson algorithm are also applied in the LM algorithm (refer to Section 4.2.1). The initial conditions are also selected in the same way as the NR algorithm Choice of Damping Parameter The selection of the damping parameter λ affects both the direction and magnitude of an iteration step. If the damping parameter is large, then the algorithm will move at short steps in the steepest descent direction. This is good when the present iteration is far away from the solution. On the other hand, if the damping parameter is small, then the algorithm approaches a Gauss-Newton type method, which exhibits good convergence in the neighbourhood of the solution. Therefore, the damping parameter should be updated at each iteration deping on whether the algorithm is far or close to the solution. Two methods for adjusting the damping parameter are described below. Gain Ratio Adjustment Marquardt suggested updating the damping parameter based on a gain ratio [12]: ρ = F(x k ) F(x k+1 ) 1 2 xt (λ x J T F(x k )) (4.5) Where x = [ J T J + λ diag(j T J) ] 1 J T F(x k ) is the correction step at iteration k. The damping parameter is adjusted deping on the value of the gain ratio

38 4. Descent Algorithms 29 as follows: λ β, if ρ < ρ 1 λ = λ, if ρ > ρ γ 2 (4.6) Where ρ 1, ρ 2, β and γ are algorithm control parameters. In this project, the algorithm control parameters used were ρ 1 = 0.25, ρ 2 = 0.75, β = 3 and γ = 3. Error Term Adjustment An alternative to using the gain ratio is to adjust the damping parameter based only on the error term (i.e. the numerator of the gain ratio). The damping parameter is therefore updated as follows: λ β, if F(x k ) F(x k+1 ) < 0 λ = λ, if γ F(xk ) F(x k+1 ) > 0 (4.7) above. Where ρ 1, ρ 2, β and γ are the algorithm control parameters as described 4.4 Damped Newton-Raphson Algorithm The damped Newton-Raphson algorithm is a variation of the conventional NR algorithm where a damping factor helps to get around problems with near-singular and/or ill-conditioned Jacobian matrices. In the damped NR algorithm, each iteration is calculated as follows: x k+1 = x k h n (J 1 + λi)f(x k ) (4.8) Where the damping parameter λ is adjusted at each iteration based on the

39 4. Descent Algorithms 30 error term as follows: λ β, if F(x k ) F(x k+1 ) < 0 λ = λ, if γ F(xk ) F(x k+1 ) > 0 (4.9) All other aspects of the damped NR algorithm are the same as per the conventional NR algorithm described in Section 4.2 (e.g. parameter constraints, adaptive step sizes, etc) 4.5 Comparison of Descent Algorithms The descent algorithms were tested on the EURODEEM motor data set (see Appix A) for both the single cage and double cage models (with core losses). In the simulations, convergence is defined as an error rate of < Single Cage Model The results of the simulations on the single cage model are shown in Table 4.1 with fixed linear restrictions (k x = 1 and k r = 0.5). Case IEC Motors NEMA Motors Convergence Convergence Newton-Raphson 3967 (99.1%) 2347 (98.7%) Levenberg-Marquardt 3030 (75.7%) 1783 (75.0%) Damped Newton-Raphson 1387 (34.7%) 424 (17.8%) Table 4.1: Comparison of descent algorithms for the single cage model with fixed restrictions k x = 1 and k r = 0.5 The simulation results show that the conventional Newton-Raphson algorithm has robust convergence. The Levenberg-Marquardt and damped Newton- Raphson algorithms exhibit worse performance than the conventional NR al-

40 4. Descent Algorithms 31 Case IEC Motors NEMA Motors Convergence Average Error 2 Convergence Average Error 2 Newton-Raphson 685 (17.1%) (31.6%) Levenberg-Marquardt (gain ratio) Levenberg-Marquardt (error term) Damped NR (maximum iterations = 30) Damped NR (maximum iterations = 50) 663 (16.6%) (25.0%) (18.5%) (32.4%) (15.7%) (23.9%) (17.2%) (28.2%) Table 4.2: Comparison of descent algorithms for the double cage model with fixed restrictions k x = 1 and k r = 0.5 gorithm, suggesting that the system of equations in the single cage model are well-conditioned and suited for steepest descent type algorithms. Because of the high level of convergence with the Newton-Raphson algorithm, no further investigations were conducted in this project for the single cage model Double Cage Model The results of the simulations on the single cage model are shown in Table 4.2 with fixed linear restrictions (k x = 1 and k r = 0.5) The simulation results show that unlike for the single cage model, the conventional NR algorithm performs performs poorly on the double cage model, both in terms of convergence and average squared error rates. The simulations suggest that the LM algorithm can lead to a higher convergence rate compared to the conventional NR algorithm, but at the cost of a higher average squared error. In terms of convergence, the adjustment of the damping parameter λ using the error term is superior compared to using the gain ratio. Adjusting the damping parameter using the gain ratio leads to worse convergence

41 4. Descent Algorithms 32 and average squared error when compared to the conventional NR algorithm. The LM algorithm works well in the neighbourhood of the solution, but does not perform very well at the early stages, particularly when the initial estimates are far from the solution. The LM algorithm can also produce spectacularly bad results when the Jacobian matrix is ill-conditioned or near-singular. The damped NR algorithm is inted to help address the issue of ill-conditioned and near-singular Jacobian matrices. Adding a damping factor λi to the Jacobian matrix makes it more likely to be invertible. However, the damped NR algorithm takes longer to converge. This is shown in the simulation results by comparing the convergence rate when the maximum number of iterations is increased from 30 (base case) to 50. When the maximum number of iterations is 30, the convergence rate is 15.7% (lower than the conventional NR algorithm). But when it is raised to 50, the convergence rate improves to 17.2% (slightly higher than the conventional NR algorithm). The average squared error of the damped NR algorithm is also significantly lower than the conventional NR and LM algorithms. Therefore, compared to the conventional NR algorithm, the damped NR algorithm can produce results with better convergence and error rates, but at higher computational cost. 4.6 Selection of Linear Restrictions It was previously shown that two linear restrictions are necessary to make the problem formulations exactly determined and thus solvable by steepest descent methods. The selection of the linear restrictions k r and k x is important because by constraining R s and X r2 with linear restrictions, we are also constraining the solution space by two degrees of freedom. Therefore, a solution to a nonconverging problem could potentially be found at different values of k r and k x.

42 4. Descent Algorithms 33 In this section, the effects of linear restriction selection on the convergence and error rates of the conventional NR algorithm are investigated. Three approaches for selecting linear restrictions will firstly be presented, followed by computer simulations on the EURODEEM motor data set Approaches for Selecting Linear Restrictions Fixed Restrictions The simplest approach is to select a set of fixed linear restrictions that are applicable for all motors. The following values of k r and k x are recommed in [9]: k r = 0.5 k x = 1 Interval Search For each motor, the NR algorithm is run multiple times with different values of k r and / or k x selected from a closed interval with discrete steps. The value that leads to algorithm convergence or error minimisation is the selected linear restriction. In this project, the interval ranges and step sizes that are used for k r and k x are as follows : {k r 0.1 k r 1.5} in steps of 0.1 {k x 0.4 k x 1.5} in steps of 0.1

43 4. Descent Algorithms 34 Heuristic for k x Based on a rudimentary cluster analysis of the k x interval search simulation results, a simple heuristic for selecting k x is proposed: IF (I lr < 6 OR cos φ fl < 0.8) AND (T b < 4) THEN select k x = 1 ELSE select k x = 0.5 Likewise, a cluster analysis was performed on the k r interval search simulation results, but this did not lead to any suitable heuristics for selecting k r. This is consistent with the fact that there is no physical relationship between the stator and rotor resistances and thus a forced linear restriction is meaningless Computer Simulation This section investigates the effects of varying the linear restrictions on the convergence and error rates of the Newton-Raphson algorithm. The algorithm was tested on the EURODEEM motor data set (see Appix A) for the double cage model (with core losses). The conventional Newton-Raphson algorithm was tested using the following approaches for selecting linear restrictions: 1. Fixed values k x = 1 and k r = Fixed values k x = 0.5 and k r = 1 3. Fixed values k x = 1 and k r = 1 4. Fixed values k x = 0.5 and k r = 0.5

44 4. Descent Algorithms Interval search on k x, fixed value k r = Interval search on k x, fixed value k r = 1 7. Heuristic k x, fixed value k r = Heuristic k x, fixed value k r = 1 9. Fixed value k x = 1, interval search on k r 10. Fixed value k x = 0.5, interval search on k r 11. Heuristic k x, interval search on k r The results of the simulations are shown in Table 4.3. It can be seen that the choices of k r and k x significantly affect both the rate of convergence and the average squared error. With the IEC motors, the average error is almost 10 times lower when selecting k x with a heuristic and k r with an interval search over using fixed linear restrictions (k x = 1 and k r = 0.5). When fixed linear restrictions are used, the default values of k x = 1 and k r = 0.5 produce the worst results. Convergence is increased by simply adopting different fixed linear restrictions, for example k x = 0.5 and k r = 1 Varying k x yields modest improvements over using a fixed k x. The selection of k x with the simple heuristic proposed in this project leads to marginally worse convergence compared to an interval search on k x, but the average squared error is almost doubled. Notwithstanding, it is still a reasonable improvement over using fixed linear restrictions with little additional cost in terms of computation. While the results of the interval search on k x are superior, the process is considerably more computationally intensive. The results show that varying k r yields much more significant gains than varying k x, in terms of both convergence and average error rates. Even though

45 4. Descent Algorithms 36 Case IEC Motors NEMA Motors Convergence Average Error 2 Convergence Average Error 2 k x = 1, k r = (17.1%) (31.6%) k x = 0.5, k r = (24.3%) (39.3%) k x = 1, k r = (22.3%) (36.4%) k x = 0.5, k r = (19.5%) (33.8%) k x interval search, k r = 0.5 k x interval search, k r = (20.4%) (34.9%) (26.7%) (40.2%) k x heuristic, k r = (19.4%) (33.5%) k x heuristic, k r = (25.0%) (38.6%) k x = 1, k r interval search k x = 0.5, k r interval search k x heuristic, k r interval search 1140 (28.5%) (42.2%) (30.4%) (44.1%) (30.7%) (47.1%) Table 4.3: Conventional Newton-Raphson algorithm results for double cage model using diffrent methods for selecting linear restrictions the convergence rate is still under 50% in both the IEC and NEMA data sets, the errors are relatively low for the majority of motors. 4.7 Selection of Initial Conditions So far, the descent algorithms presented in this chapter have used the baseline initial conditions suggested by Pedra [5] and described in section In this section, alternative approaches for calculating initial estimates are reviewed their performance against the baseline methodology are investigated. A consolidated methodology is then adopted and incorporated into the baseline descent algorithms.

46 4. Descent Algorithms Methods for Calculating Initial Conditions Stator Resistance R s Rogers and Shirmohammadi suggested that the losses in an induction machine can be formulated as follows (ignoring core losses) [2]: P in (1 η) = I 2 s R s + I 2 r R r (4.10) At rated full load, P in = cos φ fl, η = η fl, I s = 1.0 pu and T n = I2 r R r s f. Substituting these quantities into the equation above: cos φ fl (1 η fl ) = R s + s f T n (4.11) Rated torque is also equivalent to T n = η fl cos φ fl n fl. Substituting this in yields: Finally, solving for R s yields: cos φ fl (1 η fl ) = R s + s fη fl cos φ fl n fl (4.12) R s = cos φ fl 1 η fl n fl (4.13) Stator Reactance X s Pedra assumes that the stator reactance is calculated based on the magnetising reactance [5]: X s = 1 20 X m (4.14) In the formulation of Rogers and Shirmohammadi, the total leakage reactance of the motor is first calculated. Neglecting the magnetising branch, we can

47 4. Descent Algorithms 38 approximate the equivalent circuit at locked rotor (s = 1) as per Figure 4.2. Figure 4.2: Simplified motor equivalent circuit at locked rotor Applying Kirchhoff s voltage law yields the following equation: U I lr (R s + R r ) 2 + X 2 L = 0 (4.15) Assuming nominal voltage U = 1.0 pu and solving for X L yields: 1 X L = I 2 lr (R s + R r ) 2 (4.16) Assuming that the leakage reactances are split evenly across the stator and rotor circuits, then: X s = X L 2 (4.17) Rotor Resistances R r, R r1 and R r2 Pedra estimates the rotor resistance by using the following approximation for motor active power [5]: P m,fl U 2 Based on nominal voltage U = 1.0pu and solving for R r : R r (4.18) s f

48 4. Descent Algorithms 39 Figure 4.3: Approximate breakdown of stator current R r s f P m,fl (4.19) For the double cage model, Pedra assumes that R r1 = R r and R r2 = 5R r1. Rogers and Shirmohammadi estimate the rotor resistance by first assuming that the stator current can approximately be broken down into a real rotor current component and a reactive magnetising current component (see Figure 4.3), i.e. I r = I s cos φ (4.20) I m = I s sin φ (4.21) Knowing that: P m,fl = η fl cos φ fl = T n n fl = I 2 r We substitute I r = I s cos φ to yield: R r s f n fl (4.22) η fl cos φ fl = (I s cos φ) 2 R r s f n fl (4.23) At full load, I s = 1.0 pu. Therefore, solving for R r :

49 4. Descent Algorithms 40 R r = η fls f cos φn fl (4.24) For the double cage model, the locked rotor characteristics of the motor are used. At locked rotor s = 1, the torque developed by the machine can be given by T lr = I 2 lr R lr, where R lr is the equivalent rotor resistance at locked rotor. Therefore: R lr = T lr I 2 lr (4.25) Using a machine design factor m such that it is defined as follows: m = R r1 + R r2 X r (4.26) The rotor resistances can then be calculated as follows: R r1 = R lr (1 + m 2 ) R r m 2 (4.27) R r2 = R r1r r R r1 R r (4.28) A typical value of m = is suggested by Rogers and Shirmohammadi [2]. Rotor Reactances X r1 and X r2 Pedra assumes that the inner cage rotor reactance is related to the stator reactance as follows [5]: X r1 = 1.2X s (4.29)

50 4. Descent Algorithms 41 The outer cage reactance is calculated by the linear restriction X r2 = k x X s. Similarly, Rogers and Shirmohammadi estimate the total rotor reactance X r based on the leakage reactance being evenly split between stator and rotor, i.e. [2] X r = I 2 lr (R s + R lr ) 2 (4.30) Magnetising Reactance X m Pedra estimates the magnetising reactance by using the following approximation for motor reactive power [5]: Q fl U 2 X m (4.31) Based on nominal voltage U = 1.0pu and solving for X m : X m 1 Q fl (4.32) Using the same assumption shown in Figure 4.3, we can see that: tan φ = I m I r = R r s f X m (4.33) Substituting I r = I s cos φ and I m = I s sin φ, we get: I s sin φ I s cos φ = R r s f X m (4.34) Solving for X m :