Optimal State Reference Computation with

1

Optimal State Reference Computation with Constrained MTPA Criterion for PM Motor Drives Matthias Preindl, IEEE Student Member, Silverio Bolognani, IEEE Member

Abstract—This research proposes a procedure that maps a PMSM torque request onto optimal state (current) references. Combining the procedure with a dynamic (current) controller yields a torque controller. The maximum torque per ampere (MTPA) criterion is used to minimize conduction and switching losses. This research extends the concept to field weakening (FW) operation to obtain high efficiency at any machine speed. The resulting constrained MTPA criterion is formalized as optimization problem. Since it is difficult to solve directly, the maximum and intersection torque subproblems are identified. An algorithm is obtained that maps a torque onto an optimal state reference and it is sufficiently efficient for real-time implementation. This method is compatible with a variety of state (current) controllers with/without PWM; SPM and IPM machines with saliency and reverse saliency; and a variable DC-link voltage. The proposed procedure relies on a sufficiently accurate torque model that may not be provided using rated machine parameters. Thus, an approach to compute locally optimized machine parameters is proposed that takes magnetic saturation into account. The concept is developed on a software-in-the-loop platform and evaluated on an experimental test bench. Index Terms—Drive System, Internal Permanent Magnet Synchronous Motor, Optimal Control, Field Weakening, Maximum Torque per Ampere

I. I NTRODUCTION PM synchronous motor drives consist of a PM synchronous motor, inverter, and control system. They are designed to apply a reference torque to a drive shaft, that can be generated locally, e.g. by a speed controller, or it can be communicated via field bus to control a larger system, e.g. a wind turbine or a traction system. A PMSM produces torque driving a current (control state) in the stator windings that is controlled by the terminal voltage (control input). The current vector that generates a given torque is not unique and a control system can achieve secondary control goals besides actuating a torque. A typical setup uses the maximum torque per ampere (MTPA) tracking below rated speed [1]–[3]. This minimizes conduction and switching losses [4], [5], i.e. yields operation at high efficiency, and is necessary to achieve the rated torque. To achieve high speed operation, field weakening (FW) is used to limit the terminal voltage such that it does not exceed its rated value [1], [6], [7]. FW concepts can be classified into feedback and feedforward methods. Feedback methods use a feedback of the Manuscript received March 3, 2014; accepted August 19, 2014. M. Preindl is with Department of Industrial Engineering, University of Padova, via Gradenigo 6/A, 35131 Padova, Italy, e-mail: [email protected] S. Bolognani is with Department of Industrial Engineering, University of Padova, via Gradenigo 6/A, 35131 Padova, Italy

terminal voltage [8], [9]. They are widely used but have also potential issues that need to be addressed during design and implementation: (i) the transition between MTPA and FW operation needs to be smooth; (ii) voltage feedback ensures feasibility at high speed; this does not necessarily imply operation with low current magnitude (low losses) when the reference torque is small; (iii) a voltage loop is (typically) an outer loop that needs to be decoupled in dynamics, i.e. have a slower transient response, than the inner loop due to the cascade structure; (iv) a voltage loop can yield an outer (e.g. speed) controller unstable if it drives the current vector beyond the MTPV trajectory; there an increase of the current magnitude (along the isoflux trajectories) corresponds to a decrease of torque; (v) voltage control tuning is demanding due to the nonlinear equations involved. These points are inherently avoided using feedforward methods [10], [11] that act instantaneously (no dynamics involved) on the expense of relying on the machine model. In fact, this latter approach is interesting for applications where the machine model is reasonably well known (or at least some operation points as shown later). This is the case for applications where it is worth investigating the real machine behavior to produce good models, e.g. in drive systems that are produced in large series (electric vehicles) or high power applications (railway traction or wind power). Also, good models are typically already available if model predictive current control [5], [12] is used. Present feedforward methods are written for SPM [13] or IPM with Ld ≤ Lq ; use approximations to simplify the (nonlinear) equation set [13]–[15]; and/or use iterative methods [11], [13], [15], [16]. Moreover, no attempt has been made to "invert" the PMSM torque equation online for torque control; existing method compute a reference vector from a q-current [17] or current magnitude [10], [11]. This research proposes an unified way to compute optimal state references from a torque reference at all machine speeds based on the feedforward idea. The MTPA concept is extended to speeds where field weakening is required introducing the constrained MTPA criterion. The problem is written in form of a constrained optimization problem, that minimizes the current magnitude taking the system constraints into account. The constrained MTPA criterion defines the optimal state vector that generates a desired, i.e. reference, torque. It is a static function that depends on a single operation parameter named normalized speed, that is the machine speed divided by the rated terminal voltage. It results in a parametrized static function, that maps torque onto optimal states and can be interpreted as a variable (nonlinear) gain that acts between the

2

+

DC

vc

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Solid-state Inverter

Electrical Machine

AC

gd T(ref)

Torque Control Reference Torque

Fig. 1.

Optimal State Reference Computation Current

Dynamic (Current) Control

Measurement

ω/v- r id(ref) gq

iq(ref)

Fig. 2. Interpretation: optimal state (current) references [id , iq ]′ are computed from the reference torque T based on the machine model and the normalized speed ω/¯ vr parameter; this behaves like a variable (nonlinear) gain

Block diagram

torque and state references as shown in Fig. 2. Combining the (static) state reference generation procedure with a (dynamic) current controller yields a (dynamic) torque controller as it is shown in Fig. 1. The transient behavior (the trajectory of the state to its reference) and transient constraint satisfaction of the resulting controller depends only on the current controller an is not analyzed in this text. Since torque feedback is not practicable in most applications due to cost and reliability issues, the resulting torque controller relies on a (optimized) machine model. The constrained MTPA optimization problem is a nonconvex quadratically constrained quadratic program that is difficult to solve in general. Thus, it is divided into subproblems. The maximum torque problem defines whether a reference torque is feasible or not. The intersection torque problem yields information if the optimal operation happens on the MTPA or the isoflux trajectory. Each of these subproblems can be solved using the low dimensionality of the problem and system properties. The solution is obtained analytically intersecting the relevant trajectories [11], [18]. Relying on the machine model is a potential issue [19]– [21] and needs to be addressed when designing a feedforward method. In particular, saturation and cross-saturation can be addressed using lookup tables (LUT) [11]. However, flux or inductance planes may not be available for a machine and do prevent the (nonlinear) equation set from having a closed solution. Thus, a procedure is proposed to optimize the torque model locally with respect to important operation points, i.e. the rated operation point and the demagnetization or maximum speed operation point. A variable DC-link voltage is interesting for many applications to minimize switching losses, i.e. increase efficiency, [22], [23]. Thus, the method is written to be compatible with a (reasonably slow) varying DC link that is e.g. obtained when the DC link is connected directly to a battery pack. The reference generation procedure is evaluated on a Softwarein-the-Loop (SiL) platform, where the electical machine is simulated by Matlab/Simulink, and an experimental test bench. The algorithm is shown to be sufficiently efficient for online implementation and key aspects of the algorithm are evaluated on these platforms. II. E LECTROMAGNETIC M ODEL In this section, a model is introduced to formally describe the electrical machine. PMSM are typically described in the dq reference frame to simplify the treatment [25]. There,

the dynamic of a three-phase neutral-point isolated armature winding is given by the state-space system [18] λ˙ dq (t) = −ω(t)Jλdq (t) + vdq (t) − Rs idq (t),

(1)

where λdq (t) ∈ ℝ2 is the stator flux, vdq (t) ∈ ℝ2 is the terminal voltage, and Rs idq (t) is the resistive voltage drop. The parameter Rs ∈ ℝ+ is the resistance per phase and idq (t) ∈ ℝ2 is the phase current. The angular velocity ω(t) ∈ ℝ is the rotational velocity of the dq reference frame with respect to the stationary αβ reference frame with respect to the transformation angle ϵ(t) ˙ = ω(t) and J is the rotation ′ matrix J = [[0, −1]′ , [1, 0]′ ] . This system can be simplified introducing the compensated voltage v¯dq (t) = vdq (t) − Rs idq (t),

(2)

that yields the state-space system λ˙ dq (t) = −ω(t)Jλdq (t) + v¯dq (t).

(3)

Changing the system input from vdq (t) to v¯dq (t) means compensating the voltage drop Rs idq (t) at each time instant with the terminal voltage vdq (t). The flux-current dependence of a PMSM is a static nonlinear map, where the nonlinearity is introduced by saturation and cross-saturation [26]–[28]. A nonlinear function l : ℝ2 → ℝ2 is necessary to describe this relation in general but it yields a model, which is difficult to analyze. In most cases, the complexity is not justified since the local system behavior can Table I N OMECLATURE Symbol

Description

1 .† ∥.∥ ◦ ∂ ⊕ and ⊖ ∩ and ∪ ℝ and ℝ+ ∅

Vector of ones Moore-Penrose pseudoinverse Euclidean norm Convolution Boundary of a set Minkowski sum and Pontryagin difference [24] Intersection and union of sets Real and nonnegative real numbers Empty set

idq and λdq vdq L and ψdq I and Ir Λ and v¯r T , Tm , Ti χr and χp χi and χm

Stator current and flux Terminal voltage Inductance matrix and PM flux Set of feasible currents and rated current Set of feasible fluxes and rated voltage Actual, maximum, and intersection torque Normalized rated and rated power speed param. Normalized intersection and max. speed param.

2

2

1

1

iq=−1.2 i =−.6

q

0

λ [pu]

d

λ [pu]

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id=−1.2 i =−.6

q

iq=0

−1

i =.6

d

id=.0

−1

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−2

i =1.2 q

−2

0 i [pu]

2

where Ir ∈ ℝ+ is the rated current. The set I can be projected def onto the flux space with I¯ = L ◦ I + ψdq . A voltage source inverter can provide a terminal voltage with a finite maximum magnitude. A inverter can apply a voltage that lies within a set with hexagonal shape. However, this set is backwards rotating in the dq system. Thus, the largest time-invariant approximation of the input set is used  def  vdq ∈ V = vdq ∈ ℝ2 | ∥vdq ∥ ≤ vr , (8)

d

−2

i =1.2 d

−2

d

0 i [pu]

2

q

(a) d-axis

(b) q-axis

Fig. 3. Example of a measured nonlinear flux-current characteristic and the linear approximation using rated parameters

be captured with good approximation (see Section VIII) by the linear function [29]–[31] λdq (t) = l(idq (t)) ≈ Lidq (t) + ψdq , that is characterized by the parameters     L 0 ψ L= d ; ψdq = ; 0 Lq 0

(4)

(5)

where Ld ∈ ℝ+ and Lq ∈ ℝ+ are the auto-inductance of the d and q axis, respectively and ψdq , i.e. ψ ∈ ℝ+ , is the flux generated by the PM. An example of the approximation is shown in Fig. 3. Electric machines produce torque when currents are driven in armature winding (stator of a PMSM). There, the current interacts with the magnetic field, i.e. flux, and produces the electromagnetic torque T ∈ ℝ (with exception when the stator current and flux vector are parallel). The torque equation is obtained as power balance of the electromechanical energy conversion neglecting the resistive voltage drop and the angle dependent variation of the magnetic coenergy [31], [32] 3 ′ p i Jλdq , (6) 2 dq where the parameter p ∈ ℕ+ is the number of pole pairs of the electrical machine. T =

III. C ONSTRAINTS An electric machine cannot be operated with arbitrarily large voltages and currents due to the finite terminal voltage that can be applied to the machine terminals (and supplied by an inverter) and thermal constraints. In this section, operation constraints are highlighted that have to be satisfied (at least) in steady-state conditions. Time dependencies (t) are omitted for compactness. The magnitude of the stator current needs to be limited since a winding is able to transmit only a finite amount of heat, i.e. power losses, to the ambient without exceeding its maximum operating temperature. Similarly, a high current magnitude leads to excessive thermal cycling of the solid-state valves of the inverter and their early failure. Thus, we obtain the current constraint  def  idq ∈ I = idq ∈ ℝ2 | ∥idq ∥ ≤ Ir , (7)

where vr ∈ ℝ+ is the rated voltage. This set depends √ on the def DC link voltage vc ∈ ℝ+ and is defined by vr = vc / 3 for a two-level inverter. Throughout this research, it is assumed that the DC-link, i.e. vr , is not necessarily constant and may vary within certain bounds, e.g. due to voltage variations in the power grid or in a distributed DC-link system. Using the compensated terminal voltage (2) as input, it must be ensured that v¯dq can be applied to the system. In other words, the existence of a terminal voltage vdq ∈ V for all idq ∈ I is required. Thus, the compensated terminal voltage has to satisfy the voltage constraint   v¯dq ∈ V ⊖ Rs I = v¯dq ∈ ℝ2 | ∥¯ vdq ∥ ≤ vr − Rs Ir . (9) Clearly, some of the available terminal voltage is lost above all if ∥idq ∥ ≪ Ir . However, this approach is justified for drive systems where the resistive voltage drop is small compared to the rated voltage, i.e. Rs Ir ≪ vr . The compensated voltage approach can be generalized to take model uncertainties and nonlinear inverter behavior into account (dead-times and forward voltage drop tend to decrease the maximum available voltage). A safety factor ρv ∈ (0, 1) is introduced to define rated compensated terminal voltage set   def def v¯dq ∈ V¯ = v¯dq ∈ ℝ2 | ∥¯ vdq ∥ ≤ v¯r = ρv vr , (10) The factor ρv can be chosen heuristically such that v¯r ≤ vr − Rs Ir ; typical values are ρv = (0.8, 1). For applications, where generator operation is significant, it is possible to choose different safety factors (ρv+ and ρv− ) dependent whether the mechanical power of the PMSM is positive or negative. Since ρv− (generator operation) can be selected larger (ρv− > ρv+ ), both the base speed and the maximum torque above base speed can be increased in generator operation. Throughout this paper, a single safety factor is used for simplicity. In steady-state conditions, i.e. λ˙ dq = −ωJλdq +¯ vdq = 0, the input constraint maps onto a state constraint. In other words, a state can be maintained by the system if and only if ωJλdq ∈ ¯ Since J−1 ◦ V¯ = V, ¯ the steady-state constraint becomes V. ωλdq ∈ V¯ or  def  λdq ∈ Λ = λdq ∈ ℝ2 | |ω|∥λdq ∥ ≤ v¯r . (11) def ¯ = The set Λ can be projected onto the current space with Λ −1 L ◦ (Λ − ψdq ). The constraints (7) and (11) define also the high speed behavior of a PMSM. The location of the current and voltage constraint with respect to each other defines whether a machine has an electrically limited maximum speed or not [1]. The def ¯ is the so called demagnetization current ic = center of Λ ′ [−ψ/Ld, 0] . If ic ∈ / I, the PMSM has a limited maximum

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Fig. 5.

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idq ,λdq

subject to ∥idq ∥ ≤ Ir ; |ω|∥λdq ∥ ≤ v¯r ; λdq = Lidq + ψdq ; 3/2p i′dq Jλdq = T ;

0.4

infeasible operation points

||i*dq||

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(b) Current magnitude vs. torque

Optimal field weakening operation (constant power mode)

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Optimal field weakening operation (reduced power mode)

¯ = ∅ for |ω| → ∞. Contrarily, a PMSM speed ωm since I ∩ Λ with ic ∈ I has the maximum speed ωm = ∞ since there exists a current vector (ic ) that satisfies the system constraints for all ω ∈ ℝ. Thus, a PMSM is subject to the speed constraint |ω| def ≤ χm = v¯r



1 |ψ−Ld Ir |

∞

An electrical machine is designed to apply an electromagnetic torque to the drive shaft. However, the current, i.e. flux, which produces a desired torque T , is not unique in general and provides an additional degree of freedom. The available degree of freedom can be exploited to increase the drive system efficiency. The drive system efficiency is typically maximized by minimizing the resistive losses, i.e. the current magnitude [1], [3], [7], [33]. Thus, the operation is said to be optimal, if a torque T is generated by the current i⋆dq and flux λ⋆dq that solve minimize ∥idq ∥

0 0

1

(a) Current space

Fig. 6.

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Tm=Ti =Tr

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dq

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id [pu]

ii

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Fig. 4.


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IV. O PTIMAL O PERATION

1

1.2

if Lψd > Ir , otherwise.

(12)

where χm is the maximum speed normalized by the voltage v¯r . This approach is convenient since the maximum speed ωm = v¯r /|ψ − Ld Ir | of a machine varies with v¯r . On the other hand, whether a machine has a maximum speed or not is independent of v¯r . Machines with χm = ∞ and χm < ∞ are depicted in Fig. 7 and Fig. 8, respectively. It can be shown that references that satisfy these constraints can be reached by a controller [18]. If a specific controller does converge to a reference is a current control property that needs to be verified when designing and implementing the controller.

|T | ≤ Tm (ω, v¯r )

(13a) (13b) (13c) (13d) (13e) (13f)

This constrained optimization problem uses the current magnitude (13a) as objective function that should be minimized. The current idq ∈ I (13b) and flux λdq ∈ Λ (13c) constraint needs to be considered when solving for the optimal states. The relation between currents and fluxes is defined by (13d). The constraint (13e) defines the desired torque T , which is a parameter that has to satisfy the (speed-dependent) maximum torque Tm (ω, v¯r ) constraint (13f). The problem (13) is nonconvex due to the nonlinear equality constraint (13e). Thus, it is NP hard to solve in general [34] and can be infeasible dependent on the parameters. Thus, it is advantageous to break (13) down into subproblems that can be solved exploiting the low dimensionality and system properties. In the next sections, an approach is proposed that computes a solution using the characteristic trajectories and loci of PMSM, in particular: the Maximum Torque per Ampere (MTPA) trajectory [3]; the Maximum Torque per Volt (MTPV) trajectory [7]; the isoflux locus ∂Λ (boundary of Λ); and the isocurrent locus ∂I (boundary of I). Examples of optimal operation are shown in Fig. 4, Fig. 5, and Fig. 6 in base mode (constant torque mode), field weakening constant and reduced (apparent) power mode, respectively.

V. M AXIMUM T ORQUE Prior to solve for optimal states given a desired torque, it must be ensured that (13) has a solution. Since state and input constraints apply to the PMSM, the maximum (and minimum) available torque is limited as well. In this section, an approach to find the maximum (steady-state) torque Tm ∈ ℝ+ , which can be provided by the PMSM, is shown. For compactness the dependencies (ω, v¯r ) are omitted. Due to the symmetries of the torque equation and constraints, finding Tm is equivalent to finding the minimum torque, which is the additive inverse

5

of Tm , i.e. −Tm . Thus, the maximum torque is defined to be

(14b) (14c) (14d)

The problem uses the torque equation (14a) as objective function to maximize. The steady-state current (14b) and voltage (14c) limit define the set where the current, i.e. fluxes, can lie. Due to the |ω| dependence of the voltage limit (14c), the maximum torque Tm depends on the machine speed and the rated voltage. The linear equality constraint (14d) defines the relation of currents and fluxes. The problem (14) can be written as quadratic constrained quadratic program (qcqp) in standard form. However, the resulting matrices are indefinite and the problem is NP hard to solve in general [34]. Thus, rather than solving the problem directly, a solution is proposed using the low dimensionality and system properties. Examples of the maximum torque are shown in Fig. 7 and Fig. 8 for a machine with infinite χm = ∞ and finite χm < ∞ maximum speed, respectively. At low velocities (ω ≈ 0), the machine is said to work in base mode or constant torque mode [1], [3], [33]. There, the voltage constraint (14c) is always true. The torque Tm is produced by the current im ∈ ℝ2 , which lies on the MTPA trajectory and is denoted as im ∈ MTPA. Moreover, im lies on the largest admissible isocurrent locus, i.e. the border of I, which is denoted as im ∈ ∂ I. Thus, im ∈ MTPA ∩ ∂ I. This intersection defines a set containing two current vectors, which produces the maximum and minimum torque. The solution iq > 0 is named rated operation point , where the machine def uses the rated current ir ∈ ℝ2 , i.e. rated flux λr = Lir + ψdq , def ′ to produce the rated torque Tr = 3/2p ir Jλr . The maximum torque Tm = Tr can be achieved iff λr ∈ Λ. Since Λ shrinks increasing |ω|, the machine works in constant torque mode if 1 |ω| def ≤ χr = , v¯r ∥λr ∥

(15)

where χr is the rated speed ωr = v¯r /∥λr ∥ normalized by the voltage v¯r . This approach is convenient since ωr and the rated power Pr = Tr ωr of a drive system vary with v¯r . The PMSM is said to work in field weakening, if the speed is higher than the rated one (|ω|/¯ vr > χr ) [1], [7], [35], [36]. In this condition, the rated flux λr ∈ / Λ cannot be achieved and the updated maximum torque Tm < Tr must be found. Considering the positive halfplane (iq ≥ 0, λq ≥ 0), the torque increases along the largest isoflux locus ∂ Λ starting at the d axis until its peak is achieved when intersecting the MTPV trajectory. Consequently, Tm is obtained at the intersection MTPV ∩ ∂ Λ if this point satisfies idq ∈ I. Otherwise, Tm is obtained at the intersection idq ∈ ∂ I with λdq ∈ ∂ Λ. To simplify the treatment, the rated power operation point def is introduced, which defines the current ip and the flux λp = Lip + ψdq . It is defined to be the intersection of the MTPV trajectory with the isocurrent locus ∂ I, if it exists. It does exist for machines with χm = ∞, which is shown in Fig. 7.

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Optimal operation on MTPA trajectory

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|T| [pu]

subject to ∥idq ∥ ≤ Ir ; |ω|∥λdq ∥ ≤ v¯r ; λdq = Lidq + ψdq

(14a)

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Fig. 7. Maximum and intersection torque characteristic of a machine with infinite maximum speed χm = ∞ 1

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Fig. 8. Maximum and intersection torque characteristic of a machine with finite maximum speed χm < ∞

Otherwise MTPV ∩ ∂ I = ∅ and the rated power operation point is defined to be ip = [−Ir , 0]′ , which is shown in Fig. 8. If λp ∈ Λ (and λr ∈ / Λ), the machine is said to work in constant (apparent) power mode [1], [7]. In this mode, the maximum torque Tm is obtained at the intersection of idq ∈ ∂ I with λdq ∈ ∂ Λ. Similar to the constant torque mode, the machine works in constant power mode if χr
χp , which is named reduced (apparent) power mode [1], [3], [7], [33]. In reduced power mode, the intersection MTPV ∩ ∂ Λ satisfies idq ∈ I and defines therefore the current, i.e. flux, which provides Tm . The maximum torque Tm provided by a machine with χm = ∞ and χm < ∞ are shown in Fig. 7 and Fig. 8, respectively. VI. I NTERSECTION T ORQUE After ensuring that a desired torque value T is feasible |T | ≤ Tm , the optimal operation points can be computed. The optimal operation point ∥i⋆dq ∥, i.e. ∥λ⋆dq ∥, are computed minimizing the current magnitude. This approach leads to two cases: operation on the MTPA trajectory or operation on ∂ Λ locus. To distinguish the two cases the intersection torque

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Inputs: T, vr, ω

Ti ∈ ℝ+ is introduced (17a)

subject to ∥idq ∥ ≤ Ir ; |ω|∥λdq ∥ ≤ v¯r ; λdq = Lidq + ψdq ; idq ∈ MTPA

(17b) (17c) (17d) (17e)

The problem (17) corresponds to problem (14) adding the constraint (17e) that requires the state to lie on the MTPA trajectory. Therefore (17) is more restrictive than (14) such that Ti ≤ Tm . Moreover, the problem (17) has a solution if and only if ¯ ̸= ∅ that is achieved when 0 ∈ Λ ¯ and yields the MTPA ∩ ∂ Λ condition |ω| def 1 ≤ χi = . (18) v¯r ψ Otherwise, no MTPA operation point is available and the intersection torque is defined to be Ti = 0. Examples of the intersection torque are shown in Fig. 7 and Fig. 8 for a machine with infinite χm = ∞ and finite χm < ∞ maximum speed, respectively. In base mode, the maximum torque Tr is achieved on the MTPA trajectory and corresponds therefore with the intersection torque Ti = Tm . Thus, the PMSM is operated exclusively on the MTPA trajectory if |ω|/¯ vr ≤ χr . This behavior is expected since per definition, the MTPA states are the ones with minimum current magnitude and can produce any torque |T | ≤ Tm in base mode. This operation is shown in Fig. 4. In field weakening (|ω|/¯ vr > χr ), the maximum torque Tm is produced by states off the MTPA trajectory and as a consequence Ti < Tm . In this case, the intersection torque defines if a desired torque can be produced by a state that lies on the MTPA trajectory. If T ≤ Ti , then this is possible, i.e. ¯ This operation is shown i⋆dq ∈ MTPA; otherwise i⋆dq ∈ ∂ Λ. ¯ does not contain in Fig. 5. At high speeds (|ω|/¯ vr > χi ), Λ states of the MTPA trajectory and MTPA ∩ ∂ Λ = ∅. Thus ¯ locus, i.e. i⋆ ∈ ∂ Λ ¯ operation happens exclusively on the Λ dq and is achieved setting Ti = 0. This operation is shown in Fig. 6. VII. R EFERENCE G ENERATION P ROCEDURE In this section, a state reference generation procedure is outlined based on the concepts introduced in the former sections. This procedure defines an efficient approach to compute the optimal current reference vector i⋆dq from a desired torque reference value T and the parameters v¯r and ω. The obtained vector i⋆dq can then be used as reference for a dynamic controller. The reference generation is carried out in four steps and is described by the block diagram in Fig. 9. The first step is to check the measurements to ensure that there exists a suitable reference i⋆dq . The DC link voltage is required to be positive ¯ exists. Moreover, v¯r ≥ 0. Otherwise, Λ = ∅ and no i⋆dq ∈ Λ |ω|/¯ vr is required to not exceed the maximum normalized velocity χm of the machine. This check is only necessary

check

STOP there exists no sutitable i*dq

Fail

Pass |ω|/vr ≤χr Find Tm

i′dq Jλdq

max

idq ,λdq

No

Yes: base mode MTPA ∩ ∂I defns im=ir; Tm=Tr

Find Ti

3 p 2

* Find idq

def

Ti =

|ω|/vr ≤χp

∂I ∩ ∂Λ defns im; Tm Yes

* idq ∈MTPA (Ti=Tm)

|T| Tm . In this case, it is assumed that the drive system should apply the largest available torque value ±Tm . The second step is identifying the operation mode (constant torque mode, constant power mode, reduced power mode) and computing the maximum torque Tm . The rated operation point λr (and ir , Tr ) and rated power operation point λp are parameters of the PMSM and can be computed offline. If |ω|/¯ vr ≤ χr the drive system works in base mode (constant torque mode) and the maximum torque is Tm = Tr and is generated by im = ir . Otherwise, the drive system works in field weakening and the maximum torque depends on ω and v¯r . If χr < |ω|/¯ vr ≤ χp , the drive system works constant power mode and the maximum torque Tm is obtained at ¯ ∩ ∂ I (with iq > 0). If |ω|/¯ im ∈ ∂ Λ vr > χp , the drive system works in field weakening (reduced power mode) and ¯ the maximum torque Tm is obtained at im ∈ MTPV ∩ ∂ Λ (with iq > 0). The third step is identifying whether optimal operation is ¯ achieved on the MTPA trajectory or largest isoflux locus ∂ Λ by computing the intersection torque. If |T | < Ti , the drive system is operated with i⋆dq ∈ MTPA and if Ti ≤ |T | < Tm , ¯ Otherwise, the drive system is operated with i⋆dq ∈ ∂ Λ. the torque reference |T | ≥ Tm corresponds or exceeds the maximum torque and the drive system applies the maximum

7

q

B

* [pu] i dq

C 0 i

d

−0.5

A C

i

q

0.5 0

A B

−0.5

B

C

−1 −0.5

0 T [pu]

0.5

(a) Ld < Lq ; χm = ∞

1

−0.5

0

5 0

−0.5

0 T [pu]

0.5

MTPA

MTPV

0

0

1

(b) Ld = Lq ; χm = ∞

Fig. 10. Optimal current i⋆dq as function of T ∈ [−Tm , Tm ] at different speeds ω

0

−0.5 −0.5 −0.5

−1 −1

0.5

0.5

−0.5

0.

0.5

0.5

0

MTPV 5

0.5

0

−0.5

MTPA

−0.5

−0 .5

0 i [pu]

0.5

−1 1

−1

−0.5

q

0

0

0

−0.5

−1

−1

−1

0

−0.5

−0.5

5

1 1

0.5

−1

−1 −1

0.5

0

d

1

1 0.5

0.5

A,B,C

i

1

id [pu]

i 0.5 * [pu] i dq

1

1 A

id [pu]

1

−0 .

−1− 1

5

0 i [pu]

0.5

1

q

(a) Rated Parameters

(b) Optimized Parameters

Fig. 11. Comparison of the modeled (solid) and measured (dashed) PMSM characteristics 1

VIII. L OCAL O PTIMIZATION OF THE T ORQUE M ODEL The presented procedure relies on a sufficiently accurate PMSM torque model and its main characteristics (isotorque locus, MTPA and MTPV trajectory). However, using the rated parameters1 of an electrical machines can provide mediocre results, i.e. a rough approximation of the real behavior. An example is shown in Fig. 11(a), where the modeled characteristics are compared to the measured ones. In this section, an approach is proposed to improve the model consistency based on known operation points. The model does not need to describe the machine globally but has to capture the local machine behavior (the area on the left hand side of the MTPA trajectory and on the right hand side of the ∂I locus). Thus, the parameters can be optimized to increase the model correspondence in this region. The parameters can be settled with respect to the rated operation point with the rated current ir = [idr , iqr ]′ , the rated flux λr = [λdr , λqr ]′ producing the torque Tr using the equations • the torque equation 3/2(ψ + (Ld − Lq )idr )iqr − Tr /p; 2 2 • the MTPA trajectory ψid + (Ld − Lq )(idr − iqr ) = 0; • the rated current-flux relation Lir + ψdq = λr . The high speed behavior of a PMSM depends whether a machine has infinite (χm = ∞) or finite (χm < ∞) maximum speed and so does the parameter assessment procedure for the torque plane on the left hand side of the MTPA trajectory. For machines with χm = ∞, the location of the MTPV 1 The rated L and L parameter defines the slope of the d and q axis q d flux in the origin, respectively, where the current on the other axis is set to zero. The rated ψ defines the offset of the d axis flux in the origin. These rated parameters provide the linear approximations shown in Fig. 3.

0.8 T [pu]

torque. It is observed that the maximum and intersection torque can be computed independent from each other. Therefore, step two and three can be easily implemented using parallel computation if the control hardware supports it. In the fourth step, the optimal current i⋆dq ∈ MTPA, ¯ which produce the reference torque T are i.e. i⋆dq ∈ ∂ Λ, computed. The optimal states can be computed either directly (solving a quartic equation) or using numerical methods. The equations are given in the appendix of [18]. The resulting i⋆dq are shown in Fig. 10, where the trajectory A refers to base mode (constant torque mode) and the trajectories B and C refer to operation in field weakening.

0.6 0.4

Tm Ti

0.2 0 0

2

4 6 |ω|/v [pu]

8

10

r

Fig. 12. Torque characteristics Ti and Tm : model with safety factor (solid) and measure (dashed)

trajectory needs to be settled. This can be achieved using the demagnetization current ic = [idc , 0]′ that yields • the short-circuit current-flux relation Lic + ψdq = 0. For machines with χm < ∞, the high speed behavior can be settled using the rated power operation point ip = [idp , 0]′ and λp = [λdp , 0]′ where the machine achieves its maximum speed that yields • the rated power current-flux relation Lip + ψdq = λp . Combining the equations, the following overdetermined equation system is achieved for a PMSM with χm = ∞ 3    3  − 32 idr iqr −Tr  0 2 iqr 2 idr iqr ψ 2 2 2 2  id  0   − i i − i i qr qr dr dr    Ld     1     idr    Lq  = λdr  ,  λqr   iqr 1/p 0 1 idc (19) where the last row needs to be replaced for a PMSM with χm < ∞. The system can be written compactly as Mρ = K with the least squares solution ρ = M† K, that tends to give the best results when the equations are normalized (pu). An example of an optimized model computed with this approach is shown in Fig. 11(b). The presented approach aims to improve the model locally where the machine is operated. It is capable of delivering models that are locally exact to measurement precision. The result is a typically good approximation based on (4) but may not characterize well PMSM with arbitrarily large saturation and cross-saturation. The motor parameters can also vary with temperature (typically the PM flux). If the variation is significant, a low and high temperature optimized model can be produced such that the parameters can be computed via

1

0.5

0.5

T [pu]

1

0 30

10 20 Time [sample]

1

0.5 0 0

10 20 Time [sample]

0 0

30

1 0.5

T [pu]

T [pu]

1 0.5

30

1

0 10 20 Time [sample]

30

0 1 [pu]

0

dq

0

10 20 Time [sample]

30

10 20 Time [sample]

30

10 20 Time [sample]

30

0

i

[pu]

10 20 Time [sample]

(b) NLC experimentation

0

10 20 Time [sample]

30

1

dph [pu]

1 0.5 0 0

−1 0

dph [pu]

−1 0

0.5

10 20 Time [sample]

0 0

30

(c) CCS-MPC simulation

(d) CCS-MPC experimentation

1

1

0.5

0.5

T [pu]

T [pu]

30

0.5

(a) NLC simulation

0

0 1

10 20 Time [sample]

0

30

1 [pu]

0

i

dq

0

10 20 Time [sample]

30

10 20 Time [sample]

30

10 20 Time [sample]

30

0

i

[pu]

10 20 Time [sample]

0 −1 0

30

dph [pu]

1

dq

30

dph [pu]

−1 0

i

2 A later optimization of the c-code has reduced the execution times by approximately 50% on the SiL platform but was not evaluated on the experimental test bench.

10 20 Time [sample]

i

dq

0

1

10 20 Time [sample]

1

0.5 0 0

−1 0

30

sph [pu]

−1 0

sph [pu]

IX. E VALUATION The reference generation procedure is evaluated on a Software-in-the-Loop (SiL) platform, where the electrical machine is simulated by Matlab/Simulink, and an experimental test bench with the (optimized) parameters shown in Table II. The generated current reference is applied to the system by a state (current) controller. Three different types of controllers have been evaluated: a nonlinear controller (NLC), convex control set (CCS) model predictive control (MPC), and finite control set (FCS) MPC [18]. All controllers are executed at the same sampling frequency. A dSpace 1104 is used as control hardware on the experimental test bench. It uses internally a 250MHz PowerPC for computation and a 20M Hz TI TMS320F240 for communication (A/D, PWM). Due to the (dSpace) communication between PowerPC and TMS320F240, only 50% of a sampling period is available to compute the control code. On the experimental test bench, the minimum, mean, and maximum execution time2 of the reference generation procedure is 34.7µs, 46.5µs, and 54.2µs, respectively. The variation is caused since some paths of the reference generation procedure (Fig. 9) are significantly simpler to compute than others. The behaviour of the reference generation procedure is explained on a torque step at standstill. At t = 0s a torque step from 0pu to 0.75pu is applied. The reference generation procedure provides the optimal current references instantaneously. The d and q current reference value are applied to the inner current controller, which reduces the (current) control error applying a voltage to the machine terminals. It is noted that the trajectory of the states (currents) towards their reference depends on the current controller. The results are shown in Fig. 13. A speed reference step from standstill to 3pu is applied to the machine using an anti-windup proportional-integral

0 1

i

dq

[pu]

1

0 10 20 Time [sample]

[pu]

0

dq

interpolation online. However, the MTPA, i.e. MTPV, is a flat optimum such that operation close but not on the trajectory does not increase the current magnitude, i.e. decrease the maximum torque in reduced power operation, by a significant amount. Applying the torque T may produce a higher flux magnitude ∥λdq ∥ than predicted by the model. Thus, it is good practice to use the voltage safety factor ρv ∈ (0, 1) as explained in Section III. The modeled intersection and maximum torque characteristics (with safety factor ρv = 0.95) are compared to the measured ones in Fig. 12. The safety factor essentially decreases the available voltage. Thus, the speed limit of the rated torque and rated power region shifts to the left (by a limited amount) and the intersection and maximum torque is reduced (by a limited amount above rated speed). This section is concluded observing that parameters that are provided by (19) optimize the correspondence of the modeled and real torque plane and may be suboptimal for other purposes. For example, the resulting number pole pairs p is a nonnegative real number (not an integer) in general.

T [pu]

8

0.5

10 20 Time [sample]

(e) FCS-MPC simulation

30

0 0

(f) FCS-MPC experimentation

Fig. 13. Torque step from 0pu to 0.75pu at standstill (dph is the three-phase duty cycle and sph is the discrete switching function)

speed controller. Since the reference step is large, the speed loop saturates and the maximum available torque is applied approximately until the speed reference is reached. At low speed, the maximum torque and acceleration are constant. The current producing maximum torque lies on the intersection of the MTPA and the isocurrent locus. Increasing the speed beyond rated speed, this operation point is not available anymore due to the voltage limit. Thus, the current moves along the isocurrent locus until the reference speed is achieved. Then the current moves along the isoflux locus to the operation point which provides the torque to maintain the machine at

9

3

0 0

0.1

0.2 Time [s]

0.3

0 0

0.4

0 0

0.4

0.1

0.2 Time [s]

0.3

0.2 Time [s]

0.3

0.1

0.2 Time [s]

0.3

0

−1 0

0.4

0 0

0.1

0.2 Time [s]

0.3

0.5

1 1.5 Time [k−sample]

0 0

2

0 0.5


2

0.5

0.5

0.5

0.5

−0.5

0.5

1

−0.5

0 id [pu]

0.5

−1

1

−1

1

Electric Machine

0 0

0.5


0 id [pu]

0.5

1

1 0 0

0.5


2

0.5


2

0.5


2

1

T [pu]

0.5

0.5 0 0

0.5


0 0

2

1

1 0 0.5


0

−1 0

2

1

1

0.5

0.5

0

q

i [pu]

−0.5

2

2

1

0

−0.5

−0.5 −1

IPMSM 10A 8.0N m 142.5mW b 9.1mH 14.6mH 636mΩ 88.3mW b 5.3, 5 6.4 · 10−3 N ms 5.0 · 10−3 kgm2 7.0W b−1 45.3W b−1 11.3W b−1 ∞

−1

3

−1 0

2L-VSI diode bridge 120V 0.95 dSpace 1104 200µs SVM

2

(b) CCS-MPC experimentation

1

Inverter and Control


d

0.5

2

idq [pu]

Table II D RIVE S YSTEM PARAMETERS

Type Rated current Is Rated torque Tr Rated flux λr Inductance (d-axis) Ld Inductance (q-axis) Lq Stator resistance Rs PM rotor flux ψ Pole pairs p (model, physical) Shaft friction B Shaft inertia J Norm. rated speed χr = ωr /¯ vr = 1/λr Norm. rated power speed χp = ωp /¯ vr Norm. intersection speed χi = ωi /¯ vr = 1/ψ Norm. max. speed χm

0 id [pu]

3

ω = 1.0pu with T = {0pu, 0.25pu, 0.5pu, 0.75pu};

Type Grid interface DC-link voltage vc Voltage safety factor ρv Embedded control platform Sampling time Ts Modulation strategy

−0.5

(a) CCS-MPC simulation

Speed step form standstill to 3pu using NLC

reference speed. The results are presented in Fig. 14 and Fig. 15. To check the convergence to the optimal operation points, different load torques are applied such that the machine provides a constant torque T at different speeds ω. The cases •

−1


ω [pu]

Fig. 14.

−1

ω [pu]

(a) NLC simulation

−1

0.5

T [pu]

0 id [pu]

2

0

idq [pu]

−0.5


q

−1

0.5

−0.5

−0.5

i [pu]

−1

0

q

q

q

−0.5

i [pu]

1

i [pu]

1

i [pu]

1

0

2

0

−1 0

1

0


1

−1 0

0.4

0.5

0.5

1 idq [pu]

0

−1 0

1

1

0 0

0.4

2

2

T [pu] 0.1

1 idq [pu]

1


0.5

0 0

0.4

0.5

1

0.5

0 0

idq [pu]

0.3

T [pu]

0.5

i [pu]

0.2 Time [s]

1

1

T [pu]

1

0.1

ω [pu]

1

2

T [pu]

1

3

3

2

ω [pu]

2

idq [pu]

ω [pu]

ω [pu]

3

−1

−0.5

0 id [pu]

0.5

(c) FCS-MPC simulation Fig. 15.

• •

1

−1

−1

−0.5

0 iq [pu]

0.5

1

(d) FCS-MPC experimentation

Speed step form standstill to 3pu using MPC

ω = 2.0pu with T = {0pu, 0.25pu, 0.5pu}; and ω = 3.0pu with T = {0pu, 0.25pu};

are evaluated and the simulation and experimental results are shown in Fig. 16. Ideally, steady-state operation with given conditions (torque, speed) would lead to a single point on the state plane. In practice, the states stay in the neighborhood

10

1

0.5

−0.25

−0.2 5

−0.5 −0.75

25

−1 −1

−1

−0.5

−0

0 id [pu]

0.5

0.5

0

q

−0.2 5

−0.5

−0.5

−0.75

5

−1 −1

−1

−0.5

−0

0 id [pu]

7

1

0

0

q

−0.25

−0.2 5

−0.5

−0.5

−0.75

5

−1 −1

−1

−0.5

−0

−0. 5 −0

0 id [pu]

7

1

(e) FCS-MPC simulation Fig. 16.

−1

−0.5

0

0.25

A PPENDIX : T RAJECTORIES AND L OCI

0

In this section, the trajectories and loci are defined using either the current or flux space. They can be transformed between the spaces using (4), i.e.

−0. 5 −0

7

0 id [pu]

0.5

−0

1

λd = Ld id + ψ

0.5

5

−0.75

−1 −1

2

(¯ vr /|ω|)

−0.2

−0.5

−1

−0.5

−0. 5 −0

0 id [pu]

7

0.5

(20)

−0

1

(f) FCS-MPC experimentation

Steady-state operation points

of this point due to power converter switching, measurement noise, etc. The results show that the drive system applies states according to the reference generation procedure. The controller operates on the MTPA trajectory at ω = 1.0pu. At ω = 2.0pu and ω = 3.0pu, the system operates on the isoflux locus, which is defined by the speed ω. X. C ONCLUSION The present research proposes an approach to compute an optimal current reference vector from a torque reference value. Optimality is achieved with respect to the constrained maximum torque per ampere criterion that is a generalization of maximum torque per ampere tracking. Since the optimization problem is difficult to solve directly, the maximum and intersection torque subproblems are identified. These concepts are used to define an algorithm that provides the optimal current reference vector and is sufficiently efficient for online imple-

(21)

and the isoflux locus in defined by an ellipsis (circle) in the flux space λ2d

0

0

−0.25

−0.5

0

0.25

0.25

0

i2q i2d + = 1, Ir2 Ir2

0.5

0.75

and λq = Lq iq .

The isocurrent locus is defined by an ellipsis (circle) in the current space

5 0.7

1

5

0.5

0.5

5

−0.75

0.5

0.25

0.25

0

1

−0.2

−0.5

0

i [pu]

q

i [pu]

0.5

0.5

5 0.7

0

−0.25

1

0.5

0.75

0.5

7

mentation. The method supports a variety of state (current) controllers with/without PWM; SPM and IPM machines with saliency and reverse saliency; and a variable DC-link voltage. To improve the correspondence between the model and the real machine, an approach is proposed to optimize the machine parameters locally with respect to important operation points. The concepts are developed on a software-in-the-loop platform and evaluated on an experimental test bench with good results. In further work, the proposed procedure will be combined with high performance current controllers. The combination with model predictive control will result in a next generation model predictive torque control with advantages in terms of design (stability guarantees) and efficient implementation (online computation time).

(d) CCS-MPC experimentation

5 0.7

1

0

−1 −1

(c) CCS-MPC simulation 1

0.5 0.25

5

0.5

−0

0 id [pu]

0.75

−0.5

−0

−0. 5

−0

−0. 5

1

0.5

0

0

−0.25

−0.5

0

0.25

0.25

−1

1

i [pu]

q

i [pu]

0.5

5

−0.75


0.5

0.75

−0.2

−0.5

−1 −1

1

5 0.7

1

0

0

−0.25

25

7

0

0.25

0.25

0

(a) NLC simulation 1

0.5

−0.5

−0

−0. 5

0.5

0.75

0.5 iq [pu]

0

0

−0.5

0

5 0.7

1

5

0.25

0.25

0

1

0.5

0.75

0.5 iq [pu]

5 0.7

1

5

+

λ2q 2

(¯ vr /|ω|)

= 1.

(22)

The maximum torque per ampere trajectory (MTPA) is defined by a hyperbola in the current plane [3], [18]  2 id + 2(Ldψ−Lq ) i2q (23)  2 −  2 = 1, ψ 2(Ld −Lq )

ψ 2(Ld −Lq )

for the IPM and id = 0A for the SPM. The maximum torque per volt trajectory (MTPV) is defined by a hyperbola in the flux plane [7], [18]  2 L ψ λd + 2(Ldq−Lq ) λ2q − (24)  2 2 = 1. Lq ψ 2(Ld −Lq )

Lq ψ 2(Ld −Lq )

for the IPM and λd = 0W b for the SPM. Both, (23) and (24) define two trajectories each. The hyperbola on the left, i.e. right, hand side is the correct solution for a machine with Ld < Lq , i.e. Ld > Lq , respectively. The MTPV trajectory crosses the d axis in λd = 0W b, i.e. id = −ψ/Ld . Observing Fig. 7 and Fig. 8) it is immediately clear that for a SPM and IPM with Ld < Lq , an intersection of the MTPV and isocurrent exists iff ψ/Ld ≤ Ir . However, observing the same graphs for a machine with Ld > Lq (Fig. 17 and Fig. 18), this result is less clear. One could suspect the existence of a case where the MTPV trajectory intersects the isocurrent locus in Fig. 18(a).

11

id [pu]

0.5 0

ii

Λ at |ω|/vr=χp

Λ at |ω|/vr=χr

0.6 0.4

−0.5

Ti

Tm

−0.5

0

Λ at 0 |ω|/vr=χp

0 i [pu]

0.5

1

0 0

χr χi

q

ii

Λ at 0 |ω|/vr=χr

−0.5

0.6

8

10

−1.5

−1

−0.5

0

i [pu]


0.2 2

0.5

0 0

1

q


Tm

Ti

−0.6

−0.6

r

(a) Current space

0

−1

4 6 |ω|/v [pu]


0.8

0.6

−0.6

χp 2

0.6

0.4

0.2 −1.2

−1

im

0.5


−0.6

−0.6

−1 −1.5


0.8

0.6

1

0.6

0.6

|T| [pu]

im

1.2

1

1

|T| [pu]

1.2

id [pu]

1

χr

χ p=χm

χi 2

4 6 |ω|/v [pu]

8

10

r

(a) Current space


Fig. 17. Maximum and intersection torque characteristic of a reverse saliency machine (Ld > Lq ) with infinite maximum speed χm = ∞

Fig. 18. Maximum and intersection torque characteristic of a reverse saliency machine (Ld > Lq ) with finite maximum speed χm < ∞

Proposition 1. The MTPV trajectory (24) never penetrates the feasible current space ∥idq ∥ ≤ Ir if ψ/Ld > Ir .

where L∆ = Ld − Lq . The q axis component is recovered substituting id into (21) or (23). The intersection of the isoflux locus and MTPV trajectory is obtained at λd given by

Proof: This result is well known for SPM and IPM with Ld < Lq , where no intersection of the MTPV trajectory and isocurrent locus exists if ψ/Ld > Ir . Let ψ/Ld > Ir and Ld > Lq , then the MTPV trajectory needs to intersect the isocurrent locus twice on the λq ≥ 0W b halfplane (an entry and exit point) to penetrate the current space ∥idq ∥ ≤ Ir . The d axis components of the intersections of the MTPV trajectory and the isocurrent locus are defined by (32) that yields  (25) λd = −b ± b2 + a, with b=

Lq ψ(L2d − 2Ld Lq + 2L2q ) , 2(Ld − Lq )(L2d + L2q )

(26)

such that there are two possible solutions on the left and right hand side of λd = −b, which we call intersection asymptote for simplicity. The MTPV (24) has the vertical asymptote [18] Lq ψ . λd = −c = − 2(Ld − Lq )

(27)

The intersection asymptote lies on the left hand side of the MTPV asymptote since after a few manipulations −b < −c, i.e. b > c results in (for Ld > Lq ) (Ld − Lq)2 + L2q > 0.

(28)

Since the MTPV asymptote lies on the λd ≤ 0W b halfplane and ∥idq ∥ ≤ Ir lies on the λd ≥ 0W b halfplane (for Ld > Lq ), the MTPV trajectory has at maximum one intersection with the isocurrent locus. The above result is sufficient such that the proposed analysis and method (Fig. 9) is suitable for machines with Ld > Lq . A PPENDIX : C OMPUTATION The maximum and intersection torque are obtained intersecting the relevant trajectories and loci defined in Figure 9. Each intersection is obtained substituting the relevant equations that yield second order polynomials and can be solved analytically. The intersection of the isocurrent locus and MTPA trajectory is obtained at id given by 2L∆ i2d + ψid − L∆ Ir2 = 0,

(29)

def

¯ 2 = 0, 2L∆ λ2d + Lq ψλd − L∆ λ r

(30)

¯ r = v¯r /|ω|. The q axis component is recovered where λ substituting λd into (22) or (24). The intersection of the isoflux locus and MTPA trajectory is obtained at at id given by   ¯ 2 = 0. L∆ (L2d + L2q )i2d + ψ(L2d + L2∆ )id + L∆ ψ 2 − λ r (31) def

The q axis component is recovered substituting id into (22) or (23). The intersection of the isocurrent locus and MTPV trajectory is obtained at λd given by   L∆ (L2d + L2q )λ2d + Lq ψ(L2q + L2∆ )λd + L2q L∆ ψ 2 − L2d Ir2 = 0. (32) The q axis component is recovered substituting λd into (21) or (24). The intersection of the isocurrent locus and isoflux locus is obtained at id given by ¯ 2 + L2 I 2 = 0, (L2d − L2q )i2d + 2Ld ψid + ψ 2 − λ r q r

(33)

The q axis component is recovered substituting id into (21) or (22). The states on the MTPA trajectory i.e. isoflux locus, which produce a given torque, are obtained substituting (22), i.e. (23), into the torque equation (6). This yields forth order polynomials that can be solved analytically with Ferrari’s method. Solving the fourth order polynomial (one per iteration) is the most demanding step of the reference generation procedure in terms of computation effort. Given a torque T , the states on the MTPA trajectory that produce T are defined by id given by L3∆ i4d + 3L2∆ ψi3d + 3L∆ ψ 2 i2d + ψ 3 id − L∆ T¯2 = 0,

(34)

where T¯ = T /(3/2p). The q axis component is recovered substituting id into (23). Given a torque T , the states on the isoflux locus that produce T are defined by id given by −L2d L2∆ i4d − 2Ld ψL∆ (2Ld − Lq )i3d ¯ 2 L2 − ψ 2 (6L2 − 6Ld Lq + Lq ))i2 +(λ r

∆

d

d

¯ 2 L∆ − ψ 2 (2Ld − Lq ))id +2ψ(λ r ¯ 2 − ψ 2 ) − L2 T¯2 = 0. +ψ 2 (λ r q

(35)

The q axis component is recovered substituting id into (22).

12

R EFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11] [12]

[13]

[14] [15]

[16]

[17]

N. Bianchi and S. Bolognani, «Parameters and volt-ampere ratings of a synchronous motor drive for flux-weakening applications», IEEE Transactions on Power Electronics, vol. 12, pp. 895–903, 1997. A. Consoli, G. Scelba, G. Scarcella, and M. Cacciato, «An effective energy-saving scalar control for industrial IPMSM drives», IEEE Transactions on Industrial Electronics, vol. 60, pp. 3658–3669, 2013. M. Preindl and S. Bolognani, «Model predictive direct torque control with finite control set for PMSM drive systems, part 1: maximum torque per ampere operation», IEEE Transactions on Industrial Informatics, vol. 9, pp. 1912–1921, 2013. J. Beerten, J. Verveckken, and J. Driesen, «Predictive direct torque control for flux and torque ripple reduction», IEEE Transactions on Industry Applications, vol. 57, pp. 404–412, 2010. M. Preindl, E. Schaltz, and P. Thøgersen, «Switching frequency reduction using model predictive direct current control for high power voltage source inverters», IEEE Transactions on Industrial Electronics, vol. 58, pp. 2826–2835, 2011. K. Kamiev, J. Montonen, M. Ragavendra, J. Pyrhonen, J. Tapia, and M. Niemela, «Design principles of permanent magnet synchronous machines for parallel hybrid or traction applications», IEEE Transactions on Industrial Electronics, vol. 60, pp. 4881–4890, 2013. M. Preindl and S. Bolognani, «Model predictive direct torque control with finite control set for PMSM drive systems, part 2: field weakening operation», IEEE Transactions on Industrial Informatics, vol. 9, pp. 648–657, 2013. Y. Inoue, S. Morimoto, and M. Sanada, «Comparative study of PMSM drive systems based on current control and direct torque control in flux-weakening control region», IEEE Transactions on Industry Applications, vol. 48, pp. 2382–2389, 2012. S. Bolognani, S. Calligaro, and R. Petrella, «Adaptive fluxweakening controller for interior permanent magnet synchronous motor drives», IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 2, pp. 236–248, 2014. S. Morimoto, Y. Takeda, T. Hirasa, and K. Taniguchi, «Expansion of operating limits for permanent magnet motor by current vector control considering inverter capacity», IEEE Transactions on Industry Applications, vol. 26, pp. 866–871, 1990. S.-Y. Jung, J. Hong, and K. Nam, «Current minimizing torque control of the IPMSM using ferrari’s method», IEEE Transactions on Power Electronics, vol. 28, pp. 5603–5617, 2013. M. Preindl and E. Schaltz, «Sensorless model predictive direct current control using novel second-order PLL observer for PMSM drive systems», IEEE Transactions on Industrial Electronics, vol. 58, pp. 4087–4095, 2011. T. Miyajima, H. Fujimoto, and M. Fujitsuna, «A precise model-based design of voltage phase controller for IPMSM», IEEE Transactions on Power Electronics, vol. 28, pp. 5655– 5664, 2013. M. Kunter, T. Schoenen, W. Hoffmann, and R. De Doncker, «IPMSM control regime for a hybrid-electric vehicle application», in Emobility - Electrical Power Train, 2010. J. Lemmens, P. Vanassche, and J. Driesen, «Pmsm drive current and voltage limiting as a constraint optimal control problem», IEEE Journal of Emerging and Selected Topics in Power Electronics, 2014, Early Access. K. Hoang, J. Wang, M. Cyriacks, A. Melkonyan, and K. Kriegel, «Feed-forward torque control of interior permanent magnet brushless AC drive for traction applications», in International Electric Machines and Drives Conference (IEMDC), 2013. C. Mademlis and V. Agelidis, «A high-performance vector controlled interior PM synchronous motor drive with extended

[18] [19]

[20]

[21]

[22]

[23] [24] [25]

[26] [27]

[28]

[29] [30]

[31] [32] [33]

[34] [35]

[36]

speed range capability», in Annual Conference of the IEEE Industrial Electronics Society (IECON), 2001. M. Preindl, «Novel model predictive control of a PM synchronous motor drive», PhD thesis, University of Padua, 2014. S. Morimoto, M. Sanada, and Y. Takeda, «Effects and compensation of magnetic saturation in flux-weakening controlled permanent magnet synchronous motor drives», IEEE Transactions on Industry Applications, vol. 30, pp. 1632–1637, 1994. Y. Chen, Z. Zhu, and D. Howe, «Influence of inaccuracies in machine parameters on field-weakening performance of PM brushless AC drives», in International Electric Machines and Drives Conference (IEMDC), 1999. Y. F. Shi, Z. Zhu, Y. Chen, and D. Howe, «Influence of winding resistance on flux-weakening performance of a permanent magnet brushless AC drive», in International Conference on Power Electronics, Machines and Drives, 2002. T. Schoenen, M. Kunter, M. Hennen, and R. De Doncker, «Advantages of a variable DC-link voltage by using a DC-DC converter in hybrid-electric vehicles», in Vehicle Power and Propulsion Conference (VPPC), 2010. M. Preindl and D. Bagnara, «Wind turbine for generating electric energy», WO2013001496, 2013. L. Montejano, «Some results about minkowski addition and difference», Mathematika, vol. 43, pp. 265–273, 1996. R. H. Park, «Two-reaction theory of synchronous machines generalized method of analysis - part i», Transactions of the American Institute of Electrical Engineers, vol. 48, pp. 716– 727, 1929. F. Parasliti and P. Poffet, «A model for saturation effects in high-field permanent magnet synchronous motors», IEEE Transactions on Energy Conversion, vol. 4, pp. 487–494, 1989. C. Mademlis and V. Agelidis, «On considering magnetic saturation with maximum torque per current control in interior permanent magnet synchronous motor drives», IEEE Transactions on Energy Conversion, vol. 16, pp. 246–252, 2001. B. Stumberger, G. Stumberger, D. Dolinar, A. Hamler, and M. Trlep, «Evaluation of saturation and cross-magnetization effects in interior permanent-magnet synchronous motor», IEEE Transactions on Industry Applications, vol. 39, pp. 1264–1271, 2003. N. Bianchi and T. M. Jahns, Design, Analysis, and Control of Interior PM Synchronous Machines. Cleup (Padua), 2006. N. Bianchi, S. Bolognani, and B. Ruzojcic, «Design of a 1000 HP permanent magnet synchronous motor for ship propulsion», in European Conference Power Electronics and Applications (EPE), 2009. M. Barcaro, «Design and analysis of interior permanent magnet synchronous machines for electric vehicles», PhD thesis, University of Padua, Italy, 2011. D. White and H. Woodson, Electromechanical Energy Conversion. John Wiley & Sons, 1959. M. Preindl and S. Bolognani, «Model predictive direct speed control with finite control set of PMSM drive systems», IEEE Transactions on Power Electronics, vol. 28, pp. 1007–1015, 2013. S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. M. Zordan, P. Vas, M. Rashed, S. Bolognani, and M. Zigliotto, «Field-weakening in high-performance PMSM drives: a comparative analysis», in Industry Applications Conference (IAS), 2000. S. Bolognani, S. Calligaro, R. Petrella, and F. Pogni, «Fluxweakening in IPM motor drives: comparison of state-of-art algorithms and a novel proposal for controller design», in European Conference Power Electronics and Applications (EPE), 2011.

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Matthias Preindl was born in Brixen, Italy, in 1986. He is working towards the PhD degree in Energy Engineering at University of Padova, Italy and is currently visiting scolar at University of California, Berkeley, USA. He received the BSc degree (summa cum laude) from the University of Padova, Italy in 2008 and the MSc from ETH Zürich, Switzerland in 2010, both in Electrical Engineering. Moreover, he has been visiting student at Aalborg University, Denmark where he wrote his diploma thesis. From 2010 to 2012, M. Preindl was with Leitwind AG, Italy, where he was a R&D engineer. Before that, he worked for Energy.dis GmbH, Italy and the Italian National Research Council (CNRRFX). His research interests include control and design of power electronic systems with applications in drive systems, renewable energy generation, and vehicular systems.

Silverio Bolognani (M’76) received the Laurea degree in electrical engineering from the University of Padova, Padova, Italy, in 1976. In 1976, he joined the Department of Electrical Engineering, University of Padova, where he is currently a Full Professor of electrical converters, machines, and drives, and was the Founder of the Electrical Drives Laboratory. His research on brushless and induction motor drives was carried out in the framework of European and National Research Projects. His research interests include advanced control techniques for motor drives and motion control and design of ac electrical motors for variable-speed applications. He has been an Invited Speaker at International Conferences. He is the author or coauthor of more than 200 papers on electrical machines and drives. He holds three patents. Prof. Bolognani is currently the Chairman of the IEEE North Italy IAS/IES/PELS Joint Chapter. He has been a member of the Steering or Technical Committees.