Current Minimizing Torque Control of the IPMSM Using ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 12, DECEMBER 2013

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Current Minimizing Torque Control of the IPMSM Using Ferrari’s Method Sung-Yoon Jung, Student Member, IEEE, Jinseok Hong, Student Member, IEEE, and Kwanghee Nam, Member, IEEE

Abstract—For the torque control of an interior permanent magnet synchronous motor (IPMSM), it is necessary to determine a current command set that minimizes the magnitude of the current vector. This is known as the maximum torque per ampere. In the field-weakening region, current minimizing solutions are found at the intersection with the voltage limits. However, the optimal problem yields fourth-order polynomials (quartic equations), and no attempt has been made to solve these quartic equations online for torque control. Instead, premade lookup tables are widely used. These lookup tables tend to be huge because it is necessary to create separate tables on the basis of the dc-link voltage and motor temperature. In this study, we utilize Ferrari’s method, which gives the solution to a quartic equation, for the torque control. Further, a recursive method is also considered to incorporate the inductance change from the core saturation. A simulation and some experiments were performed using an electric vehicle motor, which demonstrated the validity of the proposed method. Index Terms—Electric vehicle (EV), Ferrari’s method, interior permanent magnet synchronous motor (IPMSM), maximum torque per ampere (MTPA), torque control, voltage limit.

I. INTRODUCTION N interior permanent magnet synchronous motor (IPMSM) is known to be a superior solution in terms of efficiency, power density, and wide speed operating range. Because the reluctance torque can be utilized in the fieldweakening region, the speed range of an IPMSM can be extended while maintaining a constant power. This wide constantpower speed range gives the virtue of high power density. In an IPMSM, the torque is determined using not only the q-axis current iq , but also the d-axis current id . Therefore, there has been a question of optimality in combining the two current components. The solution is to produce the desired torque while minimizing the current magnitude. The maximum torque per ampere (MTPA), developed by Morimoto et al. [1], is the current minimizing solution that is used under the base speed. Above the base speed, the MTPA is not feasible because of the limit in the dc-link voltage. In general, the optimal combination is found at the intersection of the torque and voltage limit

A

Manuscript received September 26, 2012; revised January 11, 2013; accepted January 28, 2013. Date of current version June 6, 2013. Recommended for publication by Associate Editor N. Kar. The authors are with the Department of Electrical Engineering, Pohang University of Science and Technology, Gyeongbuk 790-784, Korea (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2013.2245920

curves. There are two problems that hinder pursuing the optimality in a practical situation: finding the intersection of the two curves is a highly nonlinear problem and the inductances change along with the current as a result of core saturation [2], [3]. Even cross-coupling phenomena cannot be neglected in some specific applications [4], [5]. Thus, lookup table methods are widely used in the IPMSM torque control. They calculate the optimal current commands a priori, put them into a table, and then draw current commands from this table on the basis of the required torque and speed. Nalepa and Kowalska [6] effectively demonstrated the difficulty in finding the optimal solution owing to magnetic saturation and temperature variations. They proposed the use of a 3-D table in generating a feed-forward compensation term for the d-axis current command based on the dc-link voltage, q-axis current, and speed. It should be noted that the field strength of a permanent magnet, for example a neodymium magnet, depends on the temperature, and its change cannot be neglected in electric vehicle (EV) applications because the ambient temperature is in the range of −40–150 ◦ C [7]. Monajemy and Krishnan [8] derived a fourth-order polynomial that resulted from torque and voltage limit equations and proposed the use of solution lookup tables based on flux linkage and torque commands. Instead of using the lookup table, analytic approaches based on the mathematical model were attempted in [1] and [2]. However, those approaches were limited in practical use, because no explicit current set was obtainable from a given desired torque. Jeong et al. [9] formulated current cost functions with torque and voltage constraints, and proposed the use of Newton’s method online to find the minimizing solutions. However, the convergence and computational burden would be problematic in realtime applications. Lee et al. [10] separated the cases on the basis of whether or not the solution was within the voltage limit and derived fourth-order polynomials in both cases. They found solutions by using a technique of approximating the fourth-order polynomial by a second-order one. Bolognani et al. [11] proposed a local MTPA search method involving the injection of an additional pulsating current. Based on the magnitude of torque ripple, they constructed an MTPAtracking loop. The torque ripple was measured indirectly via speed variation in the experiment. Therefore, the performance was limited by resolution of the speed sensor. Kim et al. [12] also injected a high-frequency signal, but utilized a power relation that could be calculated from the stator voltage and current. Zhu et al. [13] developed an online optimal control strategy to achieve maximum efficiency, which adjusted the d-axis current

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incrementally by exploiting the dc-link current, q-axis current error, and speed as optimization objectives. Direct torque control (DTC) for IPMSM drives has been investigated since the 1990s. DTC does not require an accurate motor model and parameters, except for the armature resistance. The torque and stator flux linkage are directly controlled using both a hysteresis comparator and a switching table [14], [15]. However, there are problems, such as an unfixed switching frequency and a large torque ripple. To solve these problems, space vector modulation was used, along with a reference flux vector calculator (RFVC) [16]. Relevant DTC methods that considered the voltage and current limits were followed in [17] and [18]. For a surface-mounted permanent magnet synchronous motor (SPMSM), selecting the optimal current set is simpler than for an IPMSM. Chen et al. [19] proposed a current controller for an SPMSM that adjusted the d-axis current on the basis of the voltage saturation. Because the current set was found on the voltage boundary, it was claimed to be optimal in the sense of minimizing the copper loss. Liu et al. [20] proposed an improved method of tracking the maximum torque per voltage line in an SPMSM by including the stator resistance and inverter nonlinearities. In this paper, an online torque control method based on analytic solutions is proposed that covers the entire speed region. Fourth-order polynomials are derived from the optimality conditions. Two optimality conditions are considered: within the base speed region and in the field-weakening region where the voltage limit is active. A discriminant is used to discern these two cases. Ferrari’s method is used to find the solution of quartic equations. Through repeating the computations, variations in the d- and q-axes inductances are considered. This paper is organized as follows. Section II provides preliminaries on the IPMSM model, inductance saturation effect, and motor losses. In Section III, current minimizing conditions are derived with and without the voltage limit. Then, Ferrari’s method is used to solve quartic polynomials in Section IV, and a current minimizing algorithm is proposed in Section V. The entire torque control algorithm is constructed in Section VI. Finally, a simulation and some experiments are shown in Sections VII and VIII. II. PRELIMINARIES A. IPMSM Model and Voltage/Current Limits Fig. 1 shows a 2-D FEM model of an IPMSM used in this experiment. The motor was developed as a propulsion motor for C-class passenger EVs. The outer diameters of stator and rotor are 249 and 171 mm, respectively. The stack length is 120 mm and the air-gap height is 0.8 mm. It has eight poles and two permanent magnets are arranged in a v-shape in each pole. Cavities were designed around the edges of permanent magnets to reduce the leakage flux. Considering core saturation effects, the stator flux linkage of the IPMSM in the synchronous reference frame is described by λd = Ld (id , iq )id + ψm

(1)

λq = Lq (id , iq )iq

(2)

Fig. 1.

2-D mesh model of the IPMSM.

where id and iq are the d- and q-axes currents; Ld and Lq are the d- and q-axes inductances; and ψm is the rotor flux linkage due to the permanent magnets. The stator voltage equations of the IPMSM in the synchronous frame are did − ωe Lq iq (3) vd = rs id + Ld dt diq + ωe Ld id + ωe ψm vq = rs iq + Lq (4) dt where vd and vq are the d- and q-axes voltages; ωe is the electrical angular frequency; and rs is the stator resistance. The electromagnetic torque equation of the IPMSM is given by 3 P (ψm iq + (Ld − Lq )id iq ) (5) 4 where P is the pole number. Neglecting the ohmic drop across the stator resistance, the current and voltage limits are described by T =

L2d

id +

ψm Ld

2

i2d + i2q ≤ Is2 + L2q i2q ≤

Vs2 ωe2

(6) (7)

where Is is the peak value of the maximum stator current and Vs is the peak value of the maximum phase voltage. As shown in Fig. 2, the current limit is the dashed circle centered at the origin with radius Is . The voltage limit appear as the dotted ellipse centered at (− ψLmd , 0), and the major and minor radii are Vs /(Ld ωe ) and Vs /(Lq ωe ), respectively. Therefore, the voltage limit ellipse shrinks as ωe increases. B. Inductance Change Due to the Core Saturation In most torque control applications, torque sensors are not used directly. Instead, torque estimates based on the stator current measurements are used for feedback. Therefore, accurate inductance information is indispensable for a better torque precision and for an optimal current command selection. However, due to the core saturation, the inductances vary nonlinearly depending on the load condition and current angle. Therefore, lookup tables are used in practice to accommodate inductance changes.

JUNG et al.: CURRENT MINIMIZING TORQUE CONTROL OF THE IPMSM USING FERRARI’S METHOD

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Fig. 2. Current limit circle, voltage limit ellipse, and constant torque curve on d–q axes.

To estimate inductances, we use voltage measurements fd (id , iq ) = rs id − ωe Lq (id , iq )iq

(8)

fq (id , iq ) = rs iq + ωe Ld (id , iq )id + ωe ψm .

(9)

Specifically, we measure the d- and q-axes voltages two times in the steady state, altering the polarity of the q-axis current, i.e., we measure fd (id , iq ), fd (id , −iq ) and fq (id , iq ), fq (id , −iq ) under regulated current conditions, and estimate inductances in such way that [3], [5] fq (id , iq ) + fq (id , −iq ) − 2ωe ψm Ld (id , iq ) = 2ωe id

(10)

fd (id , −iq ) − fd (id , iq ) . Lq (id , iq ) = 2ωe iq

(11)

In practice, the voltages are measured indirectly from the space vector modulation process after compensating the dead-time and IGBT on-drop. This inductance estimation method includes the dependence on currents obviously, while eliminating a dependence on the ohmic drop of the stator coil. Fig. 3 shows the 3-D plots of Ld and Lq measurements of an EV motor used in this experiment. In Fig. 4, the measured data are compared with the computed data by an FEM tool, JMAG [21]. The 2-D mesh model shown in Fig. 1 was utilized for inductance computation. The computed Ld and Lq were adjusted to include the end-turn effects by multiplying by 1.27 and 1.08, respectively. Inductances were measured in the current range, 0 to 250 A. It needs to be emphasized that the measured values by (10) and (11) match well with the FEM computed results in the overall current range. C. Motor Losses The losses of motor consist of copper loss, iron loss, stray loss, and mechanical loss. The copper loss refers to the joule loss of the stator coil, and is described by Pcu =

3 rs (i2d + i2q ). 2

(12)

Fig. 3.

Variations in L d and L q depending on currents. (a) L d . (b) L q .

The iron loss means the loss of the iron core for a time-varying field. It consists of two parts: hysteresis loss and eddy current loss. Since the loop area of a B–H curve signifies an energy per volume, the hysteresis loss is proportional to the frequency and the loop area. The loop area does not grow linearly with the α , maximum field Bm . Experimentally, it is estimated by kh f Bm where kh = 40–55 is a constant depending on the silicon contents in the steel, f is a frequency of the field, and α is in the range of 1.8–2.2 [22]. The latter is often called Joule loss, since it is caused by induced current in the conductive magnetic ma2 , where ke is a terial. It is predicted analytically by ke f 2 Bm constant which depends on the thickness of the steel lamination and the silicon percentage. Putting together, the iron loss under sinusoidal excitation is modeled as α 2 + ke f 2 Bm . PF e = kh f Bm

(13)

The mechanical and stray losses are small compared with the copper and iron losses. Fig. 5 shows copper and iron loss of the IPMSM used in this experiment. The iron losses were calculated over a wide speed range under various current conditions using a commercial FEM tool, JMAG [21]. The results state that copper loss is most dominant especially in the low-speed area. In the extreme high speed range, the iron loss grows significantly. However, it is still smaller than the copper loss.

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Fig. 4. Inductance comparison between measured and FEM data: (a) L d and (b) L q .

Numerous research works were performed regarding the total loss minimization of an IPMSM [10], [23], [24]. Morimoto et al. [23] established a loss minimizing control based on an equivalent circuit which contained an iron loss model as well as a copper loss model. Utilizing a derivative of the loss function, a loss-minimizing d−axis current was determined. Cavallaro et al. [24] developed an online loss minimizing algorithm based on the loss model of Morimoto [23]. Lee et al. [10] proposed a method of finding the loss minimizing solution using an approximation technique. The total loss minimization is not an easy task because a target cost function itself is a high-order polynomial if an iron loss model is included. Furthermore, the coefficients vary depending on current and frequency [25]. Thereby, an explicit form of analytical solutions is hardly obtainable. Furthermore, the computational load should not be high for practical applications. In this study, we narrow down the focus to the copper loss minimization like the MTPA. But we consider voltage and current limits, while accepting inductance variations. III. ANALYTIC SOLUTION FOR CURRENT MINIMIZATION In order to establish a torque control loop, it is necessary to find the current command values, (i∗d , i∗q ). There are numerous (id , iq ) choices for a given torque, as shown in Fig. 2. However, each choice needs to be evaluated from the perspective of loss minimization. It should be noted that the winding copper loss is the dominant one among many loss components. Thus, a simple optimization method is to focus on just reducing the stator current, and the MTPA is such a current minimizing solution. If a torque curve intersects the MTPA line within the voltage and current limits, then the intersection point will be a desired current command. However, the MTPA solution may be located

Fig. 5. Copper and iron loss of the IPMSM depending on mechanical speed and stator current: (a) 100 A. (b) 200 A.

outside the voltage limit, as shown in Fig. 6. In such cases, a suboptimal solution is found at the intersection between the torque curve and the voltage limit ellipse. A. Case i): (MTPA) The MTPA solution curve is given by [1] 2 + 8I 2 (L − L )2 ψm ± ψm q d id = 4(Ld − Lq ) iq = ± I 2 − i2d

(14) (15)

where 0 ≤ I ≤ Is . Therefore, it is depicted as the two sets of parabolas shown in Fig. 2. However, finding the MTPA solution, (14) and (15) for a given torque, T = T0 is not straightforward. It is formulated as a constrained minimization problem Minimize

i2d + i2q

3P (ψm iq + (Ld − Lq )id iq ) = T0 . 4 The Lagrangian for this problem is defined as 3P 2 2 (ψm iq + (Ld − Lq )id iq ) − T0 L(id , iq , λ) = id + iq + λ 4 subject to

where λ is a Lagrangian multiplier. Then the necessary conditions for minimization are 3P ∂L λ(Ld − Lq )iq = 2id + 0= ∂id 4 0=

∂L 3P λ (ψm + (Ld − Lq )iq ) = 2iq + ∂iq 4

0=

3P ∂L = (ψm iq + (Ld − Lq )id iq ) − T0 . ∂λ 4


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2 2 ψm Vs2 2ψm ψm − C2 = + Ld (Ld − Lq )2 Ld (Ld − Lq ) ωe2 Ld (Ld − Lq ) 4 2 L2q 16T02 1 Vs2 ψm ψm + 2 · − 2 · 2 . D2 = (Ld − Lq )2 L2d Ld 9P 2 ωe Ld IV. FERRARI’S SOLUTION TO QUARTIC EQUATIONS In the above, two quartic polynomials are derived: from either the MTPA or the voltage limit. It is known that a general solution always exists for quartic polynomials [26]. Lodovico Ferrari invented a systematic procedure to solve all quartics [27]. In this section, analytic solutions are obtained utilizing Ferrari’s method. A. Case i): (MTPA) The general solution to (16) is given as [28] η1 μ1 A1 id = − ±s ±t (18) 4 2 2 where 1 3A1 C1 − 12D1 − B12 α1 = (19) 3 1 β1 = (−2B13 + 9A1 B1 C1 + 72B1 D1 − 27C12 − 27A21 D1 ) 27 (20)

3 3 α3 α3 β12 β12 B1 β1 β1 + − + + 1 + − − + 1 γ1 = 3 2 4 27 2 4 27 Fig. 6. (a) Case i): intersection between voltage limit ellipse and torque curve (G 1 , G 2 ) and (b) Case ii): Tangential Intersection—tangential intersection between voltage limit ellipse and torque curve (H ).

Solving the necessary conditions simultaneously for id , we obtain the quartic polynomial i4d

+

A1 i3d

3ψ m (L d −L q ) , 16T 02 − 9P 2 (L d −L 2 . q)

where A1 = D1 =

+

B1 i2d

B1 =

+ C1 id + D1 = 0

2 3ψ m (L d −L q ) 2

, C1 =

3 ψm (L d −L q ) 3

(16) , and

i4d + A2 i3d + B2 i2d + C2 id + D2 = 0

Lq 2− Ld

μ1 =

A21 − B1 + γ1 4

(22)

3 2 1 A1 − η12 − 2B1 ±s (4A1 B1 − 8C1 − A31 ). (23) 4 4η1

It should be noted that two ±s should have the same sign, while the sign of ±t is independent. When r1 , r2 , r3 , and r4 are the solutions of a quartic polynomial, the discriminant is defined as (r2 − r4 )2 (r3 − r4 )2 .

As the speed increases, the voltage ellipse shrinks. If T0 is high in a medium speed range, the intersection between the MTPA and the torque curve may occur outside the voltage limit. Therefore, we need to find the simultaneous solutions of (5) and (7), which lead to the following quartic polynomial:

2ψm A2 = Ld − Lq

η1 =

(21)

D = (r1 − r2 )2 (r1 − r3 )2 (r1 − r4 )2 (r2 − r3 )2

B. Case ii): (Intersection With Voltage Limit)

where

2 2 2 ψm ψm 4 · ψm Vs2 + B2 = + − 2 (Ld − Lq )2 Ld (Ld − Lq ) Ld ωe2 L2d

(17)

(24)

The discriminant tells us about the nature of its roots [29]. Noting that all of the coefficients of (16) are real, if at least two of the roots coincide, then D = 0; if all four roots are real or complex conjugates, then D > 0; if two roots are real and two are complex conjugates, then D < 0. For quartic polynomial (16), the discriminant is equal to [30] D = 256D13 − 192A1 C1 D12 − 128B12 D12 + 144B1 C12 D1 − 27C14 + 144A21 B1 D12 − 6A21 C12 D1 − 80A1 B12 C1 D1 + 18A1 B1 C13 + 16B14 D1 − 4B13 C12 − 27A41 D12 + 18A31 B1 C1 D1 − 4A31 C13 − 4A21 B13 D1 + A21 B12 C12 . (25)

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Substituting A1 , B1 , C1 , and D1 into (25), it follows that D=−

T4 256 4 (Ld − Lq )10 06 (4096T02 (Ld − Lq )2 + 243ψm P 2) 729 P < 0. (26)

It should be noted that the discriminant, (26) is always less than zero for all T0 . This corresponds to the case of two real roots and two complex conjugates. The existence of two real roots is justified graphically by two intersection points F1 and F2 as shown in Fig. 2. Of the two real roots, we are concerned with the one in the second quadrant. Therefore, the desired solution, F1 is obtained with the minus signs from ±s and ±t 4T 0 η1 μ1 A1 3P − − , (idα , iq α ) = − . 4 2 2 ψm + (Ld − Lq )idα (27) B. Case i): (Intersection With Voltage Limit) The solutions will have the same form as (18) except the coefficients. Fig. 6(a) shows that the two curves intersect at G1 and G2 . Because G1 is closer to the MTPA, the desired solution is G1 with the minus sign from ±s and the positive sign from ±t . Thus, the final solution of Case ii) is 4T 0 η2 μ2 A2 3P − + , (idβ , iq β ) = − . 4 2 2 ψm + (Ld − Lq )idβ (28) where η2 and μ2 are defined in the same manner as η1 and μ1 . That is, η2 and μ2 are obtained from (22) and (23) by replacing A1 , B1 , C1 , and D1 by A2 , B2 , C2 , and D2 , respectively. C. Case ii): (Tangential Intersection) In the high-speed region, the voltage ellipse shrinks toward a point (− ψLmd , 0). It is assumed here that ψLmd < Is . Then, the torque curve intersects the voltage limit tangentially, and the intersection point lies inside the current limit circle. Fig. 6(b) shows an example. Tangential intersection points are calculated in the following manner [31]. Substituting id = λd L−ψd m and (7) into (5), it follows that 2 2 3P λd − ψm (Vs /ωe )2 − λ2d . (29) λd − Lq T2 = 4 Ld L2q We take differentiation with respect to λd and make it equal to zero. Then 2 + 8(L − L )2 (V /ω )2 −Lq ψm ±v L2q ψm d q s e ∗ . (30) λd = 4(Ld − Lq ) Because the solution converges to (− ψLmd , 0) as ωe increases, the plus sign is chosen from ±v , so that λ∗d converges to zero. Substituting (30) into (7), the desired tangential solution is obtained as (Vs /ωe )2 − λ∗2 λ∗d − ψm d , . (31) (id tan , iq tan ) = Ld Lq

It should be noted that, on the other hand, the tangential intersection can also be obtained from the discriminant. The discriminant for (17) is D=

T2 256 L2q (Ld − Lq )4 · · 6 0 8 ζT04 + δT02 + σ 10 729 Ld P ωe

(32)

where ζ = 4096ωe8 L4d L4q (Ld − Lq )2 4 4 4 2 2 2 2 2 δ = −144P 2 ωe4 L2d L2q (−ψm ωe Lq + 20ψm ωe Vs Ld Lq 8Vs4 L4d

− 32Vs4 L3d Lq + 48Vs4 L2d L2q − 32Vs4 Ld L3q 2 2 2 2 2 2 4 − 40ψm ωe Vs Ld L3q + 20ψm ωe Vs Lq + 8Vs4 L4q )

σ = 81P 4 Vs2 (Vs Ld − ωe ψm Lq − Vs Lq )3 (Vs Ld + ωe ψm Lq − Vs Lq )3 . Because D = 0 at a tangential point, we find from (32) that

−δ + δ 2 − 4ζσ Ttan = . (33) 2ζ Equation (31) give the location of the tangential intersection in the current plane, whereas (33) gives the magnitude of the tangential torque. Therefore, T0 = Ttan is the maximum available torque for a given speed. Correspondingly, a demanded torque should always be less than Ttan for a given ωe , i.e., T0 ≤ Ttan . In a later section, Ttan is used as a checking criterion to determine whether the requested torque can be produced. If (31) is substituted into the torque equation (5), then the result should be the same as (33). However, although the identity was checked through numerical evaluations, it is too complex to show. V. DETERMINATION OF I 2 -MINIMIZING SOLUTION As shown in the previous section, the current minimizing solution is found on the intersection with the MTPA or a voltage limit, depending on the speed and torque. Thus, before calculating the solution, it is necessary to classify the cases. A method suggested here is to find a solution candidate from Case ii), and check whether it is the (sub) optimal solution or not. Specifically, for a given T0 , determine the intersection with the voltage limit. Then, find the location of the MTPA solution. If the MTPA solution is within the voltage limit, we need to use the MTPA solution. Otherwise, the precalculated solution (Case ii)) is determined to be the desired one. Because testing is conducted before finding the MTPA solution, unnecessary calculation effort can be avoided. This a priori determining method is based on the local geometric analysis at the intersection N (idβ , iq β ). Let χ = (χ1 , χ2 )T be a tangent vector of the constant torque line, T (id , iq ) = T0 . The differential one-form, dT = ( ∂∂ iTd did , ∂∂ iTd diq ) is orthogonal to the tangent vector χ as shown in Fig. 7 dT, χ ≡

∂T ∂T χ1 + χ2 = 0 ∂id ∂iq

(34)


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Fig. 7. Index determining location of MTPA point: (a) d(I 2 ), χ > 0 for Case ii), (b) d(I 2 ), χ < 0 for Case i).

where < ·, · > means an inner product. Straightforwardly, we can choose χ as 4 ∂T ∂T (χ1 , χ2 ) ≡ , − 3P ∂iq ∂id = − ψm − (Ld − Lq )id , (Ld − Lq )iq . In the second quadrant, χ points to the third quadrant because Ld − Lq < 0. Now, we take a differentiation of I 2 = i2d + i2q with the vector, χ at the intersection N (idβ , iq β ). Then 1 1 ∂I 2 ∂I 2 2 d(I ), χ = χ1 + χ2 2 2 ∂idβ ∂iq β (i d β , i q β ) = (Ld − Lq )(i2q β − i2dβ ) − ψm idβ . This is a directional derivative of the constant torque line T (id , iq ) = T0 in the direction of χ at (idβ , iq β ). It should be noted that first d(I 2 ), χ = 0 at the intersection with the MTPA, because the minimum length of I 2 is achieved when the torque line and circle meet tangentially. In other words, d(I 2 ) and χ are orthogonal on the MTPA line. It is obvious from Fig. 7(a) that d(I 2 ), χ > 0 in the left region of the MTPA line, i.e., d(I 2 ) and χ form an acute angle. Therefore, I 2 increases along the χ direction, indicating that the MTPA solution exists in the right section of N1 . This implies that if d(I 2 ), χ > 0 at N1 (idβ , iq β ), then the MTPA solution,

Fig. 8. Segmentation of operation region depending on optimal conditions: (a) torque versus speed profile and (b) current contour.

M1 lies outside the voltage limit. Therefore, the intersection, N1 is the desired solution. In other words, N1 is the minimizing solution of I 2 within the voltage limit. On the other hand, it can be noted from Fig. 7(b) that d(I 2 ), χ < 0 in the right region of the MTPA line, i.e., d(I 2 ) and χ form an obtuse angle. Therefore, I 2 decreases along the χ direction, indicating that the MTPA solution exists in the left section of N2 . This implies that the MTPA solution, M2 lies inside the voltage limit. Therefore, the intersection, M2 is the desired minimizing solution. The aforementioned statements can be summarized as I 2 minimizing (i∗d , i∗q ) =

N : intersection with voltage limit, Case ii),

if dI 2 , χ > 0,

M : intersection with the MTPA, Case i),

if dI 2 , χ < 0. (35)

A typical torque–speed curve is shown in Fig. 8(a), and the curve is divided into four segments. The corresponding current contour is shown in Fig. 8(b). In Segment I, the torque remains constant. This torque excursion is depicted as a point, “P” in the

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Fig. 10. Torque control block diagram with proposed current command selection algorithm. TABLE I IPMSM PARAMETERS

Fig. 9.

Flowchart of current command selection algorithm.

current plane. Obviously,“P” is a solution on the MTPA line. As soon as the voltage limit is touched, the solution should follow the intersections with the voltage limits (see Segment II). Correspondingly, the current contour is depicted by P ∼ Q. Above the speed, ω2 (see Segment III), the torque should be reduced. Otherwise, no solution is found within the current limit. In Segment III, the maximum torque is found at the intersection between the voltage and current limits. Because the torque and voltage limit curves also meet, the solutions in Segment III belong to). In Segment IV, the maximum torque is found at the tangential intersection between the torque and the voltage limit curves. The contour (R–S) of these intersection points is called maximum torque per flux (MTPF) line [31]. It converges to (−ψm /Ld , 0) as the speed increases to infinity. The MTPF solutions of Segment IV correspond to Case ii)—tangential intersection of equal roots. VI. CURRENT COMMAND SELECTION AND TORQUE CONTROL ALGORITHMS Here, we consider the variation of Ld and Lq , which affects the coefficients of the quartic polynomials, (16) and (17). This variation will make it difficult to find the right solution as a result of the circular reasoning: the values of Ld and Lq are not correctly estimated without the proper current information, and the current, (i∗d , i∗q ) of (27), (28), and (31) cannot be evaluated correctly without proper values for Ld and Lq . To resolve this problem, we use a step-by-step recursion method: the nominal values of Ld and Lq are used to find an optimal current solution. Using this solution, we recalculate Ld and Lq and then these updated Ld and Lq values are used to obtain the next current solution.

Variable Number of poles (P) Maximum speed Maximum power Maximum torque Maximum current (Is ) DC-link voltage (VDC ) Permanent magnet flux (ψm ) Nominal d-axis ind. (Ld ) Nominal q-axis ind. (Lq ) Stator resistance (rs )

Value 8 10000 RPM 70.0 kW 185 Nm 176.7 A (rms) 360 V 0.099 Wb 0.312 mH 0.606 mH 16.9mΩ

A. Current Command Selection The proposed current command selection algorithm is depicted in the flow chart shown in Fig. 9. For a given torque reference T0k , the current command set (ikd ∗ , ikq ∗ ) is given at the end of the flowchart. It starts with reading Ld and Lq from the lookup table with reference to the previous current command (k −1)∗ (k −1)∗ (k −1)∗ (k −1)∗ , iq ), i.e., it reads Lkd −1 ≡ Ld (id , iq ) set (id (k −1)∗ (k −1)∗ k −1 and Lq ≡ Lq (id , iq ). The next step is to check whether or not the command torque is greater than the tangential torque Ttan which is the maximum k , then it torque available within a voltage limit. If T0k ≤ Ttan k k proceeds to the next step. If T0 > Ttan , it stops and requests a reduced torque command. In what follows, the intersection with the voltage limit (idβ , iq β ) is calculated by utilizing (28). Then, on the basis of criterion, (35), it determines whether or not the MTPA solution is feasible. If dI 2 , χ < 0, it calculates the MTPA solution (idα , iq α ) using (27) and uses it as the current command. Otherwise, Case ii) solution, (28) is selected as the current command. Finally, it checks whether or not the solution is inside the current limit. If the solution is outside the current limit, the flowchart ends and requests a reduced torque command. Otherwise, either (idα , iq α ) or (idβ , iq β ) is output as the current command set (ikd ∗ , ikq ∗ ). B. Torque Control Block Diagram The torque control block diagram is shown in Fig. 10. It contains the previously described current command selection algorithm. For the current loop, a conventional proportional and


Fig. 12.

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Experimental equipment: dynamometer, IPMSM, and inverter.

where Kp and Ki are the PI-gains of the current controller.

VII. SIMULATION RESULTS A simulation was performed using the MATLAB Simulink. Table I lists the parameters of an IPMSM developed for EV propulsion. In the simulation, the torque control loop was constructed like the block diagram shown in Fig. 10. In this simulation scenario, the motor speed was 2000 r/min initially, and it increased to 8000 r/minat the end. The MTPA operating points were selected from 0 to 0.06 s corresponding to Case i). This operating period is marked with “P” in Fig. 11(e). Above 3500 r/min, the voltage limit was active. Thus, the intersections with the voltage limits were selected as the operating points corresponding to Case ii). Fig. 11(d) shows that the voltage limit was hit at t = 0.06 s. The constant torque operation contour is described by a curve P ∼ Q shown in Fig. 11(e). From 6700 to 8000 r/min, the torque should be reduced because of the current limit. Thus, the available maximum torque is determined by the intersection between the voltage and the current limit curves. The torque drops to 82 Nm at the end. Here, the plot of d(I)2 , χ is shown in Fig. 11(c). It should noted that d(I)2 , χ = 0 correctly indicates the boundary between Case i) and Case ii), as was illustrated in (35). It is worthwhile to note that the solutions of the fourth-order equations were obtained by using Ferrari’s method.

VIII. EXPERIMENTAL RESULTS Fig. 11. Simulation results: (a) speed and torque, (b) d- and q-axes currents, (c) dI 2 , χ, (d) current and voltage magnitudes, and (e) contour of (id , iq ).

integral controller is used along with decoupling and back EMF compensation Ki vdk = Kp + (36) (ikd ∗ − ikd −1 ) − ωek Lkq −1 ikq −1 s Ki k vq = K p + (ikq ∗ − ikq −1 ) + ωek Lkd −1 ikd −1 + ωek ψm (37) s

Experiments were performed using an EV motor with the parameters shown in Table I. The same Ld and Lq lookup tables were used in the experiments. Fig. 12 shows the dynamometer, in which the load motor governed the shaft speed, and the EV motor controlled the shaft torque. The torque and speed were monitored through the instruments installed in the dynamometer, and the inverter output voltage and current were monitored via a power analyzer. A Freescale MPC5554 was used as the inverter microcontroller, in which floating point calculations were executed to find the solutions of the fourth-order polynomials.

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Fig. 13.


Comparison of the analytic solutions with the current minimizing points found experimentally: (a) 1500 r/min. (b) 3200 r/min. (c) 6000 r/min.

Fig. 14. Total losses of the motor along constant torque curves: (a) 1500 r/min. (b) 3200 r/min. (c) 6000 r/min. Analytic solutions obtained by the proposed method were compared with the minimum loss points searched experimentally.

A. Current Minimizing Performances and Loss Minimization Efforts Fig. 13 showed several constant torque curves found experimentally. Also, the mark, “ ” indicates the point of minimum magnitude in each constant torque line that is found experimentally. The mark, “ ” is the point obtained from the proposed current minimizing algorithm. It should be emphasized that two sets of points coincide in all operating range. Fig. 13(a) shows a case when the motor speed was running at 1500 r/min. Since the voltage limit did not act, the marked points denote the contour of MTPA line. Fig. 13(b) shows shows a hybrid case at 3200 r/min, in which some points were found on the MTPA line, the others on the intersections with the voltage limit. Fig. 13(c) shows a case of 6000 r/min in which all the current minimizing solutions were found on the intersections with the voltage limit. Fig. 14 shows the total losses of the motor under the same operating conditions as in Fig. 13. The losses were measured by taking the difference between√the supplied electric power and the shaft power, i.e., Ploss = 3VLL I cos φ − T · P2 ωe , where VLL is the line-to-line voltage, I is the phase current, and φ is the power factor angle. The losses were plotted versus d-axis current. The minimum loss points found through repeated experiments were marked with “”, whereas the analytic solutions

resulted from the proposed algorithm were marked with “ ”. Note that they coincide well in these experiments except one point. The exception occurred at 3200 r/min with 50 Nm, in which the minimum loss point was discovered with a slightly lager current. The main reason seemed to be related with the iron loss. Fig. 15 shows an overall efficiency map of the IPMSM obtained from experiments. The efficiency was calculated ac√ cording to T · P2 ωe / 3VLL I cos φ. The maximum efficiency is 0.975, and the average efficiency in the frequent operating region is about 0.95. In order to demonstrate the capability of finding the MTPA points, we made an experimental scenario. In the first part, the proposed algorithm was switched OFF, and current angle was controlled manually in a constant torque control mode. Current angle was selected according to iq : 90◦ → 130◦ → 90◦ . tan−1 id It should be noted that the current magnitude is minimized at the MTPA point. Note also that the MTPA solutions locate i when current angles are tan−1 ( i dq ) = 107.3◦ and 114.5◦ for T = 50 and 100 Nm, respectively. In the second part, the proposed


Fig. 15.

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Efficiency map of the IPMSM used in the experiments.

Fig. 17. Execution times of current command selection algorithm: (a) dI 2 , χ > 0 (Case ii)). (b) dI 2 , χ < 0 (Case i)).

Fig. 16. Comparison with the MTPA points before and after the proposed algorithm turned on at 1500 r/min: (a) 50 Nm and (b) 100 Nm.

algorithm was turned ON suddenly. Then, the MTPA point was calculated, and currents were regulated to that point in a step manner. The corresponding results are shown in Fig. 16 for T = 50 and 100 Nm. Fig. 17 shows the execution times for the solutions of the quartic equations. As shown in Fig. 9, there are two paths in the current command selection algorithm: one without the MTPA (Case ii)) and the other with the MTPA (Case i)). Each execution time is divided into three parts. The first part includes

the calculation time for the tangential torqueTtan intersection, (idβ , iq β ) with the voltage limit, and dI 2 , χ. The second part indicates the calculation time for the MTPA solution, (idα , iq α ). The third part is the time required for the remaining vector control operations such as A/D conversion, current control, and space vector modulation. To see the computation time interval, we used flags to designate the start and end of the computation blocks. Fig. 17(a) shows a case where dI 2 , χ > 0. The overall computation time for the current control is 41.7 μs, out of which the computation time for (idβ , iq β , ) is 23.2 μs. Fig. 17(b) shows a case where dI 2 , χ < 0. In this case, the MTPA solution is also required. The computation time for (idβ , iq β , ) is the same as before (23 μs). The additional time for the MTPA solution is 16.3 μs. Thus, the overall time is 57.8 μs. Considering the PWM interval (143 μs), the overall computation time is sufficiently short (40.4% of the PWM interval). From these results, it can be concluded that solving quartic equations online in the torque-current control loop is not cumbersome at all. To see the importance of Ld , Lq updates depending on the current, two torque control experiments were performed at 6000 r/min. Fig. 18(a) shows the large torque error when the fixed values of Ld = 0.312 mH and Lq = 0.606 mH were used. Current offsets were caused by voltage saturation. Fig. 18(b) shows a case with no error when Ld and Lq were updated depending on the current change using the proposed algorithm. This shows the necessity of updating Ld and Lq in the quartic polynomials.

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Fig. 18.


Torque control accuracy: (a) with fixed inductance values and (b) with L d and L q adjustments based on lookup table.

An experiment was performed under the same operation scenario as the simulation except for the time scale. The torque reference was set at 100 Nm and the shaft speed was accelerated from 2000 to 8000 r/min. The current command selection algorithm and the control block were constructed according to Figs. 9 and 10, respectively. Initially, the voltage did not affect the current command selection. Thus, dI 2 , χ < 0 and the MTPA solution should be sought. As soon as the voltage limit became active, the current contour followed the sequence of intersections, P ∼ Q. The active sign of the voltage limit was indicated by dI 2 , χ ≥ 0. Specifically, it should be noted that vd2 + vq2 reached the maximum (phase) voltage (200 Vp eak ) at

the moment when dI 2 , χ = 0, as shown in Fig. 19(d) and (e). Finally, the torque was reduced along the current limit circle. In that period, the current magnitude was maintained at 250 A peak, as shown in Fig. 19(f) and (g). The shaft torque was kept at 95 Nm before reaching Q. That is, there was a 5 Nm offset in the shaft torque. This difference was caused by the acceleration torque resulting from rotor inertia during the speed change. B. Comparative Study The proposed method was compared with a conventional lookup table method. From the lookup table, a pair of current command, (i∗d , i∗q ) were read for each torque and speed. It contained 10 × 10 data, which were formed through a priori

experiments. The speed step was 1000 r/min and the torque step was 20 Nm. A linear interpolation was used to find the desired current command values. Fig. 20(a) showed locations of current commands based of n the lookup table and the proposed algorithm. The differences tended to be large as the q-axis current grew. This kind of discrepancy might result from the coarse lookup table data. Power losses were compared in Fig. 20(b). In all cases, the proposed method resulted in less losses than the lookup table method. C. Parameter Robustness To demonstrate a robustness of the proposed algorithm against the parameter variation, we utilized inexact inductance values intentionally. During the experiments, motor shaft speed were fixed by the dynamometer motor at 2000 r/min. Since erroneous parameters were utilized in the torque control algorithm, a torque offset took place. In such a case, we adjusted the torque command level until the shaft torque reached a desired value, 100 Nm. At that time, we read the varied current command levels that came out of the current minimizing torque control ∗2 algorithm, and evaluated I = i∗2 d + iq . We repeated experiments for different levels of inductance errors: (Ld − L∗d )/L∗d = −0.3 ∼ 0.3 and (Lq − L∗q )/L∗q = −0.3 ∼ 0.3. Fig. 21 shows an aspect of robustness test results. Specifically, it shows the magnitudes of current commands and its percent variation versus


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Fig. 19. Experimental results: (a) speed profile set by dynamometer load motor, (b) shaft torque, (c) d- and q-axes currents, (d) dI 2 , χ, (e) current magnitude (peak value), (f) voltage magnitude (peak value), and (g) contour of (id , iq ).

inductance errors. The current minimizing algorithm itself performed well even with ±30% inductance errors. As can be expected, the magnitude of current command is minimum (MTPA) when there is no parameter error, and the command level tends to increase as the inductance error grows. Note also that the controller is more sensitive to the variation of Lq than that of Ld . In the second experiment, we applied suddenly a false value of dc-link voltage to the algorithm by letting to VDC = 300 V, when the real dc-link voltage was fixed at 360 V. Fig. 22 shows plots of dc-link voltage, torque, current magnitude, and angle when the motor was running at 5000 r/min with a 100-Nm load. The situation corresponded to Case II, where the current commands were on the intersection with a voltage limit. Note that the current magnitude was increased from 186 to 229 A at the time of the false value injection, while the torque was kept at 100 Nm. It demonstrated a proper reaction of the algorithm, since the both current magnitude and angle were increased against a voltage drop. In other words, a larger negative d-axis current was chosen to counteract the voltage drop. On the other hand, a reverse phenomenon would occur if a larger VDC is used than the real one. In such a case, a larger voltage vector will be produced in the current controller, and will be limited eventually by a voltage limiting block.

Fig. 20. Comparison between a lookup table method and the proposed method: (a) Locations of current commands and (b) power losses for given speeds and torques.

IX. CONCLUDING REMARKS The current minimizing torque control method was considered for an IPMSM. In producing a desired torque, the current minimizing solution was sought. If the voltage limit was not active, it was possible to use the MTPA solution. Otherwise, it was necessary to use the intersection value between the voltage limit and torque curves. In both cases, quartic polynomials were induced, and Ferrari’s method was used to obtain general solutions for all of the quartic polynomials. The proposed method was applied to a 70 kW, eight-pole IPMSM developed for EV propulsion. The EV propulsion motors are characterized by a wide torque-speed range. They need to generate a high torque in a low speed region for starting, and extend a constant power mode to more than 10 000 r/min. Under this requirements, the current vector should cover a wide range, even saturating the stator and rotor cores. Correspondingly, the inductance variation is significant. Hence, lookup table methods are widely used to accommodate the inductance change. In this

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Fig. 21.


Current command variation against L d and L q errors: (a) (L d − L ∗d )/L ∗d = −0.3, (b) L d = L ∗d , and (c) (L d − L ∗d )/L ∗d = 0.3.

REFERENCES

Fig. 22. 300 V.

Current command variation against dc-link voltage error; 360 V →

study, we proposed a method of solving the current minimizing problem analytically by utilizing the Ferrari’s method in all operating region. The proposed torque control method looks attractive based on the following facts. 1) It offers a current minimizing solution for a given torque command across the whole range of torque and speed, i.e., the MTPA and voltage limit solutions are integrated into a single context. 2) The variations in Ld and Lq are reflected in a recursive manner. 3) This method eliminates the use of complex current lookup tables, which should be prepared separately for different -link voltages and mechanical speeds. Thus, this analytic method simplifies the torque control algorithm and shortens the inverter development time. Through simulation and experiment results, it was shown that the current command of minimum length was obtained online by the proposed algorithm for a given torque command. The performances were checked in the field-weakening region, as well as the MTPA region. But, the proposed algorithm does not give a total loss minimizing solution, because the iron loss is not considered in the optimization cost. In the future, the Ferrari’s method needs to be extended to finding the total loss optimization.

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Sung-Yoon Jung (S’07) was born in Seoul, Korea, in 1979. He received the B.S. degree in electrical and electronic engineering from Hanyang University, Seoul, Korea, in 2005, and the M.S. and Ph.D. degrees in electrical engineering from the Pohang University of Science and Technology, Pohang, Korea, in 2008 and 2013, respectively. His main research interests include design of inverter and converter, design of ac motor, and control and drive of PMSM, especially for electric vehicle. Jinseok Hong (S’08) was born in Incheon, Korea, in 1982. He received the B.S. degrees in electrical engineering and mechanical engineering from Hongik University, Seoul, Korea, in 2007, and the M.S. degree in electrical engineering, in 2009, from Pohang University of Science and Technology (POSTECH), Pohang, Korea, where he is currently working toward the Ph.D. degree. His research interests are design, analysis and control of power electronic systems, especially ac motor drive, and electric vehicle. Kwanghee Nam (S’83–M’86) was born in Seoul, Korea, in 1956. He received the B.S. degree in chemical technology, and the M.S. degree in control and instrumentation engineering from Seoul National University, Seoul, Korea, in 1980 and 1982, respectively. He received the M.S. degree in mathematics and the Ph.D. degree in electrical engineering from the University of Texas, Austin, TX, USA, in 1986. From 1998 to 2000, he was the Director of the Information Research Laboratories and the Dean of the Graduate School of Information Technology, Pohang University of Science and Technology (POSTECH), Pohang, Korea, where he is currently a Professor in the Department of Electrical Engineering. He is an author of a book, AC Motor Control and Electrical Vehicle Applications, (CRC Press). His current research interests include ac motor control, power converters, motor design, and electric power train.