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Maximum torque per ampere control with direct voltage control for IPMSM drive systems
Electrical Power and Energy Systems 116 (2020) 105509
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Maximum torque per ampere control with direct voltage control for IPMSM drive systems
T
⁎
Mostafa Malekpoura, Rasoul Azizipanah-Abarghooeeb, , Vladimir Terzijab a b
Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran School of Electrical and Electronic Engineering, The University of Manchester, Manchester M13 9PL, UK
A R T I C LE I N FO
A B S T R A C T
Keywords: Direct voltage control Interior permanent magnet synchronous motor Maximum torque per ampere Parameter variation
This paper proposes an analytical method to determine maximum torque per ampere (MTPA) trajectories in the voltage plane for interior permanent magnet synchronous machines (IPMSMs). The MTPA is formulated as an optimization problem and its solution is obtained within a closed-form function, from the current and voltage planes. In consequent, the MTPA tracking control strategy with the novel closed-loop direct voltage control is developed for the IPMSM drive systems. Comparison studies against the well-known conventional current regulation-based MTPA tracking control strategy illustrate the superiority of the developed strategy in various aspects as well as its efficient MTPA tracking capability. Furthermore, it also shows more robustness to the IPMSM’s parameter variations than conventional strategy from voltage constraint point of view.
1. Introduction Interior permanent magnet synchronous motor (IPMSM) has been employed in many electrified transportations due to its high efficiency and power density [1]. Maximum torque per ampere (MTPA) speed control strategy has been widely adopted to achieve high efficiency of an IPMSM drives in constant torque region [2,3]. At first, the MTPA current trajectory of IPMSMs is achieved by drawing the constant torque loci in the current plane and finding the nearest points on torque loci to the origin [2]. In this analytical technique, the direct and quadrature components of the motor current are obtained to attain MTPA operation by numerical solving of a quadratic function [4,5]. Albeit IPMSM has a non-linear model, the conventional proportionalintegral (PI) is the most popular current regulator for tracking the MTPA [6]. The analytical MTPA methods usually deploy the IPMSM’s parameters to compute the MTPA trajectory. However, these parameters might change due to cross-magnetization saturation effects and temperature. As a result, the MTPA loci computed analytically has error [7,8]. In order to address this issue, model-based online estimation algorithms are utilized to estimate parameters of the motors [9,10]. In the other hand, signal injection-based control strategies are employed as parameter independent MTPA tracking schemes to meet the parameter variation issue [11,12]. In these schemes, a high-frequency
current signal is injected into the motor. After which, the MTPA point is online searched in these strategies by utilizing this observation that at MTPA points the variation in torque associated with the variation in the current angle is zero. Whilst this method is insensitive to parameters variation, measuring the instantaneous values of load torque and the complexity of it increases its cost of application. All analytical techniques for MTPA tracking of IPMSM drives need closed-loop current regulation to indirectly determine the motor’s reference voltages. The reference currents are also obtained based on reference torque and MTPA trajectory. However, the reference voltage can be directly determined from the reference torque to achieve MTPA operation [13,14]. Consequently, no explicit current regulation is required. In [13], the IPMSM’s voltage amplitude and angle are provided in terms of the electromagnetic torque and motor speed in order to obtain a relation between the reference torque and the voltage amplitude and angle. Whereas this method simplifies the control structure, any analytical method was not established to derive MTPA trajectory in the voltage plane. In particular, the torque-voltage characteristic was derived through simulations for a given motor. Moreover, the motor voltage regulation was conducted using an open-loop controller [13]. Recently, a current sensor-less MTPA controller is presented in [14] for the IPMSM motors. However, any analytical solution was not proposed to determine MTPA curve on the voltage plane and it is instead calculated using a numerical approach [14].
⁎
Corresponding author. E-mail addresses: [email protected] (M. Malekpour), [email protected] (R. Azizipanah-Abarghooee), [email protected] (V. Terzija). https://doi.org/10.1016/j.ijepes.2019.105509 Received 18 July 2019; Accepted 22 August 2019 0142-0615/ Crown Copyright © 2019 Published by Elsevier Ltd. All rights reserved.
Electrical Power and Energy Systems 116 (2020) 105509
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presented in (5) are portrayed in current plane [4]. The current limitation is a circle with the radius of Is,max. In other hand, the voltage restrictions are illustrated with discrepant ellipsoids corresponding to different speeds. For the sake of simplicity in determining the ellipsoids, an internal voltage is defined as [4]:
In this paper, the analytical method to achieve MTPA trajectory in the current plane is tersely explained. Moreover, whilst the nominal point is approximated as in [4], the accurate one is calculated for the IPMSM in this study. After that, the voltage-based MTPA trajectory expression is obtained by drawing the current and voltage restrictions of IPMSM in the voltage plane, instead of current one used in conventional approaches. Additionally, the MTPA is formulated as a constrained optimization problem to justify that MTPA trajectories can be expressed as closed-form functions either in the current or voltage planes and no numerical calculations to solve MTPA problem and store the solution to the look-up tables are needed. Furthermore, a novel MTPA tracking control strategy with closed-loop direct voltage regulation is designed. Additionally, it is noticed from the simulation results extracted from the proposed model in MATLAB Simulink environment that the proposed MTPA tracking capability based on direct voltage control is as accurate as that of traditional current regulation. Moreover, it also shows more robustness to the IPMSM’s parameter variations than conventional strategy from voltage constraint point of view, so that the stability margin around the nominal point is increased. The rest of this paper is organized as: Section 2 outlines the IPMSM modelling. The detailed analytical MTPA strategy is presented in Section 3. The conventional MTPA scheme is tersely explained. After which, the proposed strategy to achieve MTPA trajectories in the voltage plane is explained. In Section 4, the derived MTPA tracking control strategy with closed-loop direct voltage regulation is simulated in MATLAB Simulink. In Section 5, the simulation results are provided. Finally, the paper is concluded in Section 6.
Vso,max = Vs,max − Rs Is,max By employing (1) and (6), it yields:
Vqso = ωr (Ld ids + λ f ), Vdso = −ωr Lq iqs
(ωr Lq )2iqs2 , ref + (ωr Ld )2 (ids, ref + Ld−1 λ f )2 ⩽ Vo2,max
idsMTPA , ref =
Te − TL = J
idsMTPA , nom =
−λf 4(Ld − Lq )
−
(0.25λ f (Ld − Lq)−1)2 + (0.5Is,max )2
(10)
Accordingly, it can be directly calculated. Furthermore, the rated values of torque and speed can be achieved by (3) and (8). Albeit, there are some assumptions in determining nominal values, the accurate computation is deployed in this study. In other words, the value of Vso,max obtained by (6) is lower than its true one leading to the lower calculated nominal speed than its true one. In order to remove this assumption, the voltage limitation is rearranged using (1) and (5) as follows:
(1)
(2)
2 2 2 2 Vqso , nom + Vdso, nom = (Vqs, nom − Rs iqs, nom ) + (Vds, nom − Rs ids, nom )
(3)
(11)
This can be simplified as follows:
Vso2 ,max = Vs2,max + Rs2 Is2,max −
(4)
4 Rs Pin 3
(12)
with Pin as the input active power of motor. If the left hand side of (12) is substituted with that of (8) and Pin is rewritten in terms of torque, speed and loss of motor, the following polynomial can be extracted to determine rated speed.
where, J is the moment of inertia of the rotor and the connected load is in kg.m2. TL denotes the load torque. 3. The MTPA control strategy
aω ωr2, nom + bω ωr , nom + cω = 0 In this section, the conventional MTPA method is firstly explained. Then, the proposed MTPA technique and MTPA trajectory determination approach are detailed.
(13)
where,
aω = (Ld ids, nom + λ f )2 + (Lq iqs, nom )2 bω =
3.1. The conventional MTPA control strategy
8 RT , 3P s e, nom
cω = (Rs Is,max )2 − Vs2,max
(14)
The accurate nominal speed of motor deployed in this paper is 4% more than the approximated value taken from traditional technique. Moreover, while the nominal values of speed and q-d currents are placed in (1), the maximum value of voltage would be equal to the maximum voltage of motor. The current and voltage limitations beside the MTPA trajectory are portrayed in Fig. 1 using data presented in Table 1 for the studied IPMSM. Additionally, the torque loci for rated torque and its half value are illustrated in this figure. It can be seen that
The IPMSM motors can tolerate the currents and voltages less than their maximum limits. Thus, the equilibrium point of these motors should be imposed by current and voltage constraints. This can be formulated as follows: 2 2 2 2 2 iqs2 , ref + ids , ref ⩽ Is,max , Vqs, ref + Vds, ref ⩽ Vs,max
(9)
Therefore, while According to (9), iqsMTPA be determined in , ref is changed from zero the range of zero and its nominal amount. Based on the traditional method in specifying the nominal values of currents, iqs2 , ref is rearranged in (9) in terms of id and Is,max obtained from (5). In this regard, the nominal value of id in MTPA trajectory is expressed as follows:
where, P is number of poles. Finally, rotor speed can be determined by the equation of motion as follows:
(p (2P −1ωr ))
2 (0.5λ f (Ld − Lq)−1)2 + (iqsMTPA , ref )
−
is also function of iqsMTPA , ref . to its rated value, idsMTPA , ref will
with Lq and Ld as q- and d-axis inductances, respectively. p is d/dt, and ωr denotes the angular velocity of the rotor. The electromagnetic torque is expressed:
3P (λ f iqs + (Ld − Lq ) iqs ids ) 22
−λf 2(Ld − Lq )
idsMTPA , ref
where,
Te =
(8)
While iq is calculated in terms of id using (3) and substituted in the current limitation (5), then the derivation of two sides of this limitation is computed and equated to zero, is yielded in the following form to achieve MTPA [2,4]:
The voltage equations of the IPMSM in the rotor reference frame can be written as follow [3]:
λqs = Lq iqs , λ ds = Ld ids + λ f
(7)
Substitution of (7) in the voltage restriction of (5) gives its currentbased description. The centre of these ellipsoids located in horizontal axis, i.e., d-axis current. The ellipsoids’ area is decreased from infinite to a specific value by growing motor’s speed from zero to the nominal one.
2. The IPMSM model
Vqs = Rs iqs + pλ qs + ωr λ ds Vds = Rs ids + pλds − ωr λqs
(6)
(5)
In order to determine the MTPA trajectory, the constraints 2
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Fig. 1. MTPA trajectory in current plane.
Table 1 Parameters of the studied IPMSM [15]. Number of poles P
4
d-axis inductance Ld
9.67 mH
Armature resistance Rs Magnet flux linkage λf q-axis inductance Lq
0.824 ohm 78.5 mWb 24.3 mH
Max. current Is,max Max. voltage Vs,max Rotor inertia J
8.66 A 52 V 0.005 kg.m2
Current Regulation
ωr,ref +
PI −
Te,ref
ωr
q-dec
iqs,ref MTPA ids,ref
+
− + −
ωr
q-dec = ωr(Ldids+λf )
qd abc
d-dec = ωr(Lqiqs )
+
PI
+
+
Vqs,ref Vds,ref
d-dec
Fig. 3. Voltage and current constraints in voltage and internal voltage planes.
PWM Inverter
a = g (Rs2 + ωr2 Ld2), b = g (2Rs ωr (Lq − Ld ))
−
c = g (Rs2 + ωr2 Lq2), d = g (2Rs λ f ωr2 (Ld − Lq))
ωr
is
θr
d/dt
PI
e = −g (2λ f ωr (Rs2 + ωr2 Lq2))
IPMSM
f=
the MTPA trajectory is the operating points on different torque locus subject to the lowest distance to the origin. The block diagram of a conventional MTPA tracking is shown in Fig. 2. In this scheme, the speed error is passed through a PI controller to determine the reference torque. Then, the q-d reference currents corresponding to this torque is calculated by MTPA block. It is to be noticed that the MTPA trajectory is already computed and saved as Look-up table in this block. After which, a current controller using two PI controllers tune the references of inverter voltages so that the measured currents would be equal to their corresponding references.
2 2 −2 2 (ωr Lq)−2Vdso , ref + (ωr Ld ) (Vqso, ref − ωr λ f ) ⩽ Is,max
In the proposed method, the MTPA trajectory is obtained in voltage plane. It is clear-cut that the voltage limitation is a circle with the radius of Vs,max. In other hand, different ellipsoids are corresponded to the current limitations with discrepant speeds. In order to achieve the relations of these ellipsoids, the q-d currents are calculated using (1) as follows:
Rs2
+
ωr2 Ld Lq
, ids =
Vdso = K3 ωr2 [ωr λ f + K2 (Vqso − ωr λ f )]−1
+
ωr2 Ld Lq
(19)
K = 0.75P , L = Ld − Lq , K1 = −(Lq)−1K , K2 = (Ld )−1L, K3 = (K1)−1Te
(15)
(20)
After that, Vdso obtained by (18) is located in (19) to give:
Substitution of (15) in (5) results in ellipsoids’ mathematical functions as follows: 2 2 aVds , ref + bVds, ref Vqs, ref + cVqs, ref + dVds, ref + eVqs, ref + f = 0
(18)
where,
ωr Lq (Vqs − λ f ωr ) + Rs Vds Rs2
(17)
As can be seen from the bottom of Fig. 3, the ellipsoid’s rotation is eliminated. The origin of each ellipsoid has a distance equal to ωr λ f to the origin over the vertical axis. The MTPA trajectory is determined in internal voltage plane. In this context, Vdso can be calculated by deriving q-d currents from (7) and locating them in (3) as follows:
3.2. The proposed MTPA control strategy
Rs (Vqs − λ f ωr ) − ωr Vds
+ ωr2 Lq2)) − Is2,max , g = (Rs2 + ωr2 Ld Lq)−2
The voltage limitation beside the current ones is shown in the upper plot of Fig. 3 for different speeds. It is observable that the ellipsoids’ centres are moved up over the vertical axis and their areas are raised following the speed increment. However, the rotation of ellipsoids is mitigated. In order to make the calculation of MTPA trajectory easy, voltage and current limitations are plotted in internal voltage plane. To determine the current limitations, the q-d currents obtained by (7) are placed in (5) to yield:
Fig. 2. Conventional MTPA tracking implementation with current regulation.
iqs =
g (λ f2 ωr2 (Rs2
(ωr Lq)−2 (K3 ωr2 )2 (ωr λ f + K2 (Vqso, ref − ωr λ f
(16)
))2
+
(Vqso, ref − ωr λ f )2 (ωr Ld )2
⩽ Is2,max
(21)
By using the derivative operator for both sides of (21), the following relation is yielded:
where, 3
Electrical Power and Energy Systems 116 (2020) 105509
M. Malekpour, et al. 2 −2K2 (ωr Lq)−2Vdso , ref
(ωr λ f + K2 (Vqso, ref − ωr λ f ))
+
2(Vqso, ref − ωr λ f ) (ωr Ld )2
=0
(22)
The acceptable solution of (22) provides the reference of Vqso for achieving MTPA. It has a dual relationship with (9) as:
Vqso, ref =
−ωr λ f (1 − 2K2 ) 2K2
2
+
2
⎛ ωr λ f ⎞ + ⎜⎛ Ld Vdso, ref ⎟⎞ Lq ⎠ ⎝ 2K2 ⎠ ⎝
⎜
⎟
(23)
Accordingly, it is to be pointed out that there is a MTPA trajectory corresponding to each value of motor’s speed. For instance, as shown in Fig. 4, the curve located between A and B is the representative of MTPA trajectory for nominal speed. While the torque increases from zero at point B, to nominal values at point A. Based on torque loci for nominal speed, the BA curve consists of operating points on torque loci with the least distance to centre of ellipsoid of nominal speed. While the reference torque is zero, the solution of (22) is:
Vdso, ref = 0, Vqso, ref = ωr λ f
Fig. 4. MTPA trajectory for nominal speed in internal voltage plane.
(24) 1
This is illustrated by blue dotted line in Fig. 4. Indeed, while the motor’s speed increases from zero to its rated value under zero torque, the operating point moves from origin to point B. This line is corresponded to the origin in Fig. 1. It means that while the reference torque is zero, it is impossible to determine operating point of motor, at current plane. However, this is doable in voltage plane. The MTPA trajectories are portrayed in Fig. 5 for different speeds under the step of 0.2 p.u. In a given speed, Vqso and Vdso are respectively mitigated and grown following the torque increment from zero to nominal value. In other hand, both voltage amplitudes are increased after speed increment from zero to nominal value under a given torque. Furthermore, while torque is constant, the ratio of these voltages, i.e., voltage angle is constant. In [13], the applied voltage angle and amplitude variations are portrayed in terms of speed and torque variations. It is observed that the slope of the voltage-speed variation is proportional to the square of the applied voltage angle. As a result, a MTPA speed controller is designed without current or voltage regulation loops. Moreover, it is claimed that this technique is useful for all IPMSMs and it is independent of motor parameters variations. However, these claims are not valid. Since the square proportionality between the above noticed two quantities is firstly an approximation and secondly it is dependent to motor parameters. Furthermore, based on information explained for Fig. 5, it can be concluded that the MTPA trajectories change in voltage plane with the parameters variation. Consequently, their relationships with speed or torque will be varied.
Fig. 5. MTPA trajectories for different speeds in internal voltage plane.
currents’ values lower than Is,max while speed is lower than it’s nominal. Additionally, it would be observed that the solution of the problem, i.e., the MTPA trajectory is located within the circle of current limitation. As a result, the only constraint that should be taken into account is the torque relation of (3). Before solving the proposed MTPA problem, it’s better to graphically investigate and draw it. In Fig. 6, ‘OF’ is plotted for a three-dimension curve corresponding to a torque equal to the 20% of nominal one. However, ‘OF’ is normalised based on its value in the nominal operating point. It can be seen that the projection of ‘OF’ on current plane is exactly torque locus. Given the torque constant for torque locus, it is observed that the global optimum, i.e., A1, is precisely placed on the torque locus where it has lowest distance to the origin, i.e., A2. The main purpose of (25) is to find A1 points for different torque’ values from zero to nominal one. In this regard, the current square is rewritten considering the limitation (3) as follows:
3.3. The MTPA as an optimization problem Before proceeding, the MTPA is detailed as an optimization problem. To the best of authors’ knowledge, this problem has not been appropriately defined and solved in literature. Whereas the final response of the state-of-the-art approaches is correct, however, they could not obtain a closed-form solution for MTPA trajectory. This solution overcomes the problem related to look-up table requirement and online calculation to update data. According to the explanation of MTPA, current should be selected for a given torque in order to maximize the objective function (OF), i.e, torque to current ratio as follows:
OFMTPA =
Te ∝ Is
Te2 Is2
Is2 = iqs2 , ref + (Te − Kλ f iqs, ref )2 (LK iqs, ref )−2
(26)
Substitution of (26) in (25) yields:
OFMTPA =
(Te K L)2iqs2 , ref (Te K L)2 iqs4 , ref + (Te − Kλ f iqs, ref )2
(27)
By differentiating of (27) in terms of iqs,ref and equating it to zero, a very depressed quadratic function can be obtained:
ai iqs4 , ref + di iqs, ref + ei = 0
(25)
The current and voltage limitations of (5) as well as (3) constitute the constraints of the corresponding optimization problem. Based on the above explanations, the voltage restriction is satisfied for all
(28)
where,
ai = K 2L2 , di = Kλ f Te, ei = −Te2
(29)
One of the acceptable solution for (28) is as follows: 1
For interpretation of color in Fig. 4, the reader is referred to the web version of this article.
iqs, ref = −Si + 0.5 −4Si2 + qi Si−1 4
(30)
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Voltage Regulation
ωr,ref +
PI
−
ωr
Vqso,ref
Te,ref
MTPA V dso,ref V−I ids,ref iqs,ref
Vqs,ref
+
+
+
Vds,ref−
Rs
+ −
qd abc d/dt
θr
PI
+
PI
+
Vs
Vqs,ref
PWM Vds,ref Inverter
IPMSM
Fig. 7. The proposed MTPA tracking implementation with voltage regulation.
direct voltage control is proposed and designed. This is illustrated in Fig. 7. This is the dual model of the conventional technique shown in Fig. 2, so that ‘Current Regulation’ is replaced with ‘Voltage Regulation’. The ‘MTPA V-I’ block determines the reference values of current and internal voltage using (30) and (36). Then, the reference voltage of inverter’s output is specified by adding the resistance voltage drop to internal voltage. The values of reference voltages are updated through two PI controllers to guarantee the equality of the measured voltages with the reference ones. In comparison with traditional method, the proposed architecture measures the motor’s voltage instead of its current. The way of determining the output reference voltage of inverter presented in [13] is quite different from that of the proposed method in this study. Moreover, there is no voltage feedback in [13] and the voltage regulation is conducted in an open-loop manner. As it is justified in the simulation results, there is a steady-state error in the operating point of motor with respect to MTPA trajectory due to the PWM modulation delay of inverter while open-loop voltage regulation is deployed. By assuming that the electromagnetic torque is equal to its reference value, the transfer function between the actual and reference rotor speed is given by [3]:
Fig. 6. Graphical representation of the MTPA optimization problem.
where,
Δi = 256(ai ei )3 − 27(ai2 di )2 , Δi0 = 12ai ei Δi1 = 27ai di2, Qi = (3ai )−1 (Qi
Si = 0.5
3
+
0.5(Δi1 +
−27Δi )
Δi0 Qi−1) ,
qi = di ai−1
(31)
To the best of authors’ knowledge, the only reference that provided (28) is [3]. Furthermore, [3] is wrongly proposed numerical technique to solve (28) and a closed-form solution is not provided. In addition, the MTPA optimization problem is differently explained and solved in the study undertaken. In [3], the derivative of (26) is directly calculated. However, the iq is rearranged in terms of id in the current limitation (5) and this strategy needs to solve a complete quadratic function in other literature, as explained in subsection III.A. Whereas, all these techniques reach to one optimal solution, an appropriate formulation of the MTPA optimization problem is proposed and solved in this study to provide a closed-form solution and overcome all the above mentioned complicatedness. The MTPA trajectories can be achieved in the voltage plane similar to the relations (26) to (31) derived for the currents. Combination of (18) and (19) results:
Is2 =
2 Vdso , ref
(ωr Lq )2
+
Kp, sc s + Ki, sc ωr (s ) = 2 ωr , ref (s ) Js + Kp, sc s + Kp, sc Ki, sc
Placing the poles of (38) at −7.5 ± 2.5i yields Kp,sc = 0.075 and Ki,sc = 0.3125. Whereas due to the different amounts of q-d reactance, the plant is nonlinear, it is straightforward to assume similar reactance in tuning the PI controllers’ parameters in order to convert the IPMSM to a linear system. Assuming Ls is the average value of Lq and Ld, the following relation can be written for the conventional MTPA tracking [3]:
(K3 ωr − λ f Vdso, ref )2 2 (K2 Ld )2Vdso , ref
(32)
Placing (32) into (25) yields:
OFMTPA =
2 (K2 Lq Ld ωr Te )2Vdso , ref 4 (K2 Ld )2Vdso , ref
+ Lq2 ωr2 (K3 ωr − λ f Vdso, ref )
(33)
iqs (s )
By differentiating of (33) in terms of Vdso,ref and equating it to zero, a very depressed quadratic function is given: 4 a v Vdso , ref
+ d v Vdso, ref + e v = 0
iqs, ref (s )
where, (35)
Vqs (s )
One of the acceptable solution of (34) is as follows:
Vdso, ref = Sv − 0.5 −4Sv2 − qv Sv−1
Vqs, ref (s ) (36)
Δv = 256(a v e v )3 − 27(a v2 d v )2 , Δv0 = 12a v e v 3
0.5(Δv1 +
Kp, cc s + Ki, cc Ls s 2 + (Kp, cc + Rs ) s + Ki, cc
(39)
=
(Kp, vc + 1) s + Ki, vc Td s 2 + (Kp, vc + 1) s + Ki, vc
(40)
where Td is the summation of PWM modulation and voltage measurement delays. They are half of the inverse of switching frequency [16] and 0.005 s, respectively. Placing the poles of (40) at –100 and –500 results in Kp,vc = 0.23 and Ki,vc = 100.
where,
Δv1 = 27a v d v2, Q v =
=
Placing the poles of (39) at –100 and –500 yields Kp,cc = 12.5 and Ki,cc = 850. In designing the voltage regulator in the proposed framework, the PWM modulation delay and thus, the transfer function between the measured voltage and its reference is as follows:
(34)
a v = (K2 Ld )2 , d v = (K3 λ f Lq2 ωr3), e v = −K32 Lq2 ωr4
(38)
−27Δv )
Sv = 0.5 (3a v )−1 (Q v + Δv0 Qv−1) , qv = d v a v−1
5. Simulation results
(37)
In this section, the MTPA tracking capability and dynamic performance of the proposed MTPA tracking with direct voltage control are validated and compared with those of conventional strategy. In this regard, the IPMSM drive systems shown in Figs. 1 and 7 are simulated in MATLAB Simulink environment. Switching frequency of the inverter
4. Implementation of the proposed MTPA with direct voltage control In this section, a framework to implement MTPA tracking with 5
Electrical Power and Energy Systems 116 (2020) 105509
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Time (s) Fig. 8. Dynamic performance of the IPMSM drives during free-acceleration.
is 10 kHz. The measured quantities are filtered with the first-order filters which have time constant equal to 0.002 s.
5.1. First case: free-acceleration performance
Time (s)
The simulation results of free-acceleration are shown in Fig. 8. All variables are in p.u.. It is clear-cut that the motor speed of the proposed technique is coincided with that of traditional method. This is also the case for torque except the first few moment of speed increment. This is due to the transient over-shoot values of iq and id in the proposed method compared to the conventional one. However, there exist these over-shoots in Vqs of the conventional one. This sort of differences between the voltages and currents of the proposed method and traditional one through the transient condition is quite logical due to the implementation of voltage regulation and current regulation, respectively.
Fig. 9. MTPA tracking capability of the IPMSM drive systems.
5.3. Third case: open-loop voltage regulation In this scenario, the influence of open-loop voltage regulation on dynamic operation of MTPA tracking of IPMSM drive system is investigated. To this end, the coefficients of PI controllers in ‘Voltage Regulation’ of Fig. 7 are set to zero. With this assumption, the time interval of 3–4 s of Fig. 9 is reshown in Fig. 11. As can be seen, the steady-state error of 1% to 4% would be appeared in IPMSM parameters with respect to MTPA operating point by eliminating voltage feedback from the proposed strategy. This is because of inverter PWM modulation latency and could be zero if switching frequency is infinite. Albeit, the voltage controller is open-loop in [13], however, it is not pointed out. Indeed, the voltage and or current control should be closed-loop to guarantee MTPA tracking process.
5.2. Second case: MTPA tracking capability In this case study, the MTPA tracking capability of the proposed strategy is analyzed and the pertinent simulations are illustrated in Fig. 9. Firstly, motor spins with its nominal speed and then the load torque is increased to its rated one within one second. The dynamic performance of the proposed strategy is quite identical to the traditional one. In order to evaluate the applicability of the suggested method in faster variations of load torque, it is abruptly changed by ± 50% and the results are plotted in Fig. 10. It is clearly observed that the IMPSM variables provided by the proposed strategy are matched with those obtained by conventional approach. However, ids is more oscillatory for the proposed method compared to the traditional one. In other hand, Vds is more oscillatory for the traditional method compared to the proposed one.
5.4. Fourth case: parameter variations It is to be noticed that a parameter estimation is required to estimate IPMSM’s parameters in analytical MTPA tracking approaches, however, there is definitely some errors in estimation process. In this regard, the influence of IPMSM parameters’ variations like magnet flux linkage and inductances on MTPA tacking capability of the proposed strategy is investigated. In first scenario, it is assumed that the motor is working under the nominal criteria and then the magnet flux linkage is reduced by 20% in one second. The results of the proposed method and 6
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Time (s) Fig. 10. Performance of IPMSM drive systems during load torque steps.
Time (s) Fig. 12. Dynamic performance of the MTPA tracking strategies for 20% decrease in magnet flux linkage. Table 2 Current and voltage variations versus parameter variations.
Time (s)
No.
Parameter variation
Current is (Con.)
Current is (Pro.)
Voltage Vs (Con.)
Voltage Vs (Pro.)
1 2 3 4 5 6
λf λf Lq Lq Ld Ld
+7% –7% +13% –9% –4% +5%
+8% –6% +13% –9% –3% +5%
+8% –8% –8% +8% –2% +3%
–1% +1% –5% +4% –8% +8%
(–20%) (+20%) (–20%) (+20%) (–20%) (+20%)
Fig. 11. The proposed MTPA tracking with open-loop voltage regulation.
Thus, motor can’t supply load torque. However, while the voltage and current limitations are respectively considered and neglected, just the proposed method can supply load torque. If both voltage and current can violate their restrictions and motor is stable, the voltage and current stress applied to the motor in the proposed strategy is less than the conventional one. It might be misunderstood that the voltage increment in the traditional method is to achieve a new MTPA point. Whilst as the load torque is constant, the flux is reduced and the current amplitude is increased to provide the previous torque, the voltage amplitude must be constant in all stages. Furthermore, it should be under scored that the speed and torque variations of the proposed strategy are significantly lower than those of the traditional approach. The sensitivity of the current and voltage with respect to parameter variations is shown in
traditional one in response of this change are shown in Fig. 12. It is clear-cut that the new operating point of motor is not located on MTPA trajectory following the magnet flux linkage reduction due to lack of parameter estimation method. While the load torque is constant, current should be increased to compensate flux reduction. As a result, the current amplitude increases by 7% and 8% with respect to the nominal one using the traditional and proposed technique. In other hand, the voltage amplitude variations should be analyzed. Whilst the voltage amplitude is reduced by 1% using the proposed strategy, it is almost 8% more than nominal one implementing the traditional approach. If the motor current is restricted to its rated value, motor would be unstable following the flux decrement in both traditional and proposed methods. 7
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Declaration of Competing Interest The authors declare that they haven't received any research grant for this work and they have no conflict of interest regarding the publication of this article. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijepes.2019.105509.
Time (s) Fig. 13. Performance of MTPA tracking strategies for 20% decrease, increase and increase in flux linkage, q-axis and d-axis inductance, respectively.
References [1] Yang Y, et al. Design and Comparison of Interior Permanent Magnet Motor Topologies for Traction Applications. IEEE Trans Transp Electrif March 2017;3(1):86–97. [2] Jahns TM, Gerald BK, Thomas WN. “Interior permanent magnet synchronous motors for adjustable speed drives”. IEEE Trans. Ind. Appl. 1986;IA-22(4):738–47. Jul./Aug. [3] Krause Paul, et al. Analysis of electric machinery and drive systems. John Wiley & Sons; 2013. [4] Morimoto S, Sanada M, Takeda Y. Wide-speed operation of interior permanent magnet synchronous motors with high performance current regulator. IEEE Trans. Ind. Appl. 1994;30(4):920–6. [5] Uddin MN, Radwan TS, Rahman MA. Performance of interior permanent magnet motor drive over wide speed range. IEEE Trans Energy Convers Mar 2002;17(1):79–84. [6] Rahman MA, Vilathgamuwa DM, Uddin MN, Tseng King-Jet. Nonlinear control of interior permanent-magnet synchronous motor. IEEE Trans Ind Appl 2003;39(2):408–16. [7] Rabiei A, Thiringer T, Alatalo M, Grunditz EA. Improved maximum-torque-perampere algorithm accounting for core saturation, cross-coupling effect, and temperature for a PMSM intended for vehicular applications. IEEE Trans Transp Electrif June 2016;2(2):150–9. [8] Li S, Sarlioglu B, Jurkovic S, Patel NR, Savagian P. Comparative analysis of torque compensation control algorithms of interior permanent magnet machines for automotive applications considering the effects of temperature variation. IEEE Trans Transp Electrif Sept. 2017;3(3):668–81. [9] Mohamed YARI, Lee TK. Adaptive self-tuning MTPA vector controller for IPMSM drive system. IEEE Trans Energy Convers 2006;21(3):636–44. Sept. [10] Hoang KD, Aorith HKA. Online control of IPMSM drives for traction applications considering machine parameter and inverter nonlinearities. IEEE Trans Transp Electrif Dec. 2015;1(4):312–25. [11] Bolognani S, Petrella R, Prearo A, Sgarbossa L. Automatic tracking of MTPA trajectory in IPM motor drives based on AC current injection. IEEE Trans Ind Appl 2011;47(1):105–14. Jan.-Feb. [12] Sun T, Wang J, Chen X. Maximum torque per ampere (MTPA) control for interior permanent magnet synchronous machine drives based on virtual signal injection. IEEE Trans Power Electron Sept. 2015;30(9):5036–45. [13] H. Chaoui; O. Okoye; M. Khayamy, “Current Sensorless MTPA for IPMSM Drives,” IEEE/ASME Transactions on Mechatronics , vol.PP, no.99, pp.1-1. [14] Khayamy M, Chaoui H. Current sensorless MTPA operation of interior PMSM drives for vehicular applications. IEEE Trans Veh Technol 2018;67(8):6872–81. Aug. [15] Shinohara A, Inoue Y, Morimoto S, Sanada M. Maximum torque per ampere control in stator flux linkage synchronous frame for DTC-Based PMSM drives without using q-Axis inductance. IEEE Trans Ind Appl 2017;53(4):3663–71. July-Aug. [16] Holmes DG, Lipo TA, McGrath BP, Kong WY. Optimized design of stationary frame three phase AC current regulators. IEEE Trans Power Electron 2009;24(11):2417–26. Nov.
Table 2. It is clear that the absolute values of voltage and current variations with respect to the flux increment are identical to the flux decrement. It can be seen from this Table that the current variations are similar for both traditional and proposed techniques. It means that these changes are not associated with the controlling strategy and they are dependent to the load torque and motor flux. In other hand, it can be concluded that the voltage changes and their stress on the motor in the proposed technique are less than those of traditional one. For instance, the current and voltage variations of scenarios 1, 4 and 6 listed in Table 2 are shown in Fig. 13. It can be deduced that the voltage stress of the proposed technique is almost half of the conventional one under identical current stress. While the temporary voltage changes are imposed to 10%, the motor can work stably and supply load torque only using the proposed strategy. 6. Conclusions In this paper, a new MTPA control strategy is derived for IPMSMs. The voltage and current limitations are plotted in voltage plane to determine MTPA trajectories in this plane. In this regard, the MTPA problem is formulated as a constrained optimization problem. The closed-form solution is obtained without using the look-up tables and storing numerical results. Accordingly, a MTPA tracking control strategy with direct voltage control for IPMSM drive systems is developed. The MTPA tracking capability of the proposed strategy is demonstrated using a simulated IPMSM drive system developed in MATLAB Simulink environment. It is shown that while the voltage regulation is open-loop, there is a steady-state error of operating point of motor with respect to the MTPA operating point which is proportional to the inherent delay of PWM modulation. Consequently, it is compulsory to deploy the closed-loop voltage regulation to remove this error. Additionally, the suggested method is investigated and also compared against the traditional method to demonstrate its superiority. There is lower amount of voltage stress on the motor using the proposed strategy under parameter changes and their estimation error. Therefore, the stability margin of the motor is increased so that the over design in selecting inverter will be mitigated correspondingly.
8
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Maximum torque per ampere control with direct voltage control for IPMSM drive systems
T
⁎
Mostafa Malekpoura, Rasoul Azizipanah-Abarghooeeb, , Vladimir Terzijab a b
Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran School of Electrical and Electronic Engineering, The University of Manchester, Manchester M13 9PL, UK
A R T I C LE I N FO
A B S T R A C T
Keywords: Direct voltage control Interior permanent magnet synchronous motor Maximum torque per ampere Parameter variation
This paper proposes an analytical method to determine maximum torque per ampere (MTPA) trajectories in the voltage plane for interior permanent magnet synchronous machines (IPMSMs). The MTPA is formulated as an optimization problem and its solution is obtained within a closed-form function, from the current and voltage planes. In consequent, the MTPA tracking control strategy with the novel closed-loop direct voltage control is developed for the IPMSM drive systems. Comparison studies against the well-known conventional current regulation-based MTPA tracking control strategy illustrate the superiority of the developed strategy in various aspects as well as its efficient MTPA tracking capability. Furthermore, it also shows more robustness to the IPMSM’s parameter variations than conventional strategy from voltage constraint point of view.
1. Introduction Interior permanent magnet synchronous motor (IPMSM) has been employed in many electrified transportations due to its high efficiency and power density [1]. Maximum torque per ampere (MTPA) speed control strategy has been widely adopted to achieve high efficiency of an IPMSM drives in constant torque region [2,3]. At first, the MTPA current trajectory of IPMSMs is achieved by drawing the constant torque loci in the current plane and finding the nearest points on torque loci to the origin [2]. In this analytical technique, the direct and quadrature components of the motor current are obtained to attain MTPA operation by numerical solving of a quadratic function [4,5]. Albeit IPMSM has a non-linear model, the conventional proportionalintegral (PI) is the most popular current regulator for tracking the MTPA [6]. The analytical MTPA methods usually deploy the IPMSM’s parameters to compute the MTPA trajectory. However, these parameters might change due to cross-magnetization saturation effects and temperature. As a result, the MTPA loci computed analytically has error [7,8]. In order to address this issue, model-based online estimation algorithms are utilized to estimate parameters of the motors [9,10]. In the other hand, signal injection-based control strategies are employed as parameter independent MTPA tracking schemes to meet the parameter variation issue [11,12]. In these schemes, a high-frequency
current signal is injected into the motor. After which, the MTPA point is online searched in these strategies by utilizing this observation that at MTPA points the variation in torque associated with the variation in the current angle is zero. Whilst this method is insensitive to parameters variation, measuring the instantaneous values of load torque and the complexity of it increases its cost of application. All analytical techniques for MTPA tracking of IPMSM drives need closed-loop current regulation to indirectly determine the motor’s reference voltages. The reference currents are also obtained based on reference torque and MTPA trajectory. However, the reference voltage can be directly determined from the reference torque to achieve MTPA operation [13,14]. Consequently, no explicit current regulation is required. In [13], the IPMSM’s voltage amplitude and angle are provided in terms of the electromagnetic torque and motor speed in order to obtain a relation between the reference torque and the voltage amplitude and angle. Whereas this method simplifies the control structure, any analytical method was not established to derive MTPA trajectory in the voltage plane. In particular, the torque-voltage characteristic was derived through simulations for a given motor. Moreover, the motor voltage regulation was conducted using an open-loop controller [13]. Recently, a current sensor-less MTPA controller is presented in [14] for the IPMSM motors. However, any analytical solution was not proposed to determine MTPA curve on the voltage plane and it is instead calculated using a numerical approach [14].
⁎
Corresponding author. E-mail addresses: [email protected] (M. Malekpour), [email protected] (R. Azizipanah-Abarghooee), [email protected] (V. Terzija). https://doi.org/10.1016/j.ijepes.2019.105509 Received 18 July 2019; Accepted 22 August 2019 0142-0615/ Crown Copyright © 2019 Published by Elsevier Ltd. All rights reserved.
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presented in (5) are portrayed in current plane [4]. The current limitation is a circle with the radius of Is,max. In other hand, the voltage restrictions are illustrated with discrepant ellipsoids corresponding to different speeds. For the sake of simplicity in determining the ellipsoids, an internal voltage is defined as [4]:
In this paper, the analytical method to achieve MTPA trajectory in the current plane is tersely explained. Moreover, whilst the nominal point is approximated as in [4], the accurate one is calculated for the IPMSM in this study. After that, the voltage-based MTPA trajectory expression is obtained by drawing the current and voltage restrictions of IPMSM in the voltage plane, instead of current one used in conventional approaches. Additionally, the MTPA is formulated as a constrained optimization problem to justify that MTPA trajectories can be expressed as closed-form functions either in the current or voltage planes and no numerical calculations to solve MTPA problem and store the solution to the look-up tables are needed. Furthermore, a novel MTPA tracking control strategy with closed-loop direct voltage regulation is designed. Additionally, it is noticed from the simulation results extracted from the proposed model in MATLAB Simulink environment that the proposed MTPA tracking capability based on direct voltage control is as accurate as that of traditional current regulation. Moreover, it also shows more robustness to the IPMSM’s parameter variations than conventional strategy from voltage constraint point of view, so that the stability margin around the nominal point is increased. The rest of this paper is organized as: Section 2 outlines the IPMSM modelling. The detailed analytical MTPA strategy is presented in Section 3. The conventional MTPA scheme is tersely explained. After which, the proposed strategy to achieve MTPA trajectories in the voltage plane is explained. In Section 4, the derived MTPA tracking control strategy with closed-loop direct voltage regulation is simulated in MATLAB Simulink. In Section 5, the simulation results are provided. Finally, the paper is concluded in Section 6.
Vso,max = Vs,max − Rs Is,max By employing (1) and (6), it yields:
Vqso = ωr (Ld ids + λ f ), Vdso = −ωr Lq iqs
(ωr Lq )2iqs2 , ref + (ωr Ld )2 (ids, ref + Ld−1 λ f )2 ⩽ Vo2,max
idsMTPA , ref =
Te − TL = J
idsMTPA , nom =
−λf 4(Ld − Lq )
−
(0.25λ f (Ld − Lq)−1)2 + (0.5Is,max )2
(10)
Accordingly, it can be directly calculated. Furthermore, the rated values of torque and speed can be achieved by (3) and (8). Albeit, there are some assumptions in determining nominal values, the accurate computation is deployed in this study. In other words, the value of Vso,max obtained by (6) is lower than its true one leading to the lower calculated nominal speed than its true one. In order to remove this assumption, the voltage limitation is rearranged using (1) and (5) as follows:
(1)
(2)
2 2 2 2 Vqso , nom + Vdso, nom = (Vqs, nom − Rs iqs, nom ) + (Vds, nom − Rs ids, nom )
(3)
(11)
This can be simplified as follows:
Vso2 ,max = Vs2,max + Rs2 Is2,max −
(4)
4 Rs Pin 3
(12)
with Pin as the input active power of motor. If the left hand side of (12) is substituted with that of (8) and Pin is rewritten in terms of torque, speed and loss of motor, the following polynomial can be extracted to determine rated speed.
where, J is the moment of inertia of the rotor and the connected load is in kg.m2. TL denotes the load torque. 3. The MTPA control strategy
aω ωr2, nom + bω ωr , nom + cω = 0 In this section, the conventional MTPA method is firstly explained. Then, the proposed MTPA technique and MTPA trajectory determination approach are detailed.
(13)
where,
aω = (Ld ids, nom + λ f )2 + (Lq iqs, nom )2 bω =
3.1. The conventional MTPA control strategy
8 RT , 3P s e, nom
cω = (Rs Is,max )2 − Vs2,max
(14)
The accurate nominal speed of motor deployed in this paper is 4% more than the approximated value taken from traditional technique. Moreover, while the nominal values of speed and q-d currents are placed in (1), the maximum value of voltage would be equal to the maximum voltage of motor. The current and voltage limitations beside the MTPA trajectory are portrayed in Fig. 1 using data presented in Table 1 for the studied IPMSM. Additionally, the torque loci for rated torque and its half value are illustrated in this figure. It can be seen that
The IPMSM motors can tolerate the currents and voltages less than their maximum limits. Thus, the equilibrium point of these motors should be imposed by current and voltage constraints. This can be formulated as follows: 2 2 2 2 2 iqs2 , ref + ids , ref ⩽ Is,max , Vqs, ref + Vds, ref ⩽ Vs,max
(9)
Therefore, while According to (9), iqsMTPA be determined in , ref is changed from zero the range of zero and its nominal amount. Based on the traditional method in specifying the nominal values of currents, iqs2 , ref is rearranged in (9) in terms of id and Is,max obtained from (5). In this regard, the nominal value of id in MTPA trajectory is expressed as follows:
where, P is number of poles. Finally, rotor speed can be determined by the equation of motion as follows:
(p (2P −1ωr ))
2 (0.5λ f (Ld − Lq)−1)2 + (iqsMTPA , ref )
−
is also function of iqsMTPA , ref . to its rated value, idsMTPA , ref will
with Lq and Ld as q- and d-axis inductances, respectively. p is d/dt, and ωr denotes the angular velocity of the rotor. The electromagnetic torque is expressed:
3P (λ f iqs + (Ld − Lq ) iqs ids ) 22
−λf 2(Ld − Lq )
idsMTPA , ref
where,
Te =
(8)
While iq is calculated in terms of id using (3) and substituted in the current limitation (5), then the derivation of two sides of this limitation is computed and equated to zero, is yielded in the following form to achieve MTPA [2,4]:
The voltage equations of the IPMSM in the rotor reference frame can be written as follow [3]:
λqs = Lq iqs , λ ds = Ld ids + λ f
(7)
Substitution of (7) in the voltage restriction of (5) gives its currentbased description. The centre of these ellipsoids located in horizontal axis, i.e., d-axis current. The ellipsoids’ area is decreased from infinite to a specific value by growing motor’s speed from zero to the nominal one.
2. The IPMSM model
Vqs = Rs iqs + pλ qs + ωr λ ds Vds = Rs ids + pλds − ωr λqs
(6)
(5)
In order to determine the MTPA trajectory, the constraints 2
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Fig. 1. MTPA trajectory in current plane.
Table 1 Parameters of the studied IPMSM [15]. Number of poles P
4
d-axis inductance Ld
9.67 mH
Armature resistance Rs Magnet flux linkage λf q-axis inductance Lq
0.824 ohm 78.5 mWb 24.3 mH
Max. current Is,max Max. voltage Vs,max Rotor inertia J
8.66 A 52 V 0.005 kg.m2
Current Regulation
ωr,ref +
PI −
Te,ref
ωr
q-dec
iqs,ref MTPA ids,ref
+
− + −
ωr
q-dec = ωr(Ldids+λf )
qd abc
d-dec = ωr(Lqiqs )
+
PI
+
+
Vqs,ref Vds,ref
d-dec
Fig. 3. Voltage and current constraints in voltage and internal voltage planes.
PWM Inverter
a = g (Rs2 + ωr2 Ld2), b = g (2Rs ωr (Lq − Ld ))
−
c = g (Rs2 + ωr2 Lq2), d = g (2Rs λ f ωr2 (Ld − Lq))
ωr
is
θr
d/dt
PI
e = −g (2λ f ωr (Rs2 + ωr2 Lq2))
IPMSM
f=
the MTPA trajectory is the operating points on different torque locus subject to the lowest distance to the origin. The block diagram of a conventional MTPA tracking is shown in Fig. 2. In this scheme, the speed error is passed through a PI controller to determine the reference torque. Then, the q-d reference currents corresponding to this torque is calculated by MTPA block. It is to be noticed that the MTPA trajectory is already computed and saved as Look-up table in this block. After which, a current controller using two PI controllers tune the references of inverter voltages so that the measured currents would be equal to their corresponding references.
2 2 −2 2 (ωr Lq)−2Vdso , ref + (ωr Ld ) (Vqso, ref − ωr λ f ) ⩽ Is,max
In the proposed method, the MTPA trajectory is obtained in voltage plane. It is clear-cut that the voltage limitation is a circle with the radius of Vs,max. In other hand, different ellipsoids are corresponded to the current limitations with discrepant speeds. In order to achieve the relations of these ellipsoids, the q-d currents are calculated using (1) as follows:
Rs2
+
ωr2 Ld Lq
, ids =
Vdso = K3 ωr2 [ωr λ f + K2 (Vqso − ωr λ f )]−1
+
ωr2 Ld Lq
(19)
K = 0.75P , L = Ld − Lq , K1 = −(Lq)−1K , K2 = (Ld )−1L, K3 = (K1)−1Te
(15)
(20)
After that, Vdso obtained by (18) is located in (19) to give:
Substitution of (15) in (5) results in ellipsoids’ mathematical functions as follows: 2 2 aVds , ref + bVds, ref Vqs, ref + cVqs, ref + dVds, ref + eVqs, ref + f = 0
(18)
where,
ωr Lq (Vqs − λ f ωr ) + Rs Vds Rs2
(17)
As can be seen from the bottom of Fig. 3, the ellipsoid’s rotation is eliminated. The origin of each ellipsoid has a distance equal to ωr λ f to the origin over the vertical axis. The MTPA trajectory is determined in internal voltage plane. In this context, Vdso can be calculated by deriving q-d currents from (7) and locating them in (3) as follows:
3.2. The proposed MTPA control strategy
Rs (Vqs − λ f ωr ) − ωr Vds
+ ωr2 Lq2)) − Is2,max , g = (Rs2 + ωr2 Ld Lq)−2
The voltage limitation beside the current ones is shown in the upper plot of Fig. 3 for different speeds. It is observable that the ellipsoids’ centres are moved up over the vertical axis and their areas are raised following the speed increment. However, the rotation of ellipsoids is mitigated. In order to make the calculation of MTPA trajectory easy, voltage and current limitations are plotted in internal voltage plane. To determine the current limitations, the q-d currents obtained by (7) are placed in (5) to yield:
Fig. 2. Conventional MTPA tracking implementation with current regulation.
iqs =
g (λ f2 ωr2 (Rs2
(ωr Lq)−2 (K3 ωr2 )2 (ωr λ f + K2 (Vqso, ref − ωr λ f
(16)
))2
+
(Vqso, ref − ωr λ f )2 (ωr Ld )2
⩽ Is2,max
(21)
By using the derivative operator for both sides of (21), the following relation is yielded:
where, 3
Electrical Power and Energy Systems 116 (2020) 105509
M. Malekpour, et al. 2 −2K2 (ωr Lq)−2Vdso , ref
(ωr λ f + K2 (Vqso, ref − ωr λ f ))
+
2(Vqso, ref − ωr λ f ) (ωr Ld )2
=0
(22)
The acceptable solution of (22) provides the reference of Vqso for achieving MTPA. It has a dual relationship with (9) as:
Vqso, ref =
−ωr λ f (1 − 2K2 ) 2K2
2
+
2
⎛ ωr λ f ⎞ + ⎜⎛ Ld Vdso, ref ⎟⎞ Lq ⎠ ⎝ 2K2 ⎠ ⎝
⎜
⎟
(23)
Accordingly, it is to be pointed out that there is a MTPA trajectory corresponding to each value of motor’s speed. For instance, as shown in Fig. 4, the curve located between A and B is the representative of MTPA trajectory for nominal speed. While the torque increases from zero at point B, to nominal values at point A. Based on torque loci for nominal speed, the BA curve consists of operating points on torque loci with the least distance to centre of ellipsoid of nominal speed. While the reference torque is zero, the solution of (22) is:
Vdso, ref = 0, Vqso, ref = ωr λ f
Fig. 4. MTPA trajectory for nominal speed in internal voltage plane.
(24) 1
This is illustrated by blue dotted line in Fig. 4. Indeed, while the motor’s speed increases from zero to its rated value under zero torque, the operating point moves from origin to point B. This line is corresponded to the origin in Fig. 1. It means that while the reference torque is zero, it is impossible to determine operating point of motor, at current plane. However, this is doable in voltage plane. The MTPA trajectories are portrayed in Fig. 5 for different speeds under the step of 0.2 p.u. In a given speed, Vqso and Vdso are respectively mitigated and grown following the torque increment from zero to nominal value. In other hand, both voltage amplitudes are increased after speed increment from zero to nominal value under a given torque. Furthermore, while torque is constant, the ratio of these voltages, i.e., voltage angle is constant. In [13], the applied voltage angle and amplitude variations are portrayed in terms of speed and torque variations. It is observed that the slope of the voltage-speed variation is proportional to the square of the applied voltage angle. As a result, a MTPA speed controller is designed without current or voltage regulation loops. Moreover, it is claimed that this technique is useful for all IPMSMs and it is independent of motor parameters variations. However, these claims are not valid. Since the square proportionality between the above noticed two quantities is firstly an approximation and secondly it is dependent to motor parameters. Furthermore, based on information explained for Fig. 5, it can be concluded that the MTPA trajectories change in voltage plane with the parameters variation. Consequently, their relationships with speed or torque will be varied.
Fig. 5. MTPA trajectories for different speeds in internal voltage plane.
currents’ values lower than Is,max while speed is lower than it’s nominal. Additionally, it would be observed that the solution of the problem, i.e., the MTPA trajectory is located within the circle of current limitation. As a result, the only constraint that should be taken into account is the torque relation of (3). Before solving the proposed MTPA problem, it’s better to graphically investigate and draw it. In Fig. 6, ‘OF’ is plotted for a three-dimension curve corresponding to a torque equal to the 20% of nominal one. However, ‘OF’ is normalised based on its value in the nominal operating point. It can be seen that the projection of ‘OF’ on current plane is exactly torque locus. Given the torque constant for torque locus, it is observed that the global optimum, i.e., A1, is precisely placed on the torque locus where it has lowest distance to the origin, i.e., A2. The main purpose of (25) is to find A1 points for different torque’ values from zero to nominal one. In this regard, the current square is rewritten considering the limitation (3) as follows:
3.3. The MTPA as an optimization problem Before proceeding, the MTPA is detailed as an optimization problem. To the best of authors’ knowledge, this problem has not been appropriately defined and solved in literature. Whereas the final response of the state-of-the-art approaches is correct, however, they could not obtain a closed-form solution for MTPA trajectory. This solution overcomes the problem related to look-up table requirement and online calculation to update data. According to the explanation of MTPA, current should be selected for a given torque in order to maximize the objective function (OF), i.e, torque to current ratio as follows:
OFMTPA =
Te ∝ Is
Te2 Is2
Is2 = iqs2 , ref + (Te − Kλ f iqs, ref )2 (LK iqs, ref )−2
(26)
Substitution of (26) in (25) yields:
OFMTPA =
(Te K L)2iqs2 , ref (Te K L)2 iqs4 , ref + (Te − Kλ f iqs, ref )2
(27)
By differentiating of (27) in terms of iqs,ref and equating it to zero, a very depressed quadratic function can be obtained:
ai iqs4 , ref + di iqs, ref + ei = 0
(25)
The current and voltage limitations of (5) as well as (3) constitute the constraints of the corresponding optimization problem. Based on the above explanations, the voltage restriction is satisfied for all
(28)
where,
ai = K 2L2 , di = Kλ f Te, ei = −Te2
(29)
One of the acceptable solution for (28) is as follows: 1
For interpretation of color in Fig. 4, the reader is referred to the web version of this article.
iqs, ref = −Si + 0.5 −4Si2 + qi Si−1 4
(30)
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M. Malekpour, et al.
Voltage Regulation
ωr,ref +
PI
−
ωr
Vqso,ref
Te,ref
MTPA V dso,ref V−I ids,ref iqs,ref
Vqs,ref
+
+
+
Vds,ref−
Rs
+ −
qd abc d/dt
θr
PI
+
PI
+
Vs
Vqs,ref
PWM Vds,ref Inverter
IPMSM
Fig. 7. The proposed MTPA tracking implementation with voltage regulation.
direct voltage control is proposed and designed. This is illustrated in Fig. 7. This is the dual model of the conventional technique shown in Fig. 2, so that ‘Current Regulation’ is replaced with ‘Voltage Regulation’. The ‘MTPA V-I’ block determines the reference values of current and internal voltage using (30) and (36). Then, the reference voltage of inverter’s output is specified by adding the resistance voltage drop to internal voltage. The values of reference voltages are updated through two PI controllers to guarantee the equality of the measured voltages with the reference ones. In comparison with traditional method, the proposed architecture measures the motor’s voltage instead of its current. The way of determining the output reference voltage of inverter presented in [13] is quite different from that of the proposed method in this study. Moreover, there is no voltage feedback in [13] and the voltage regulation is conducted in an open-loop manner. As it is justified in the simulation results, there is a steady-state error in the operating point of motor with respect to MTPA trajectory due to the PWM modulation delay of inverter while open-loop voltage regulation is deployed. By assuming that the electromagnetic torque is equal to its reference value, the transfer function between the actual and reference rotor speed is given by [3]:
Fig. 6. Graphical representation of the MTPA optimization problem.
where,
Δi = 256(ai ei )3 − 27(ai2 di )2 , Δi0 = 12ai ei Δi1 = 27ai di2, Qi = (3ai )−1 (Qi
Si = 0.5
3
+
0.5(Δi1 +
−27Δi )
Δi0 Qi−1) ,
qi = di ai−1
(31)
To the best of authors’ knowledge, the only reference that provided (28) is [3]. Furthermore, [3] is wrongly proposed numerical technique to solve (28) and a closed-form solution is not provided. In addition, the MTPA optimization problem is differently explained and solved in the study undertaken. In [3], the derivative of (26) is directly calculated. However, the iq is rearranged in terms of id in the current limitation (5) and this strategy needs to solve a complete quadratic function in other literature, as explained in subsection III.A. Whereas, all these techniques reach to one optimal solution, an appropriate formulation of the MTPA optimization problem is proposed and solved in this study to provide a closed-form solution and overcome all the above mentioned complicatedness. The MTPA trajectories can be achieved in the voltage plane similar to the relations (26) to (31) derived for the currents. Combination of (18) and (19) results:
Is2 =
2 Vdso , ref
(ωr Lq )2
+
Kp, sc s + Ki, sc ωr (s ) = 2 ωr , ref (s ) Js + Kp, sc s + Kp, sc Ki, sc
Placing the poles of (38) at −7.5 ± 2.5i yields Kp,sc = 0.075 and Ki,sc = 0.3125. Whereas due to the different amounts of q-d reactance, the plant is nonlinear, it is straightforward to assume similar reactance in tuning the PI controllers’ parameters in order to convert the IPMSM to a linear system. Assuming Ls is the average value of Lq and Ld, the following relation can be written for the conventional MTPA tracking [3]:
(K3 ωr − λ f Vdso, ref )2 2 (K2 Ld )2Vdso , ref
(32)
Placing (32) into (25) yields:
OFMTPA =
2 (K2 Lq Ld ωr Te )2Vdso , ref 4 (K2 Ld )2Vdso , ref
+ Lq2 ωr2 (K3 ωr − λ f Vdso, ref )
(33)
iqs (s )
By differentiating of (33) in terms of Vdso,ref and equating it to zero, a very depressed quadratic function is given: 4 a v Vdso , ref
+ d v Vdso, ref + e v = 0
iqs, ref (s )
where, (35)
Vqs (s )
One of the acceptable solution of (34) is as follows:
Vdso, ref = Sv − 0.5 −4Sv2 − qv Sv−1
Vqs, ref (s ) (36)
Δv = 256(a v e v )3 − 27(a v2 d v )2 , Δv0 = 12a v e v 3
0.5(Δv1 +
Kp, cc s + Ki, cc Ls s 2 + (Kp, cc + Rs ) s + Ki, cc
(39)
=
(Kp, vc + 1) s + Ki, vc Td s 2 + (Kp, vc + 1) s + Ki, vc
(40)
where Td is the summation of PWM modulation and voltage measurement delays. They are half of the inverse of switching frequency [16] and 0.005 s, respectively. Placing the poles of (40) at –100 and –500 results in Kp,vc = 0.23 and Ki,vc = 100.
where,
Δv1 = 27a v d v2, Q v =
=
Placing the poles of (39) at –100 and –500 yields Kp,cc = 12.5 and Ki,cc = 850. In designing the voltage regulator in the proposed framework, the PWM modulation delay and thus, the transfer function between the measured voltage and its reference is as follows:
(34)
a v = (K2 Ld )2 , d v = (K3 λ f Lq2 ωr3), e v = −K32 Lq2 ωr4
(38)
−27Δv )
Sv = 0.5 (3a v )−1 (Q v + Δv0 Qv−1) , qv = d v a v−1
5. Simulation results
(37)
In this section, the MTPA tracking capability and dynamic performance of the proposed MTPA tracking with direct voltage control are validated and compared with those of conventional strategy. In this regard, the IPMSM drive systems shown in Figs. 1 and 7 are simulated in MATLAB Simulink environment. Switching frequency of the inverter
4. Implementation of the proposed MTPA with direct voltage control In this section, a framework to implement MTPA tracking with 5
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Time (s) Fig. 8. Dynamic performance of the IPMSM drives during free-acceleration.
is 10 kHz. The measured quantities are filtered with the first-order filters which have time constant equal to 0.002 s.
5.1. First case: free-acceleration performance
Time (s)
The simulation results of free-acceleration are shown in Fig. 8. All variables are in p.u.. It is clear-cut that the motor speed of the proposed technique is coincided with that of traditional method. This is also the case for torque except the first few moment of speed increment. This is due to the transient over-shoot values of iq and id in the proposed method compared to the conventional one. However, there exist these over-shoots in Vqs of the conventional one. This sort of differences between the voltages and currents of the proposed method and traditional one through the transient condition is quite logical due to the implementation of voltage regulation and current regulation, respectively.
Fig. 9. MTPA tracking capability of the IPMSM drive systems.
5.3. Third case: open-loop voltage regulation In this scenario, the influence of open-loop voltage regulation on dynamic operation of MTPA tracking of IPMSM drive system is investigated. To this end, the coefficients of PI controllers in ‘Voltage Regulation’ of Fig. 7 are set to zero. With this assumption, the time interval of 3–4 s of Fig. 9 is reshown in Fig. 11. As can be seen, the steady-state error of 1% to 4% would be appeared in IPMSM parameters with respect to MTPA operating point by eliminating voltage feedback from the proposed strategy. This is because of inverter PWM modulation latency and could be zero if switching frequency is infinite. Albeit, the voltage controller is open-loop in [13], however, it is not pointed out. Indeed, the voltage and or current control should be closed-loop to guarantee MTPA tracking process.
5.2. Second case: MTPA tracking capability In this case study, the MTPA tracking capability of the proposed strategy is analyzed and the pertinent simulations are illustrated in Fig. 9. Firstly, motor spins with its nominal speed and then the load torque is increased to its rated one within one second. The dynamic performance of the proposed strategy is quite identical to the traditional one. In order to evaluate the applicability of the suggested method in faster variations of load torque, it is abruptly changed by ± 50% and the results are plotted in Fig. 10. It is clearly observed that the IMPSM variables provided by the proposed strategy are matched with those obtained by conventional approach. However, ids is more oscillatory for the proposed method compared to the traditional one. In other hand, Vds is more oscillatory for the traditional method compared to the proposed one.
5.4. Fourth case: parameter variations It is to be noticed that a parameter estimation is required to estimate IPMSM’s parameters in analytical MTPA tracking approaches, however, there is definitely some errors in estimation process. In this regard, the influence of IPMSM parameters’ variations like magnet flux linkage and inductances on MTPA tacking capability of the proposed strategy is investigated. In first scenario, it is assumed that the motor is working under the nominal criteria and then the magnet flux linkage is reduced by 20% in one second. The results of the proposed method and 6
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M. Malekpour, et al.
Time (s) Fig. 10. Performance of IPMSM drive systems during load torque steps.
Time (s) Fig. 12. Dynamic performance of the MTPA tracking strategies for 20% decrease in magnet flux linkage. Table 2 Current and voltage variations versus parameter variations.
Time (s)
No.
Parameter variation
Current is (Con.)
Current is (Pro.)
Voltage Vs (Con.)
Voltage Vs (Pro.)
1 2 3 4 5 6
λf λf Lq Lq Ld Ld
+7% –7% +13% –9% –4% +5%
+8% –6% +13% –9% –3% +5%
+8% –8% –8% +8% –2% +3%
–1% +1% –5% +4% –8% +8%
(–20%) (+20%) (–20%) (+20%) (–20%) (+20%)
Fig. 11. The proposed MTPA tracking with open-loop voltage regulation.
Thus, motor can’t supply load torque. However, while the voltage and current limitations are respectively considered and neglected, just the proposed method can supply load torque. If both voltage and current can violate their restrictions and motor is stable, the voltage and current stress applied to the motor in the proposed strategy is less than the conventional one. It might be misunderstood that the voltage increment in the traditional method is to achieve a new MTPA point. Whilst as the load torque is constant, the flux is reduced and the current amplitude is increased to provide the previous torque, the voltage amplitude must be constant in all stages. Furthermore, it should be under scored that the speed and torque variations of the proposed strategy are significantly lower than those of the traditional approach. The sensitivity of the current and voltage with respect to parameter variations is shown in
traditional one in response of this change are shown in Fig. 12. It is clear-cut that the new operating point of motor is not located on MTPA trajectory following the magnet flux linkage reduction due to lack of parameter estimation method. While the load torque is constant, current should be increased to compensate flux reduction. As a result, the current amplitude increases by 7% and 8% with respect to the nominal one using the traditional and proposed technique. In other hand, the voltage amplitude variations should be analyzed. Whilst the voltage amplitude is reduced by 1% using the proposed strategy, it is almost 8% more than nominal one implementing the traditional approach. If the motor current is restricted to its rated value, motor would be unstable following the flux decrement in both traditional and proposed methods. 7
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Declaration of Competing Interest The authors declare that they haven't received any research grant for this work and they have no conflict of interest regarding the publication of this article. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijepes.2019.105509.
Time (s) Fig. 13. Performance of MTPA tracking strategies for 20% decrease, increase and increase in flux linkage, q-axis and d-axis inductance, respectively.
References [1] Yang Y, et al. Design and Comparison of Interior Permanent Magnet Motor Topologies for Traction Applications. IEEE Trans Transp Electrif March 2017;3(1):86–97. [2] Jahns TM, Gerald BK, Thomas WN. “Interior permanent magnet synchronous motors for adjustable speed drives”. IEEE Trans. Ind. Appl. 1986;IA-22(4):738–47. Jul./Aug. [3] Krause Paul, et al. Analysis of electric machinery and drive systems. John Wiley & Sons; 2013. [4] Morimoto S, Sanada M, Takeda Y. Wide-speed operation of interior permanent magnet synchronous motors with high performance current regulator. IEEE Trans. Ind. Appl. 1994;30(4):920–6. [5] Uddin MN, Radwan TS, Rahman MA. Performance of interior permanent magnet motor drive over wide speed range. IEEE Trans Energy Convers Mar 2002;17(1):79–84. [6] Rahman MA, Vilathgamuwa DM, Uddin MN, Tseng King-Jet. Nonlinear control of interior permanent-magnet synchronous motor. IEEE Trans Ind Appl 2003;39(2):408–16. [7] Rabiei A, Thiringer T, Alatalo M, Grunditz EA. Improved maximum-torque-perampere algorithm accounting for core saturation, cross-coupling effect, and temperature for a PMSM intended for vehicular applications. IEEE Trans Transp Electrif June 2016;2(2):150–9. [8] Li S, Sarlioglu B, Jurkovic S, Patel NR, Savagian P. Comparative analysis of torque compensation control algorithms of interior permanent magnet machines for automotive applications considering the effects of temperature variation. IEEE Trans Transp Electrif Sept. 2017;3(3):668–81. [9] Mohamed YARI, Lee TK. Adaptive self-tuning MTPA vector controller for IPMSM drive system. IEEE Trans Energy Convers 2006;21(3):636–44. Sept. [10] Hoang KD, Aorith HKA. Online control of IPMSM drives for traction applications considering machine parameter and inverter nonlinearities. IEEE Trans Transp Electrif Dec. 2015;1(4):312–25. [11] Bolognani S, Petrella R, Prearo A, Sgarbossa L. Automatic tracking of MTPA trajectory in IPM motor drives based on AC current injection. IEEE Trans Ind Appl 2011;47(1):105–14. Jan.-Feb. [12] Sun T, Wang J, Chen X. Maximum torque per ampere (MTPA) control for interior permanent magnet synchronous machine drives based on virtual signal injection. IEEE Trans Power Electron Sept. 2015;30(9):5036–45. [13] H. Chaoui; O. Okoye; M. Khayamy, “Current Sensorless MTPA for IPMSM Drives,” IEEE/ASME Transactions on Mechatronics , vol.PP, no.99, pp.1-1. [14] Khayamy M, Chaoui H. Current sensorless MTPA operation of interior PMSM drives for vehicular applications. IEEE Trans Veh Technol 2018;67(8):6872–81. Aug. [15] Shinohara A, Inoue Y, Morimoto S, Sanada M. Maximum torque per ampere control in stator flux linkage synchronous frame for DTC-Based PMSM drives without using q-Axis inductance. IEEE Trans Ind Appl 2017;53(4):3663–71. July-Aug. [16] Holmes DG, Lipo TA, McGrath BP, Kong WY. Optimized design of stationary frame three phase AC current regulators. IEEE Trans Power Electron 2009;24(11):2417–26. Nov.
Table 2. It is clear that the absolute values of voltage and current variations with respect to the flux increment are identical to the flux decrement. It can be seen from this Table that the current variations are similar for both traditional and proposed techniques. It means that these changes are not associated with the controlling strategy and they are dependent to the load torque and motor flux. In other hand, it can be concluded that the voltage changes and their stress on the motor in the proposed technique are less than those of traditional one. For instance, the current and voltage variations of scenarios 1, 4 and 6 listed in Table 2 are shown in Fig. 13. It can be deduced that the voltage stress of the proposed technique is almost half of the conventional one under identical current stress. While the temporary voltage changes are imposed to 10%, the motor can work stably and supply load torque only using the proposed strategy. 6. Conclusions In this paper, a new MTPA control strategy is derived for IPMSMs. The voltage and current limitations are plotted in voltage plane to determine MTPA trajectories in this plane. In this regard, the MTPA problem is formulated as a constrained optimization problem. The closed-form solution is obtained without using the look-up tables and storing numerical results. Accordingly, a MTPA tracking control strategy with direct voltage control for IPMSM drive systems is developed. The MTPA tracking capability of the proposed strategy is demonstrated using a simulated IPMSM drive system developed in MATLAB Simulink environment. It is shown that while the voltage regulation is open-loop, there is a steady-state error of operating point of motor with respect to the MTPA operating point which is proportional to the inherent delay of PWM modulation. Consequently, it is compulsory to deploy the closed-loop voltage regulation to remove this error. Additionally, the suggested method is investigated and also compared against the traditional method to demonstrate its superiority. There is lower amount of voltage stress on the motor using the proposed strategy under parameter changes and their estimation error. Therefore, the stability margin of the motor is increased so that the over design in selecting inverter will be mitigated correspondingly.
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