Tuning Method Aimed at Optimized Settling Time and ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 6, JUNE 2014

3041

Tuning Method Aimed at Optimized Settling Time and Overshoot for Synchronous Proportional-Integral Current Control in Electric Machines Alejandro G. Yepes, Member, IEEE, Ana Vidal, Student Member, IEEE, Jano Malvar, Student Member, IEEE, Oscar López, Member, IEEE, and Jesús Doval-Gandoy, Member, IEEE

Abstract—Implementation of proportional-integral controllers in synchronous reference frame is a well-established current control solution for electric machines. Nevertheless, their gain selection is still regarded to be poorly reported, particularly in relation to the influence of the computation and modulation delay. To fill this gap, a design procedure to set the maximum gains for an acceptable damped response, with the delay being considered, has been recently proposed. In contrast, this paper presents a simple rule of thumb to achieve nearly the minimum settling time in combination with negligible overshoot for reference changes. This conclusion is theoretically demonstrated by the analysis of root locus diagrams and of overshoot versus settling time trajectories for sweeps of gain values. The design approaches aimed at gain maximization and the one developed here are compared, revealing that the latter provides shorter settling time and much lower overshoot in the command tracking response, while allowing greater stability margins. On the other hand, the proposed tuning method leads to a worse disturbance rejection, but by including an active resistance with enhanced pole/zero cancellation as a second degree of freedom, both design procedures attain comparable and optimized attenuation of disturbances. Matching simulation and experimental results validate the theoretical study. Index Terms—Current control, digital control, machine vector control, pulse width modulation converters, variable speed drives.

I. INTRODUCTION N electric drives for high-performance industrial applications, both steady-state and transient characteristics provided by the control are crucial. Improving the control performance of ac drives has been the focus of comprehensive research during the last decades, and it still continues to receive a lot of attention from the research community and industry. Field oriented control (FOC), which is one of the most established strategies, consists in a dual-loop involving an outer regulator in charge of

I

Manuscript received March 26, 2013; revised June 13, 2013; accepted July 26, 2013. Date of current version January 29, 2014. This work was supported in part by the Spanish Ministry of Science and Innovation and in part by the European Commission, European Regional Development Fund (ERDF) under the project DPI2012-31283 and the FPI scholarship BES-2010-031334. This paper was presented in part at the IEEE Energy Conversion Congress and Exposition, Denver, CO, USA, September 2013. Recommended for publication by Associate Editor J. R. Espinoza. The authors are with the Department of Electronics Technology, University of Vigo, Vigo 36310, Spain (e-mail: [email protected]; [email protected]; [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.2013.2276059

torque/flux regulation and an inner current controller [1], [2]. The performance of the current control determines the overall system performance, so special care should be devoted to its design in order to achieve fast and accurate current regulation [3], [4]. Several current control techniques for ac drives are able to achieve a very fast response in combination with negligible steady-state error. The hysteresis controller, for instance, attains an almost instantaneous tracking of the reference and is quite robust to instability problems, at the expense of either a variable switching frequency (fixed band) or a certain complexity (variable band) [2], [5], [6]. Predictive and deadbeat regulators ideally also provide a very fast dynamic response, but they are usually very sensitive to uncertainties in the plant parameters [7]–[10]. Minimum time current control was introduced to theoretically obtain the fastest transient response by finding the optimal control voltage for tracking the current reference under the voltage limit constraint [3], [11]; nevertheless, its large computational load may be a significant drawback in industrial applications [4]. In any case, the most widely spread current control technique for FOC is that based on PI control in synchronous reference frame (SRF) [1], [2], [4], [12]–[20]. A considerable research effort has been devoted to compare and develop alternative PI controllers in SRF, with particular focus on the internal model control (IMC) [14], complex-vector analysis [12], [13], [17], and high ratios of fundamental-tosampling frequencies [15]–[18]. Some improvements have been successively incorporated in the conventional PI current control structures, such as axes cross-coupling decoupling [12]–[14], time delay compensation [15], [16], [18], active resistance for a better disturbance rejection [13], [16], [19]–[22], a reference modification for faster response under the converter voltage constraint [4], and enhanced pole/zero cancellation in the discretetime domain with the fundamental frequency being close to the sampling one [15]. However, despite the widespread usage of synchronous PI controllers, their gain tuning is still a topic that is regarded to be poorly reported [2]. Some basic design guidelines that relate the PI gains with the time- and frequency-domain specifications have been previously obtained by assuming a first-order approximation of the system [14], but the time delay should not be disregarded when attempting to obtain the best gain adjustment [23]. Furthermore, when active resistance is implemented, it has been recommended to select its value so that the poles of the disturbance rejection response are mapped in the same locations as those given by the

0885-8993 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

3042


current control gain (i.e., for command tracking), after it has been conveniently designed [19], [21]; therefore, developing an adequate gain selection strategy is also important in order to obtain a satisfying active resistance value. To fill the gap in the literature regarding the tuning of PI control in SRF, a design methodology has been recently proposed to set the maximum gains for an acceptable damped response, taking into account the computation and modulation delays [2], [23]. Nevertheless, maximizing the gains (the bandwidth) does not necessarily imply achieving the best possible (or nearly so) performance in terms of time-domain specifications [24]. This paper proposes a simple method to select the gains of PI current controllers in SRF for electric machines so that negligible overshoot and nearly the minimum achievable settling time are simultaneously obtained for changes in the current reference. The fulfillment of these conditions is theoretically proved by means of root locus diagrams and by overshoot versus settling time trajectories for gain sweeps. A comparison between the command tracking response provided by this method and the one aimed at gain maximization (i.e., the method from [2], [23]) is made, which proves that the developed design procedure provides better specifications in terms of shorter settling time and much lower overshoot, while also allowing a greater phase margin (PM). In addition, the disturbance rejection capability obtained with both design procedures is also addressed. With this regard, it is assessed that the proposed method is in principle more sensitive to disturbances, but by incorporating the active resistance concept it is possible to obtain equally good disturbance rejection capability while using any of the two tuning methods for the current controller gain. Furthermore, the dynamics are enhanced by including the interaction between the active resistance and the time delay in the PI structures, so that a better pole/zero cancellation is achieved. Matching simulation and experimental results carried out with a 1.47 kW surface-mounted permanent magnet synchronous machine (SPMSM) corroborate the theoretical analysis. The paper is organized as follows. Section II presents the fundamentals of the current controllers, the plant, and the tuning method aimed at gain maximization. Then, Section III analyzes the drawbacks of that design approach and the new gain selection technique is developed. In Section IV, the time-domain specifications obtained for command tracking by both tuning strategies are compared through overshoot versus settling time trajectories. Section V studies the disturbance rejection capability that results with both control gain tuning procedures, depending on the value of the active resistance. The simulated and experimental results are provided in Section VI. Finally, Section VII concludes the work. II. BACKGROUND

Fig. 1. Global block diagram of the current control closed-loop of a symmetric ac machine.

From Fig. 1, idq = id + jiq is the stator current space vector in the SRF. Its reference iref dq is obtained through torque/speed and flux outer controllers [1], [2]. The stator current error is defined as Δidq = iref dq − idq (not explicitly shown in the figure). The voltage source converter (VSC) introduces a gain equal to half the dc-link voltage (i.e., vdc /2), but it is compensated by the −1 placed at the output of the controller, multiplication by 2vdc which is often applied automatically in the digital device to appropriately scale the duty cycle signal. An estimate of the back EMF edq of the machine can be added as a feedforward signal edq at the output of the current controller, as shown in Fig. 1, to approximately cancel its effect [2], [12], [13], [23], [25]. Computation and modulation imply an additional delay of one and a half samples in the stationary frame Gd (s) = e−sT d [17], [18], [23], [26], [27] where Td = 1.5Ts , with Ts = 1/fs being the sampling period. Note that Ts coincides with the period of the pulsewidth modulation (PWM) carrier in case of single update mode, whereas the former is half of the latter in double update mode [18], [23], [26]. Due to the frequency shift property of the Laplace transform,1 the delay is seen in the SRF as Gd (s + jω0 ) = e−(s+jω 0 )T d . The phase lag introduced by the time delay at dc in the SRF can be canceled by including a delay compensation term ejω 0 T d in the current control loop (see the rectangular blocks in the center of Fig. 1) [15], [16], [18], [29]. The resulting delay in the SRF after this compensation coincides with the original one in the stationary frame, i.e., ejω 0 T d e−(s+jω 0 )T d = e−sT d = Gd (s). Synchronous sampling is assumed, which implies that the sampled waveforms are free from commutation harmonics, so no antialiasing filters are needed [30], [31]. Consequently, the current feedback path can be considered to have unity gain and negligible phase lag, as often done [2], [15]–[18], [23]. Given that the back EMF edq may not be completely canceled in practice by edq (due to the time delay, estimation inaccuracy, etc.), and in some cases the feedforward compensation is not implemented, it is convenient from the analysis viewpoint to replace edq by a disturbance edq that results from the combination of both signals: edq (s) = edq (s) − edq (s)Gd (s). In this manner, either assuming perfect edq decoupling or considering it as a disturbance allows a symmetric electric machine to be modeled in the SRF as a simple RL load with an axes cross-coupling term jω0 L [12], [13], [15]

A. Global Block Diagram of Current Control Closed-Loop Fig. 1 represents the block diagram of the current control closed-loop of a symmetrical ac machine (SPMSM or induction machine), with respect to an SRF rotating at the rotor flux electrical frequency ω0 = 2πf0 (rotor FOC is considered [1]).

GL (s) =

1L

1 idq (s) = . vdq (s) sL + jω0 L + R

eζ t g(t) = G(s − ζ), where ζ ∈ C [28].

(1)

YEPES et al.: TUNING METHOD AIMED AT OPTIMIZED SETTLING

3043

Fig. 2. Block diagram of the current control closed-loop showing the two types of PI controllers in detail. (a) State-feedback decoupling G P I (s). (b) G c P I (s).

The stator current idq is given in closed-loop by the sum of two components idq (s) = GCL (s)iref dq (s) + GDR (s)edq (s).

(2)

Thus, the dynamics can be studied and adjusted by means of GCL (s) (command tracking) and GDR (s) (disturbance rejection) [13], [19], [20]. To improve the rejection of the edq disturbance through GDR (s), it has been suggested to implement an active resistance Ra in the digital device, as depicted in Fig. 1 with gray dashed lines [13], [16], [19]–[22]. This technique has the same effect as increasing the resistance of (1) in a value Ra , if the time delay is neglected. As a final remark, it should be also mentioned that the stator voltage is assumed to be balanced, so it is sufficient to control only the positive-sequence current. Otherwise, in case of significant negative-sequence (e.g., in ac drives of machines with structural imbalance, in grid-connected VSCs, and so on), a different kind of current controller such as resonant ones, which are also able to provide imbalance compensation [22], [25], [32]–[34], should be selected instead. However, in that case the trajectories of the closed-loop poles and the plots of overshoot versus settling time are significantly altered with respect to the conventional approach based on positive-sequence synchronous PI controllers [25], so the conclusion derived from the following analysis would not be suitable for such situations. B. Synchronous PI Current Controllers Fig. 2(a) and (b) depicts the block diagrams of the current control closed-loop showing in detail two alternative synchronous PI current controllers that will be described subsequently: classical PI controllers with state-feedback decoupling and complexvector PI controllers, respectively. It is assumed that the vdc and e−jω 0 T d terms from Fig. 1 are canceled as above described, so they are not shown in Fig. 2. In addition, edq and edq are replaced by edq . 1) Derivation of PI Controllers From IMC: For better understanding of some fundamental concepts regarding IMC and the PI control structures that are derived from it, the time delay

Gd (s) is at this point neglected, as done in [12]–[14]; it will be taken into account from Section II-B2. The IMC method basically consists in including the inverse of the plant transfer function (excluding delays and zeros in the right-half plane) as part of the controller structure [14]. In the particular case of current command tracking with PI controllers in ac machines, it is achieved by canceling the pole of GL (s) by a matching zero in the regulator, so that the open-loop transfer function simply results in an integral term k/s, with k being the only degree of freedom of the controller [12]–[14]. In this manner, k becomes equal to the open-loop crossover frequency ωc and to the closed-loop bandwidth in rad/s, further simplifying the design procedure [14]. Moreover, IMC also assures that the imaginary term jω0 L in GL (s), which produces axes cross-coupling in the SRF and a performance degradation with increasing speed, is compensated [12]–[14]. Mainly the following well-known PI-based current control alternatives based on IMC exist for electric machines current control [12]–[17]. 1) State-Feedback Decoupling PI Control [cf., Fig. 2(a)]: It consists in implementing a classical PI controller sL + R 1 =k (3) s s in combination with a current feedback with gain jω0 L added at the output of GPI (s). As seen from the controller, the feedback cancels the imaginary part of the pole in GL (s), i.e., it replaces the plant transfer function by GL (s − jω0 ) = 1/(sL + R). Then, the resulting pole of GL (s − jω0 ) is compensated by the zero of (3). From now on, when referring to GPI (s) or (3), the complete structure including the decoupling feedback path is considered. 2) Complex-Vector PI Control [cf., Fig. 2(b)]: The complex pole of GL (s) is directly canceled by a matching zero provided by a current controller of the form GPI (s) = kp + ki

GcPI (s) = k

sL + jω0 L + R . s

(4)

Note that the substitutions kp = kL

(5)

ki = kR

(6)

made in (3) are needed for the satisfaction of the IMC theory. Consequently, in agreement with IMC, the open-loop transfer functions for command tracking of Fig. 2(a) and (b) control strategies are, respectively [12]–[14] idq (s) k (7) = GPI (s)GL (s − jω0 ) = Δidq (s) ed q ( s ) s =0 idq (s) k (8) = GcPI (s)GL (s) = . Δidq (s) ed q ( s ) s =0

In this way, if the actual plant parameters L and R coincide with the theoretical ones introduced in the controller implementations (3) and (4), both current control approaches provide identical performance in terms of command tracking [12], [13]. In case of L and R deviations, the closed-loop frequency response

3044

with respect to the SRF obtained with GPI (s) and GcPI (s) is mainly altered around respectively dc and ω0 [12]–[14]; therefore, the latter removes better the axes cross-coupling, but lowamplitude oscillations of frequency ω0 may be excited [14]. Such correspondence (to a great extent) between the theoretical and actual L and R is here assumed. In contrast to the case of command tracking response, a certain difference has been reported in terms of disturbance rejection between the GPI (s) and GcPI (s) based controllers, even if the actual R and L are known [13]. If no active resistance is implemented, GcPI (s) presents a lower capability of rejecting dc disturbances (with respect to stationary frame) in comparison to GPI (s) [13]. Nevertheless, it is known from later publications that significant Ra values are advisable in order to attain a good disturbance rejection not only for GcPI (s), but also for GPI (s) [19], [21]. As previously mentioned, implementing the active resistance Ra (the one in the idq negative feedback) in Fig. 2(a) and (b) is equivalent to replacing R by R + Ra in (1) if the time delay is disregarded. Consequently, to assure that the IMC theory is still satisfied for command tracking, under the assumption Gd (s) ≈ 1 it would be simply sufficient to substitute R by R + Ra in (3) and (4) [13], [19], [21]. This change implies the addition of the dashed lines indicated with the labels GaPI (s) and GacPI (s) in Fig. 2, where the Gd (s)/s block should be implemented simply as 1/s. Consideration of Gd (s)/s different from 1/s in the GaPI (s) and GacPI (s) schemes will be addressed later. 2) Influence of the Time Delay: If f0 becomes large with respect to the sampling frequency fs , as in high-power or highspeed applications, then also the time delay produces a noticeable discrepancy between the behavior of Fig. 2(a) and (b) control structures; in particular, the latter is more robust than the former under such conditions [15]–[17]. Moreover, the GCL (s) bandwidth deviates from k as f0 and fs become closer to each other. A relatively low value of the ratio f0 /fs is considered in this paper. In any case, even under such assumption, taking into account the time delay is essential for an optimized tuning of the current controller, because it is the main limitation that prevents the designer from setting the GCL (s) bandwidth as large as he may desire [23]. Additionally, the computation and modulation delay also cause instability if the active resistance becomes too large [13], [16], so the effect of Gd (s) should not be disregarded from the point of view of disturbance rejection optimization either.


k=

2 π − φpm fs 3 2

with φpm being the target PM (40◦ was recommended [23]), and by assuring that the integral gain ki satisfies kp π (10) ≈ . tan−1 ωc ki 2 In this manner, the maximum gains and bandwidth for acceptable damped response, taking into account the 1.5Ts delay, are obtained [23]. The successful results provided by this method in motor drives have been also proved in later works [2]. It is straightforward to check that in combination with (5), (6) usually fulfills (10), since ωc L R is then a reasonable assumption; additionally, the pair (5)–(6) is also recommendable because of the aforementioned reasons regarding IMC [12]–[14]. Consequently, a simpler and more compact expression of this design method for gain maximization [2], [23], which is assumed here mainly for comparative purposes in the following sections, can be obtained by combining φpm = 2π/9 (i.e., 40◦ [2], [23]), (5) and (9) km ax =

5π 9.3 fs ≈ 0.582fs ≈ 2πfs 27 100

2 The v −1 factor in the original equations from which (9) results [23, (21), dc (23)] is not included in (9) because that factor is instead placed in this paper at the output of the current controller so that it is canceled with the VSC gain (see Fig. 1), for simplicity. Also note that the symbol v d c is used in [23] to represent half the dc-link voltage rather than its total value. In this way, the total loop gain is the same here as in [2] and [23].

(11)

ωs

which implies, given that k = ωc because of the IMC method (see Section II-B1), that the open-loop cross-over frequency ωc becomes approximately 9.3% of the sampling frequency ωs . A slightly greater value of 10% is known as an old rule of thumb for generic digital control applications [28] km ax =

10 ωs . 100

(12)

Even though (12) has been occasionally mentioned as a limitation for the current control gain in ac drives [14], [19], the suitability of (12) or similar gain selection rules [such as (11)] for the particular case of inverter-fed electric machines has not been studied in detail until recent years [2], [23]. III. ROOT LOCUS DIAGRAMS OF COMMAND TRACKING DYNAMICS WITH TIME DELAY PADE´ APPROXIMATION The computation and modulation delay Gd (s) can be approximated with good accuracy by a second order Padé expansion as [28]

C. Maximum Gains for Acceptable Damped Response It has been proposed [23] to tune GPI (s) (and, analogously, all forms of linearized ac current regulators [23]) by setting kp according to (5), setting k in (5) as2

(9)

Gd (s) = e−sT d ≈

1 Td2 1 − s 12 Td + s2 12 . 1 1 + s 12 Td + s2 12 Td2

(13)

This approximation will be considered throughout this section. Fig. 2(b) current controller is considered rather than that in Fig. 2(a) for the study of the command tracking response, because, as mentioned in the previous section, it is equivalent with Fig. 2(a) one under the assumed constraints (known L and R values and low f0 /fs ) [12], [13], it has superior performance in case of high f0 /fs [15]–[17], and in addition its corresponding open-loop and closed-loop transfer functions


3045

idq (s) k GOL (s) = = GcPI (s)GL (s)Gd (s) = Gd (s) Δidq (s) ed q ( s ) s =0

Gd (s)k idq (s) GOL (s) = GCL (s) = ref = 1 + GOL (s) s + Gd (s)k idq (s) ed q ( s )

(14)

=0

(15) have real-valued coefficients (i.e., symmetric frequency response and root locus) and do not depend on f0 . If the active resistance is implemented in the idq feedback path and the Ra value is directly added to R in the controller gains [i.e., the Gd (s)/s block is replaced by 1/s in Fig. 2(b)], as it is usually done [13], [19], [21], the time delay prevents the pole-zero cancellation in (14) and (15) from being satisfied: GOL (s) = Gd (s)

sL + jω0 L + R + Ra k . s sL + jω0 L + R + Ra Gd (s)

(16)

Yim et al. include a one-step current prediction, based on the parameters of the machine model, in the active resistance path to overcome this problem to a certain extent [16]. Here, instead, the following small modification is introduced in the current controllers to provide cancellation between the poles and zeros in the right-hand fraction of (16). The R variable in the PI control is replaced by R + Ra Gd (s) in (3) and (4) [i.e., the Gd (s)/s block is not replaced by 1/s in Fig. 2], so that (14) and (15) are satisfied even if Ra 0. ol (which coincide with the closedThe open-loop zeros z1,2 loop zeros) and open-loop poles pol 1-3 of GOL (s) [cf., (14)] are 2 2 ol ol ol z1,2 = 2 ± j √ fs p1 = 0 p2,3 = −2 ∓ j √ fs . 3 3 (17) The mathematical expressions for the closed-loop poles pcl 1-3 are included in the Appendix. A. Root Locus Analysis of Gain Maximization Method Fig. 3(a) represents the resulting root locus diagram of GCL (s) at fs = 5 kHz. The closed-loop poles (pcl 1-3 )m ax indicated in Fig. 3 correspond tokm ax . cl and the damping ratio ξ = The cl decay rate σ = Re p σ/ p , which respectively measure the speed of attenuation and the oscillatory behavior exhibited by the transient error response associated with a closed-loop pole pcl , are two important indicators to evaluate the system performance [24]. The σ and ξ values that result for the dominant closed-loop poles pcl 1,2 when k = km ax (i.e., σm ax and ξm ax ) are specified in Fig. 3(a). Note that, as intended with this design procedure [2], [23], the closed-loop poles obtained with km ax can be indeed considered to correspond to relatively very high gain, and not as large as to become too underdamped (ξm ax = 0.32). However, on the other hand, it can be observed in Fig. 3(a) that σm ax and ξm ax are considerably far from the maximum σ and ξ feasible for pcl 1,2 , i.e., km ax does not provide the best possible (or approximately) specifications for these parameters.

Fig. 3. Root locus of G C L (s) at fs = 5 kHz. (a) Closed-loop poles at k m a x = 2909. (b) Closed-loop poles at k o p t = 1230.

B. Proposed Gain Selection for Decay Rate and Damping Ratio Optimization From Fig. 3(a), both closed-loop poles pcl 1,2 can be set further from the imaginary axis (i.e., with greater decay rate) and closer to the real axis (i.e., with damping ratio closer to one) simultaneously, than (pcl 1,2 )m ax , by lowering the gain k. Specifically, it is proposed here to choose k so that the decay rate cl σ of the dominant pole (i.e., pcl 1 for low k, and p1,2 for high k) is maximized in absolute value. This optimum gain value is defined as kopt . Fig. 3(b) illustrates the closed-loop pole locations with this gain selection (i.e., (pcl 1-3 )opt ). Both the resulting decay rate σopt and damping ratio ξopt of the dominant poles are substantially greater than those obtained with the method aimed at gain maximization: |σopt | > |σm ax | and ξopt > ξm ax . In fact, the damping ratio is made equal to unity (i.e., ideally zero overshoot [24]) and the decay rate reaches its maximum in absolute value. Any closed-loop pole pcl 1-3 changes from real- to complexvalued or vice versa when the content μ of the square roots in the generic expression of that pole (all the square roots of pcl 1-3 are identical, as shown in the Appendix) changes its sign. Therefore, a simple closed-form formula for kopt can be obtained by solving μ = 0, from the Appendix kopt ≈ 0.246fs ≈

3.9 ωs . 100

(18)

This newly developed rule of thumb suggests, thus, to select the open-loop cross-over frequency ωc as approximately 4% of the sampling frequency ωs . For instance, (18) gives kopt = 1230 at

3046


Fig. 5. Block diagram of current control closed-loop in the discrete-time domain for analysis of command tracking response (ed q = 0).

Fig. 4. 1230.

Bode diagram of G O L (s) at fs = 5 kHz for k m a x = 2909 and k o p t =

fs = 5 kHz, which is the gain value used for the closed-loop poles in Fig. 3(b). Regarding stability, comparison of Fig. 3(a) and (b) permits to observe that, besides the improvement in decay rate and damping ratio, (18) also leads to a more stable system than (11), because the closed-loop poles in Fig. 3(b) are located further from the unstable region (i.e., the right-half plane) than in Fig. 3(a). Fig. 4, which compares the open-loop Bode diagrams obtained with both gain selections confirms this fact: the PM is 69◦ with kopt and it is 40◦ with km ax . The latter PM is consistent with the design specification originally defined in [2] and [23] for the method aimed at gain maximization. Moreover, substitution of (18) in (9) and isolation of φpm gives that 69◦ does not only result when using kopt at 5 kHz, but also at other fs values. From Fig. 4, the open-loop crossover frequencies ωc coincide with k in each case, as expected from IMC (see Section II-B1). Finally, it can be noted in Fig. 4 that the secondorder Padé expansion made in (13) provides a very accurate approximation of the true delay up to about fs /3 (i.e., up to when its phase lag ∠e−sT d is close to −180◦ [28]). IV. OVERSHOOT VERSUS SETTLING TIME TRAJECTORIES OF COMMAND TRACKING RESPONSE INCLUDING COMPLETE DELAY MODEL IN THE Z-DOMAIN A. Discrete-Time Expression of Q-Axis Current Step Response The plant model expressed with respect to the SRF, when Ra = 0, in the discrete-time domain and accurately taking into account the one and a half samples delay (without approximations) is obtained by applying the ZOH discretization method to GL (s) [cf., (1)] and multiplying by the computational delay z −1 e−jω 0 T s and by the delay compensation term ejω 0 T d [15] idq (z) z −2 1 − ρ−1 ejω 0 (T d −2T s ) = GPL (z) = udq (z) ed q ( z ) R 1 − z −1 ρ−1 e−jω 0 T s =0 (19) where ρ = eR T s /L . Equation (19) is represented in block diagram form in Fig. 5. Moreover, this model also includes the slight decrease in gain caused at high frequencies by the PWM (equivalent to a ZOH in average representation) [18], which has been disregarded in the previous section as well as in other publications that study the effects of time delay on current con-

trol [2], [17], [23]. Additional delays due to dead-time, sensors, and their conditioning are neglected. The plant parameters R, L, and Ts used in this section are chosen so that they coincide with those in the experimental prototype (cf., Section VI). The time-domain closed-loop step current response ydq in the SRF at each instant t = mTs (m ∈ N), taking into account any Ra value, in accordance with Fig. 5 is given by ydq = Z −1 GCL (z)U (z) GacPI (z)GPL (z) U (z) (20) = Z −1 1 + [GacPI (z) + Ra ] GPL (z) where GacPI (z) is obtained by the application of the Tustin trans form to GacPI (s) and U (z) = j/ 1 − z −1 models a unit step in the q-axis current reference (the d-axis one is usually kept constant in the base speed region [1]). B. Analysis of Overshoot Versus Settling Time Trajectories Fig. 6(a) represents the trajectories of overshoot Mp versus settling time t1% of the iq step response (i.e., yq = Im {ydq }) for a sweep of k values, with f0 = 50 Hz and sampling frequencies of 2.5, 5, and 10 kHz. A tolerance band of δ = 1% [24] is defined for the settling time. No active resistance is considered in this figure (i.e., Ra = 0), although it has been checked that its inclusion does not produce noticeable modifications. The points that correspond to kopt and km ax are marked in the plots. The q-axis current step response yq associated with each of them is represented in Fig. 7(a). From Figs. 6(a) and 7(a), kopt leads to much lower overshoot (Mp ≈ 0%) than km ax (Mp ≈ 40%), because of the larger damping ratio of the closed-loop poles that result with the former (see Section III). Furthermore, t1% is also reduced by kopt with respect to km ax ; in fact, at kopt the settling time becomes quite close to the minimum achievable [see Fig. 6(a)], for each sampling frequency. Fig. 6(b) depicts a closer view of the kopt points from Fig. 6(a). It can be observed that even though the settling time could be reduced even more, in a small quantity, by making k slightly greater, such action would cause in contrast a certain undesired increase in overshoot. That would be the case if any of the k values indicated by squares in Fig. 6(b) were selected instead of kopt . In other words, kopt provides a critically damped response, whereas with k > kopt it is underdamped and hence Mp > 0. Furthermore, choosing those gains marked by squares is also inadvisable due to another reason: that elbow-shaped portion of the plots in the left-hand side, which corresponds to the k interval such that iq surpasses iref q but does not reach the upper limit of the tolerance band δ used for the settling time


3047

Fig. 6. Overshoot versus settling time t1 % (δ = 1%) of the iq step response for a k sweep, with f0 = 50 Hz and several fs . (a) General view. (b) Closer view of the k o p t points.

Fig. 7. Q-axis current unit step response (i.e., y q ) with f0 = 50 Hz and several fs . (a) General view. Curves with k m a x and k o p t . (b) Closer view of the k o p t f plots around ire q = 1. Those with k equal to the points marked with a square in Figs. 6(b) and 8 are also shown.

of settling time and overshoot, and certainly much better in this sense than km ax . V. STUDY OF DISTURBANCE REJECTION DYNAMICS A. Tuning of the Disturbance Rejection Response

Fig. 8. Overshoot versus settling time t0 . 5 % (δ = 0.5%) of the iq step response for a k sweep, with f0 = 50 Hz and several fs . The axes are set to the same ranges as in Fig. 6(b).

As mentioned in Section II, certain differences have been reported between the disturbance rejection behavior of GPI (s) and GcPI (s) [13]. Therefore, in this section both types of controllers are separately analyzed. The current response produced by edq (s) in the schemes shown in Fig. 2(a) and (b) is defined in the s-domain by, respectively idq (s) GDR (s) = ref edq (s) id q ( s ) =0

[see Fig. 7(b), which illustrates the corresponding time-domain responses], depends greatly on δ. For instance, although the k selections marked with squares in Fig. 6(b) achieve the minimum settling time for δ = 1%, they do not when δ = 0.5% (see Fig. 8). Consequently, kopt , which almost does not depend on δ, can be indeed considered to be an optimum choice in terms

=

−GL (s) 1 + [GaPI (s) + Ra − jω0 L] Gd (s)GL (s)

=

−s 1 s + Gd (s)k sL+R+jω0 L [1−Gd (s)]+Ra Gd (s)

GA D R (s)

GB D R (s)

(21)

3048


and

expressed in magnitude form |Δidq |. In this manner, the IE is obtained as ∞ |Δidq (t)| dt IE =

idq (s) −GL (s) GDR (s) = = edq (s) irdeqf ( s ) 1 + [GacPI (s) + Ra ] Gd (s)GL (s)

0

=0

=

1 −s . s + Gd (s)k sL + R + jω0 L + Ra Gd (s)

GA D R (s)

(22)

GB D R (s)

Note that GDR (s) has units of admittance; in fact, GDR (s) is sometimes referred to as the VSC input admittance [22]. Each of these expressions [i.e., (21) and (22)] has been written as a B multiplication of two transfer functions: GA DR (s) and GDR (s). This decomposition permits to separate the poles that depend on B k and those that depend on Ra [GA DR (s) and GDR (s) poles, respectively]. It can be also observed, by comparison between (21) and (22), that (22) presents a considerable axes cross-coupling, produced by the jω0 L term in the denominator of GBDR (s) [similarly to (1)]. This aspect is strongly related to the fact, assessed in [13], that GcPI (s) is much more sensitive at dc (with respect to stationary frame) to the disturbance than GPI (s) when Ra = 0; since the imaginary part of the GBDR (s) pole is −jω0 for GcPI (s), GBDR (s) provides a peak of very high gain (1/R) at zero hertz with respect to stationary frame. It is also worth noticing that the enhancement achieved in pole/zero cancellation in (16) by including the delay Gd (s) in the GaPI (s) and GacPI (s) controllers does not only improve the command tracking response, but also the disturbance rejection, because of the fact that GA DR (s) and GCL (s) share the same poles. 1) One-Degree-of-Freedom (1-DOF) Current Control: If no active resistance is implemented (Ra = 0), Fig. 2 structures are essentially 1-DOF current controllers, with k being the only parameter to be adjusted. Hence, k should be tuned by taking into account both command tracking and disturbance rejection dynamics [given by respectively GCL (s) and GDR (s) in (2)], unless the effect of edq (s) can be neglected (e.g., whenever edq accurately cancels the back EMF; or in SPMSMs, where edq is only given by the mechanical speed, whose rate of change is usually much slower than the electromagnetic one [1]). In fact, it is well known that in many applications both optimization problems conflict with each other, in the sense that the best k value for each of the two objectives cannot be simultaneously achieved [24]. That is the case, for instance, of speed control in ac drives [35]. Analogously to the reasoning followed in [35] for speed controllers, it can be checked that the 1-DOF conventional approach also presents this problem in the context of current control. The integrated error (IE) has been used in [35] as a measure of the performance in the disturbance rejection response, in combination with another property that gives information about the oscillatory behavior, such as the PM. However, in this case the application of the IE is not straightforward, because both (21) and (22) contain complex terms (i.e., cross-coupling). Consequently, rather than integrating the projection of Δidq on the d- and q-axes separately, the error function to be integrated is

∞

=

1

[(Re[Δidq (t)])2 + (Im[Δidq (t)])2 ] 2 dt

0

=

∞

[(L−1 {Re[GDR (s)U (s)]})2

0

+ (L−1 {Im[GDR (s)U (s)]})2 ] 2 dt 1

(23)

with U (s) = j/s representing a disturbance unit step in the qaxis. Note that the complex value of s should be ignored when applying the Re and Im operators in (23) to transfer functions, i.e., they only affect the coefficients [36]. Finding an explicit symbolic solution for (23) is quite complicated, so it is instead numerically evaluated for the parameters under consideration in this paper, and series with a time step of 10−7 s are considered rather than continuous integrals. It is found that, for Ra = 0, the expression 1 (24) kR is satisfied quite approximately by both GPI (s) and GcPI (s). In fact, in the most unfavorable conditions (fs = 2.5 kHz, f0 = 50 Hz and kopt ) the IE deviation percentage with respect to (24) is 8.5 · 10−9 % and 3.1% with each of these controllers, respectively. Thus, from (24), the disturbance rejection capability is in general enhanced by increasing the controller gain k, as in 1-DOF PI speed controllers [35]. Note that this difference between GCL (s) and GDR (s) [cf., (15), (21), and (22)], in spite of the fact that in both transfer functions k gives rise to exactly the same poles, is a consequence of the presence and the absence of k in the numerators of the former and the latter, respectively; in other words, if k were multiplying the numerator of GDR (s) as well, then the k factor in the denominator of (24) would be canceled. Consequently, it can be expected (it is verified in the following section and section) that km ax offers a greater attenuation of edq than kopt when no active resistance is implemented. In any case, from the previous sections, this advantage of km ax over kopt for 1-DOF control is obtained at the expense of a lower PM (and, consequently, also a more oscillatory response to edq changes, which is not reflected by IE [35]) and a worse command tracking response. 2) Two-Degrees-of-Freedom (2-DOF) Current Control: Inclusion of the Ra variable in Fig. 2 schemes gives rise to 2-DOF current controllers. In them, the command tracking and disturbance rejection dynamics can be independently adjusted by k and Ra , respectively. Firstly, k can be set so that GCL (s) is optimized. In our study, two cases are mainly of interest: km ax and kopt . Once k has been chosen, GA DR (s) in (21) and (22) is also established, so Ra should then be selected in order to make GDR (s) optimal by means of GBDR (s). A reasonable criterion to calculate Ra is to place the poles of GBDR (s) in the same locations as those of GCL (s), which in turn coincide with GA DR (s) ones [19], IE ≈


3049

Fig. 9. Block diagram of current control closed-loop in the discrete-time f a domain for analysis of disturbance rejection response (ire d q = 0). (a) G P I (z). (b) G ac P I (z).

[21]. This approach assures that the PM and the bandwidth, in addition to the damping ratio and decay rate associated with each pole, of GBDR (s) match those of GCL (s) and GA DR (s). If the R and jω0 L terms in the denominator of GBDR (s) are neglected in comparison to Ra , the active resistance should be set as Ra = kL to obtain identical poles in all these transfer functions [19], [21]. Thus, the active resistance values given by a Rm ax = km ax L

a Ropt = kopt L

(25)

correspond with those that a certain designer should select, for consistence, if he respectively regarded km ax or kopt as the optimum control gain for command tracking response. Concerning the IE, for the 2-DOF control an approximation is obtained by replacing R by R + Ra in (24) IE ≈

1 . k (R + Ra )

(26)

Its comparison with (24) permits to assert that the IE is much smaller with 2-DOF control for a given k than with the 1-DOF schemes; moreover, the difference between the IE values that correspond to km ax and kopt is thus also expected to be lower in the 2-DOF case. In this manner, for reasonably large k (such as kopt ), the IE becomes sufficiently small for 2-DOF current control, so the designer may focus on tuning GDR (s) only according to the position of the dominant poles (or, equivalently, the PM), as done in [19] and [21] with the Ra = kL selection. B. Overshoot Versus Settling Time Trajectories Including Complete Delay Model in the Z-Domain 1) Discrete-Time Expression of DQ-Current Response to Disturbance Step: Fig. 9(a) and (b) represents the block diagrams in the discrete-time domain, corresponding to respectively GaPI (z) and GacPI (z), for study of the current response to the edq disturbance (iref dq has been replaced by zero). GL (z) is obtained by application of the Tustin transform to (1), because there is no ZOH operation (PWM) in the path between edq and idq , unlike the case of GPL (z) [34]. The time-domain current response to an eq unit step U (z) in Fig. 9(a) and (b) is given, according to ydq = Z −1 GDR (z)U (z) , by, respectively −GL (z) −1 ydq = Z U (z) (27) 1 + [GaPI (z) + Ra − jω0 L] GPL (z) −GL (z) −1 ydq = Z U (z) . (28) 1 + [GacPI (z) + Ra ] GPL (z)

2) Analysis of Overshoot Versus Settling Time Trajectories: Equations (27) and (28) contain complex terms [as (21) and (22)], so the modulus of ydq (i.e., |ydq |) is considered for calculation of the overshoot and settling time, analogously to the use of the magnitude function in (23). Furthermore, since the steady-state current value is zero in the case of disturbance rejection (iref dq = 0), the overshoot Mp of |ydq | is not expressed as a percentage of its final value, but as the maximum absolute peak of |ydq |. In this manner, the units of Mp are A/V. Similarly, the width of the tolerance band δ that is employed to calculate the settling time tδ is here defined also in absolute terms (specifically, δ = 0.05 A) rather than as a percentage. The plots of overshoot versus settling time obtained for an Ra sweep, when an eq step of 100 V is applied to Fig. 9(a) and (b) with f0 = 50 Hz, are shown in Fig. 10(a)and (b), respectively. For both types of PI controllers and for the three sampling frequencies under consideration, the settling time and overshoot are lower with km ax than with kopt when Ra = 0, as expected from Section V-A1. Once the Ra parameter is incorporated in the design, resulting in a 2-DOF structure, the curves of km ax and kopt become closer to each other in Fig. 10, particularly with the increase in Ra (mainly as a consequence of the rapid decrease in the IE, as explained in Section V-A2) and fs . Inspection of each of the trajectories shown in Fig. 10 as a whole permits to assert that km ax and kopt potentially make possible to attain, by selecting the appropriate Ra value, slightly lower overshoot and settling time, respectively; nevertheless, in most cases such differences are negligible. Therefore, it can be stated that in general the disturbance rejection response can be made equally good with either km ax or kopt by using the second degree of freedom Ra . a a The points that correspond to Rm ax and Ropt are indicated in each of the curves of Fig. 10, so four specific combinations of Ra and k at each fs (denoted by a certain color) are marked with a a a with kopt , Rm different symbols: using Ropt ax with kopt , Ropt a with km ax , or Rm ax with km ax . It is interesting to remark that a a the possibilities Rm ax with kopt and Ropt with km ax yield quite approximately the same Mp and t0.05A , because in those cases B the poles of GA DR (s) and GDR (s) are interchanged (nearly) and hence almost identical poles are obtained in GDR (s). In addition, these options provide a good tradeoff between Mp and t0.05A , and considerably better response than when Ra = 0; the a with kopt same applies to the other two combinations (i.e., Ropt a and Rm ax with km ax ), except with km ax at fs = 2.5 kHz. The a selection of Ropt with kopt provides slightly shorter t0.05A than a Rm ax with km ax , whereas the latter gives a small advantage in Mp ; in any case, none of the two alternatives is clearly preferable over the other one. It can be also noted that the Mp associated a and kopt could be reduced by with the combination of Ropt a increasing R in a small quantity; however, it may not be desirable, since the improvement would be modest and it would mean to have a lower PM in GBDR (s) than in GCL (s). Comparison of Fig. 10(a) and (b) permits to state that, in spite of the difference in terms of cross-coupling decoupling between both PI structures, both of them provide comparable a Mp and t0.05 A ; an exception can be seen for Rm ax with km ax at

3050


Fig. 10. Overshoot versus settling time t0 . 0 5 A of the |id q | response to an eq step of 100 V for an R a sweep, with f0 = 50 Hz and several fs . (a) G aP I (z). (b) G ac P I (z).

Fig. 11.

Experimental prototype. (a) Scheme. (b) Photograph.

fs = 2.5 kHz (i.e., the largest gains and delay), where GacPI (z) attains a lower t0.05A .

TABLE I MAIN PARAMETERS OF EXPERIMENTAL PROTOTYPE

VI. EXPERIMENTAL RESULTS A. Experimental Setup The prototype is shown in Fig. 11, and its main parameters are summarized in Table I. From Fig. 11(a), an SPMSM is connected to an induction machine (IM), which is driven by an adjustable speed drive (ASD). The ASD is configured to maintain f0 at 50 Hz. The machine-side VSC (MS-VSC) controls the SPMSM so that it works as a generator, i.e., a torque is applied in the opposite direction to the rotation. A grid-side VSC (GS-VSC) is often connected to the dc-link of the MS-VSC to maintain its voltage and to supply active power to the grid [37], [38]; it is emulated here by a dc voltage source and a load resistance RL . The digital control is executed in the prototyping platform dSPACE DS1006. From Fig. 11(a), the delay compensation term ejω 0 T d from Fig. 1 is implemented by simply adding a phase lead ω0 Td to the rotor field electrical angle θ before being input to the inverse rotational transformation [16], [29].

B. Evaluation of Results 1) Command Tracking Response: In these experiments, an step is applied to test the command tracking response iref q with km ax and kopt . In agreement with Sections III and IV, a GcPI (z) controller is chosen. The feedforward signal edq is included, but the active resistance is set at zero. The measured


3051

Fig. 12. Command tracking results obtained with k m a x and R a = 0. (a) fs = 2.5 kHz (k m a x = 1454.5). (b) fs = 5 kHz (k m a x = 2909). (c) fs = 10 kHz (k m a x = 5818). Scales: current (1 A/div); time (250 μs/div, 500 μs/div and 1 ms/div from left to right).

Fig. 13. Command tracking results obtained with k o p t and R a = 0. (a) fs = 2.5 kHz (k o p t = 615). (b) fs = 5 kHz (k o p t = 1230). (c) fs = 10 kHz (k o p t = 2460). Scales: current (1 A/div); time (250 μs/div, 500 μs/div and 1 ms/div from left to right).

a Fig. 14. Command tracking results obtained with k o p t and R a = 0 at fs = 5 kHz. (a) R m a x = 51.2 with the enhanced scheme [i.e., including the G d (s)/s a block in Fig. 2(b)]. (b) R oa p t = 21.65 with the conventional scheme [i.e., replacing the G d (s)/s block in Fig. 2(b) by 1/s]. (c) R m a x = 51.2 with the conventional scheme [i.e., replacing the G d (s)/s block in Fig. 2(b) by 1/s]. Scales: current (1 A/div); time (500 μs/div).

frequency ω0 and the same iref q step as in the actual control loop are used as inputs to simulate the current behavior obtained with GPL (z) and an identical current controller GcPI (z) [see Fig. 11(a)], in accordance with (20). Then, the resulting simulated dq-current isim dq can be compared with the actual one idq to further corroborate the validity of the results. Fig. 12(a)–(c) show the results obtained with km ax for respectively fs = 2.5 kHz, fs = 5 kHz, and fs = 10 kHz. The q-axis reference step iref q has been previously stored in the memory of the oscilloscope as RefA and afterward superimposed with the other signals. Fig. 13 shows the results of analogous tests

to those applied in Fig. 12, but selecting kopt for the current controller instead of km ax . First, it can be seen that the experimental and simulated d- and q-axes currents in these figures match to a great extent. The slight differences can be mainly attributed to aspects of the actual system that have been neglected in the model [cf., (20)] such as dead-time, edq harmonics, machine saliency, saturation, and additional time delay in the sensors and their conditioning. Most importantly, the settling time and overshoot values of the q-axis step response are consistent with those theoretically

3052


a a Fig. 15. Disturbance rejection results obtained with k m a x at fs = 5 kHz. (a) No active resistance. (b) R m a x = 51.2. (c) R o p t = 21.65. Scales: current (750 mA/div); time (1 ms/div).

a Fig. 16. Disturbance rejection results obtained with k o p t at fs = 5 kHz. (a) No active resistance. (b) R oa p t = 21.65. (c) R m a x = 51.2. Scales: current (750 mA/div); time (1 ms/div).

assessed in Section IV. For all sampling frequencies, the overshoot is reduced from about 40% to almost 0% when km ax is replaced by kopt (compare Figs. 12 and 13, respectively). The time that the q-axis current takes to settle down is also shorter with the latter, although the difference is not as large as that achieved in overshoot. In this manner, the results prove that the proposed tuning method based on kopt provides an optimum combination of overshoot and settling time for a certain sampling frequency, improving in both time-domain specifications (particularly in overshoot) the performance achieved with the design technique aimed at gain maximization, i.e., km ax . The same experiments have been carried out without edq , and no noticeable differences occurred. The reason is that in SPMSMs the back EMF is only related to the speed, and in turn the mechanical rate of change is usually much lower than the electromagnetic one [1]. In an IM, operating without the feedforward compensation could be expected to have a stronger impact on the performance, because of the relation between the flux and the rotor currents. Additionally, replacing GcPI (z) by GPI (z) only causes a noticeable (although small) performance worsening at fs = 2.5 kHz (for both km ax and kopt ), due to the lower robustness of the latter to the increase in the ratio f0 /fs [15]–[17]. Finally, the influence of Ra on the command tracking response is assessed. All the previous experimental tests have

a a and Rm been also reproduced with Ropt ax , by means of the enhanced scheme proposed at the beginning of Section III [i.e., including the Gd (s)/s block in Fig. 2(b) rather than 1/s]; with this approach, almost identical results as with Ra = 0 are obtained as well. As an example, Fig. 14(a) shows the currents a obtained with kopt and Rm ax . From this oscilloscope capture, the response is practically the same as in Fig. 13(b). On the other hand, the waveforms shown in Fig. 14(b) and (c) correspond to a a and Rm kopt with respectively Ropt ax , when the conventional approach based on just increasing R as R + Ra in (3) and (4) is employed [i.e., replacing the Gd (s)/s block in Fig. 2(b) by 1/s]. From Fig. 14(b) and (c), the response is considerably worsened as Ra becomes larger. Therefore, the improvement achieved in making GCL (z) independent from Ra by taking into account the delay in GacPI (z) is proved. 2) Disturbance Rejection Response: These experiments aim to test the disturbance rejection capability with kopt and km ax , by forcing a sudden edq change while iref dq is kept at zero. Even by setting the maximum acceleration in the ASD and a speed transient of 1500 r/min, the current peak in the SPMSM stator hardly surpasses 0.5 A in the worst-case scenario. More demanding disturbances could occur in practice during fault situations, e.g., in case of a sudden stop of the shaft rotation. Due to the ASD limitations, an abrupt edq change is performed here by removing the feedforward signal edq while the machine


rotates at f0 = 50 Hz. In this manner, current peaks up to 3 A are obtained. The edq and ω0 signals are internally input to (28) in the digital device to generate the simulated currents isim dq [see Fig. 11(a)]. Given that it has been assessed in Section V that GaPI (z) and GacPI (z) provide a similar behavior in terms of overshoot and settling time for disturbance rejection, only the results obtained with GacPI (z) are shown here. All these experiments are done at fs = 5 kHz. Fig. 15(a)–(c) shows the results of the tests performed with a a km ax with respectively Ra = 0, Rm ax , and Ropt . Those obtained with kopt are shown in Fig. 16(a)–(c), which respeca a , and Rm tively correspond to Ra = 0, Ropt ax . First, it can be seen that the similarity between the simulated and experimental waveforms confirms the validity of the results and the models. Comparison of Figs. 15(a) and 16(a) verifies that, when Ra = 0, km ax is preferable over kopt in terms of disturbance rejection. If Ra is selected analogously to k [i.e., according to (25)], then, as theoretically concluded in Section V, both k values provide comparable overshoot and settling time [compare a Figs. 15(b) and 16(b)], although with Rm ax the response is more oscillatory due to its lower PM. Finally, Figs. 15(c) and 16(c) corroborate that almost identical response can be obtained with a a and Rm km ax and kopt by simply choosing respectively Ropt ax , although the minimum PM is reduced from theoretically 69◦ in Fig. 16(b) to 40◦ in Fig. 16(c). Repeating the experiments with GaPI (z) yields practically equivalent results in terms of overshoot and settling time, but with a much lower axes cross-coupling, as expected from Section V.

VII. CONCLUSION This paper proposes a simple rule of thumb (to set the gain, i.e., the open-loop crossover frequency, to 4% of the sampling frequency) to tune PI current controllers in SRF for electric machines, so that nearly the minimum achievable settling time is attained in combination with negligible overshoot in the command tracking response. Its validity is theoretically supported by analysis of root locus diagrams with a second-order Padé approximation of the computation and modulation delay and of overshoot versus settling time trajectories for sweeps of gain values. In addition, the developed procedure is compared with a recent proposal aimed at obtaining the maximum possible gains with an acceptable damped response, and it is concluded that the former provides better specifications in terms of shorter settling time and much lower overshoot for reference changes, while assuring larger stability margins. Concerning disturbance rejection, it has been concluded that setting the control gain with the proposed method yields a worse response than the previously existing tuning procedure under study. Nevertheless, it has been proved that, by introducing an active resistance as a second degree of freedom of the current control, it is possible to achieve comparable and optimized disturbance rejection response with both control gain selections. Furthermore, it has been proposed to incorporate the interaction between the active resistance and the time delay in the PI current

3053

control structures, so that a more accurate pole/zero cancellation is achieved, and hence the dynamics are greatly improved. Matching simulation and experimental results performed with a 1.47 kW SPMSM validate the theoretical analysis. APPENDIX The closed-loop poles of GOL (s) [i.e., GCL (s) poles] are √ 1 pcl −1 ± j 3 λ2 − 2 (k + 4fs ) λ 1,2 = 6 √ (A.1) − 1 ± j 3 k 2 − 20fs k λ3 pcl 3 =

1 2 λ − (k + 4fs ) λ + k 2 + 20fs k λ3 (A.2) 3

where √ λ = 2fs μ − k 3 − 30fs k 2 − 168fs2 k + 32fs3 4

3

μ = 9k + 504fs k +

6576fs2 k 2

−

2688fs3 k

+

(A.3) 256fs4 .

(A.4)

REFERENCES [1] E. Levi, “FOC: Field oriented control,” in The Industrial Electronics Handbook, (Power Electronics and Motor Drives), 2nd ed. Boca Raton, FL, USA: CRC Press, 2011. [2] D. G. Holmes, B. P. McGrath, and S. G. Parker, “Current regulation strategies for vector-controlled induction motor drives,” IEEE Trans. Ind. Electron., vol. 59, no. 10, pp. 3680–3689, Oct. 2012. [3] J.-W. Choi and S.-K. Sul, “New current control concept—Minimum time current control in the three-phase PWM converter,” IEEE Trans. Power Electron., vol. 12, no. 1, pp. 124–131, Jan. 1997. [4] J.-W. Choi and S.-K. Sul, “Fast current controller in three-phase AC/DC boost converter using d-q axis crosscoupling,” IEEE Trans. Power Electron., vol. 13, no. 1, pp. 179–185, Jan. 1998. [5] R. Ramchand, K. Gopakumar, C. Patel, K. Sivakumar, A. Das, and H. AbuRub, “Online computation of hysteresis boundary for constant switching frequency current-error space-vector-based hysteresis controller for VSIfed IM drives,” IEEE Trans. Power Electron., vol. 27, no. 3, pp. 1521– 1529, Mar. 2012. [6] M. Mohseni and S. Islam, “Transient control of DFIG-based wind power plants in compliance with the Australian grid code,” IEEE Trans. Power Electron., vol. 27, no. 6, pp. 2813–2824, Jun. 2012. [7] S.-M. Yang and C.-H. Lee, “A deadbeat current controller for field oriented induction motor drives,” IEEE Trans. Power Electron., vol. 17, no. 5, pp. 772–778, Sep. 2002. [8] K.-H. Kim and M.-J. Youn, “A simple and robust digital current control technique of a PM synchronous motor using time delay control approach,” IEEE Trans. Power Electron., vol. 16, no. 1, pp. 72–82, Aug. 2001. [9] H.-T. Moon, H.-S. Kim, and M.-J. Youn, “A discrete-time predictive current control for PMSM,” IEEE Trans. Power Electron., vol. 18, no. 1, pp. 464–472, Jan. 2003. [10] T. Laczynski and A. Mertens, “Predictive stator current control for medium voltage drives with LC filters,” IEEE Trans. Power Electron., vol. 24, no. 11, pp. 2427–2435, Nov. 2009. [11] J.-W. Choi and S.-K. Sul, “Generalized solution of minimum time current control in three-phase balanced systems,” IEEE Trans. Ind. Electron., vol. 45, no. 5, pp. 738–744, Oct. 1998. [12] F. Briz, M. Degner, and R. Lorenz, “Analysis and design of current regulators using complex vectors,” IEEE Trans. Ind. Appl., vol. 36, no. 3, pp. 817–825, May/Jun. 2000. [13] F. Briz, M. Degner, and R. Lorenz, “Dynamic analysis of current regulators for AC motors using complex vectors,” IEEE Trans. Ind. Appl., vol. 35, no. 6, pp. 1424–1432, Nov./Dec. 1999. [14] L. Harnefors and H. P. Nee, “Model-based current control of AC machines using the internal model control method,” IEEE Trans. Ind. Appl., vol. 34, no. 1, pp. 133–141, Jan./Feb. 1998. [15] H. Kim, M. Degner, J. Guerrero, F. Briz, and R. Lorenz, “Discrete-time current regulator design for AC machine drives,” IEEE Trans. Ind. Appl., vol. 46, no. 4, pp. 1425–1435, Jul./Aug. 2010.

3054

[16] J.-S. Yim, S.-K. Sul, B.-H. Bae, N. Patel, and S. Hiti, “Modified current control schemes for high-performance permanent-magnet AC drives with low sampling to operating frequency ratio,” IEEE Trans. Ind. Appl., vol. 45, no. 2, pp. 763–771, Mar./Apr. 2009. [17] J. Holtz, J. Quan, J. Pontt, J. Rodriguez, P. Newman, and H. Miranda, “Design of fast and robust current regulators for high-power drives based on complex state variables,” IEEE Trans. Ind. Appl., vol. 40, no. 5, pp. 1388– 1397, Sep./Oct. 2004. [18] B.-H. Bae and S.-K. Sul, “A compensation method for time delay of fulldigital synchronous frame current regulator of PWM AC drives,” IEEE Trans. Ind. Appl., vol. 39, no. 3, pp. 802–810, May/Jun. 2003. [19] L. Harnefors, K. Pietilainen, and L. Gertmar, “Torque-maximizing fieldweakening control: design, analysis, and parameter selection,” IEEE Trans. Ind. Electron., vol. 48, no. 1, pp. 161–168, Feb. 2001. [20] H. Kim and T. Jahns, “Current control for AC motor drives using a single DC-link current sensor and measurement voltage vectors,” IEEE Trans. Ind. Appl., vol. 42, no. 6, pp. 1539–1547, Nov./Dec. 2006. [21] A. Petersson, L. Harnefors, and T. Thiringer, “Evaluation of current control methods for wind turbines using doubly-fed induction machines,” IEEE Trans. Power Electron., vol. 20, no. 1, pp. 227–235, Jan. 2005. [22] L. Harnefors, L. Zhang, and M. Bongiorno, “Frequency-domain passivitybased current controller design,” IET Power Electron., vol. 1, no. 4, pp. 455–465, Dec. 2008. [23] D. G. Holmes, T. A. Lipo, B. McGrath, and W. Kong, “Optimized design of stationary frame three phase AC current regulators,” IEEE Trans. Power Electron., vol. 24, no. 11, pp. 2417–2426, Nov. 2009. [24] K. Ogata, Modern Control Engineering. Englewood Cliffs, NJ, USA: Prentice-Hall, 2009. [25] A. G. Yepes, A. Vidal, F. D. Freijedo, J. Malvar, O. Lopez, and J. DovalGandoy, “Transient response evaluation of resonant controllers for AC drives,” in Proc. IEEE Energy Convers. Congr. Expo., Sep. 2012, pp. 471– 478. [26] S. Buso and P. Mattavelli, in Digital Control in Power Electronics, J. Hudgins, Ed. The Netherlands: Morgan and Claypool Publishers, 2006. [27] A. G. Yepes, F. D. Freijedo, O. Lopez, and J. Doval-Gandoy, “Highperformance digital resonant controllers implemented with two integrators,” IEEE Trans. Power Electron., vol. 26, no. 2, pp. 563–576, Feb. 2011. [28] R. H. Middleton and G. C. Goodwin, Digital Control and Estimation: A Unified Approach. Englewood Cliffs, NJ, USA: Prentice-Hall, 1990. [29] P. Mattavelli, “A closed-loop selective harmonic compensation for active filters,” IEEE Trans. Ind. Appl., vol. 37, no. 1, pp. 81–89, Jan./Feb. 2001. [30] H. Fujita, “A single-phase active filter using an H-bridge PWM converter with a sampling frequency quadruple of the switching frequency,” IEEE Trans. Power Electron., vol. 24, no. 4, pp. 934–941, Apr. 2009. [31] D. Dujic, C. Zhao, A. Mester, J. Steinke, M. Weiss, S. Lewdeni-Schmid, T. Chaudhuri, and P. Stefanutti, “Power electronic traction transformerlow voltage prototype,” IEEE Trans. Power Electron., vol. 28, no. 12, pp. 5522–5534, Dec. 2013. [32] A. Tani, M. Mengoni, L. Zarri, G. Serra, and D. Casadei, “Control of multiphase induction motors with an odd number of phases under opencircuit phase faults,” IEEE Trans. Power Electron., vol. 27, no. 2, pp. 565– 577, Feb. 2012. [33] F. Gonzalez-Espin, G. Garcera, I. Patrao, and E. Figueres, “An adaptive control system for three-phase photovoltaic inverters working in a polluted and variable frequency electric grid,” IEEE Trans. Power Electron., vol. 27, no. 10, pp. 4248–4261, Oct. 2012. [34] A. Vidal, F. D. Freijedo, A. G. Yepes, P. Fernandez-Comesana, J. Malvar, O. Lopez, and J. Doval-Gandoy, “Assessment and optimization of the transient response of proportional-resonant current controllers for distributed power generation systems,” IEEE Trans. Ind. Electron., vol. 60, no. 4, pp. 1367–1383, Apr. 2013. [35] L. Harnefors, S. Saarakkala, and M. Hinkkanen, “Speed control of electrical drives using classical control methods,” IEEE Trans. Ind. Appl., vol. 49, no. 2, pp. 889–898, Mar./Apr. 2013. [36] L. Harnefors, “Modeling of three-phase dynamic systems using complex transfer functions and transfer matrices,” IEEE Trans. Ind. Electron., vol. 54, no. 4, pp. 2239–2248, Aug. 2007. [37] S. Li, T. Haskew, R. Swatloski, and W. Gathings, “Optimal and directcurrent vector control of direct-driven PMSG wind turbines,” IEEE Trans. Power Electron., vol. 27, no. 5, pp. 2325–2337, May 2012. [38] K.-H. Kim, Y.-C. Jeung, D.-C. Lee, and H.-G. Kim, “LVRT scheme of PMSG wind power systems based on feedback linearization,” IEEE Trans. Power Electron., vol. 27, no. 5, pp. 2376–2384, May 2012.


Alejandro G. Yepes (S’10–M’12) received the M.Sc. and Ph.D. degrees from the University of Vigo, Vigo, Spain, in 2009 and 2011, respectively. Since 2008, he has been with the Department of Electronics Technology, the University of Vigo. His main research interests include control of switching power converters and ac drives.

Ana Vidal (S’10) received the M.Sc. degree from the University of Vigo, Vigo, Spain, in 2010, where she is currently working toward the Ph.D. degree in the Department of Electronics Technology. Since 2009, she has been with the Department of Electronics Technology, University of Vigo. Her research interests include control of grid-connected converters and distributed power generation systems.

Jano Malvar (S’10) received the M.Sc. degree from the University of Vigo, Vigo, Spain, in 2007, where he is currently working toward the Ph.D. degree in the Department of Electronics Technology. Since 2007, he has been with the Department of Electronics Technology, University of Vigo. His research interests include power electronics, multiphase systems, ac drives, and harmonics.

Oscar López (M’05) received the M.Sc. and the Ph.D. degrees from the University of Vigo, Vigo, Spain, in 2001 and 2009, respectively. Since 2004, he has been an Assistant Professor with the Department of Electronics Technology, the University of Vigo. His research interests include the areas of ac power switching converters technology.

´ Doval-Gandoy (M’99) received the M.Sc. deJesus gree from Polytechnic University of Madrid, Madrid, Spain, in 1991, and the Ph.D. degree from the University of Vigo, Vigo, Spain, in 1999. From 1991 till 1994 he worked at industry. He is currently an Associate Professor at the University of Vigo. His research interests include the areas of ac power conversion.