IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 7, JULY 2014
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Three-Vectors-Based Predictive Direct Power Control of the Doubly Fed Induction Generator for Wind Energy Applications Yongchang Zhang, Member, IEEE, Jiefeng Hu, Student Member, IEEE, and Jianguo Zhu, Senior Member, IEEE
Abstract—This paper proposes a simple but very effective method to achieve the predictive direct power control (PDPC) for doubly fed induction generator (DFIG)-based wind energy conversion systems. The novel approach is able to operate at low switching frequency and provides excellent steady-state and dynamic performances, which are useful for high-power wind energy applications. Three vectors are selected and applied during one control period to reduce both active and reactive power ripples. Compared to prior three-vectors-based art using two switching tables, the novel approach only needs one unified switching table to obtain the three vectors. Furthermore, the duration of each vector is obtained in a much simpler and straightforward way. The switching frequency can be significantly reduced by appropriately arranging the switching sequence of the three vectors. The influence of one step delay caused by digital implementation is also investigated. The possibility of operating the proposed PDPC under unbalanced grid voltage is also briefly discussed. The novel PDPC is compared with prior three-vectors-based art and its effectiveness is confirmed by the simulation results from a 2-MW-DFIG system and the experimental results from a scaled-down laboratory setup. Index Terms—Direct power control, doubly fed induction generator (DFIG), predictive control, wind energy.
I. INTRODUCTION HE wind energy conversion system (WECS) has evolved from the early fixed-speed wind turbine system into the modern variable speed constant frequency (VSCF) system, due to its merits of low mechanical stress and power fluctuations, quick dynamic response, high wind energy capture ability, and flexible decoupled control of active/reactive power [1]. Among various configurations of VSCF systems, doubly fed induction generator (DFIG)-based WECS is very popular, because it can achieve the merits aforementioned with a power converter rated at only 25%–30% of the generator rating [2].
T
Manuscript received October 9, 2012; revised March 7, 2013, May 18, 2013, and August 3, 2013; accepted September 14, 2013. Date of current version February 18, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 51207003 and by Beijing Nova Program under Grant xx2013001. The part of this paper was presented at the Energy Conversion Congress and Exposition, Raleigh, NC, USA, September 15–20, 2012. Recommended for publication by Associate Editor Z. Chen. Y. Zhang is with the Power Electronics and Motor Drives Engineering Research Center of Beijing, North China University of Technology, Beijing 100144, China (e-mail:
[email protected]). J. Hu and J. Zhu are with the Faculty of Engineering and Information Technology, University of Technology Sydney, Ultimo, N.S.W. 2007, Australia (e-mail:
[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.2013.2282405
The conventional control method for the DFIG is vector control (VC) with the d-axis of synchronous rotary frame attached to the stator flux vector [3] or the stator voltage vector [4]. The rotor currents are decomposed into the torque (corresponding to active power) component and rotor flux (corresponding to reactive power) component, and they are regulated separately using the linear controller such as PI. A modulator such as space vector modulation is needed to generate the final gating pulses. Although a good performance is achieved by the orientationbased VC, it needs appropriate decoupling, introduces rotary transformation, and requires much tuning work to ensure system stability over the whole operating range due to the multiple loops in the controller [5]. To overcome the large tuning work and reduce the control complexity in VC, direct control methods, such as direct power control (DPC) [5], [6] and direct torque control (DTC) [7], were proposed in recent years. DTC/DPC is characterized by the quick dynamic response, simple structure, and low parameter dependence, so it has become an interest of both the academic and industry communities throughout the world [8]–[10]. However, there are large ripples in torque/flux or active/reactive powers at steady state, and the switching frequency is variable with operating point due to the use of the hysteresis controller and a predefined switching table. It is suggested to use duty cycle control in direct control methods [11]–[19] to improve the steady-state performance and maintain the simplicity and robustness. Two vectors are applied during one control period, usually the active vector from the switching table followed by a zero vector. By adjusting the duty ratio of the active vector, the ripple in the torque or active power can be reduced. Various methods have been proposed to obtain the duty ratio [11], [13], [15], [17], [18]. However, generally these methods only considers the ripple reduction of one variable, such as the torque for DTC or active power for DPC, and fail to take the flux or reactive power ripple reduction into account. The formulas to obtain the duty ratio are usually complicated and require much knowledge of system parameters [11], [13], [15], [17], [18], except [12], [16], [19]. Furthermore, although the commutation frequency can be decreased by exchanging the sequence of the active vector and zero vector [16], [19], this kind of method still cannot be applied for high-power applications operating only at a few hundreds hertz. Recently, a novel strategy using two active vectors and a zero vector during one control period was proposed for the control of the dc/ac converters [20], DFIGs [21]–[23], and permanent magnet synchronous machines (PMSM) [24]. This approach is
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an extension of the duty cycle direct control methods in [11], [13], and [15]–[19] by using three rather than two vectors to achieve ripples reduction in both torque and flux [23], [24] or active and reactive powers [20], [21]. Furthermore, the threevectors-based approach is reported to have the ability of operating at a low switching frequency, which is useful for high-power wind energy applications. Two switching tables are required in [21] and [23] for the dynamic process and the steady-state operation, respectively. The information of the rotor speed is also necessary to distinguish the status of subsynchronism or hypersynchronism. The formulas to obtain the duration of each vector in [20]–[23], are complicated, which are mainly caused by the analytical principle of ripple minimization and the complexity in slopes calculation of torque/flux or active/reactive powers. To reduce the switching losses, two candidate vector sequences are considered in [23]. However, the switching between the two adjacent vector sequences is not considered; hence, the switching frequency is not reduced to the minimum value. The influence of one-step delay caused by digital implementation is not investigated in [20], [21], and [23], although compensating the one-step delay will bring the benefit of performance enhancement [25], [26]. This paper presents a novel three-vectors-based strategy for the predictive direct power control (PDPC) of DFIGs, which uses one unified switching table to obtain the three vectors [27]. The information of the rotor speed is not needed in the process of vector selection; hence, the robustness to speed variations is improved. The durations of each vector are obtained in a much simpler and straightforward way and has less parameter dependence than that in [21]. The switching frequency reduction strategy is implemented in the novel PDPC by considering all feasible vector sequences and the smooth connection between two adjacent vector sequences. The influence of one-step delay and its compensation are investigated. A preliminary study of the proposed PDPC has been carried out in [27]. However, only the simulation results are presented in [27]. This paper presents more simulation results from a 2-MW-DFIG system to validate the effectiveness of the novel PDPC and compare it with the prior art in [21]. Furthermore, the operation of the proposed PDPC under unbalanced grid voltage is also briefly discussed. Finally, the experimental results from a scaled-down laboratory setup are also shown. The results prove that the proposed PDPC can achieve active/reactive power ripples reduction and quick dynamic response at a low switching frequency.
2) Flux equations:
A. Machine Equations A mathematical model of a DFIG described by space vectors in stationary stator-oriented reference frame is expressed as [23], [28], [29] 1) Voltage equations: dψ s dt dψ r ur = Rr ir + − jωr ψ r dt
(3)
ψ r = Lm is + Lr ir
(4)
where us , is , ur , ir , ψ s , and ψ r are the stator voltage vector, stator current vector, rotor voltage vector, rotor current vector, stator flux linkage vector, and rotor flux linkage vector, respectively; Rs , Rr , Ls , Lr , and Lm are the stator resistance, rotor resistance, stator inductance, rotor inductance, and mutual inductance, respectively; ωr is the electrical rotor speed. From (3) and (4), the stator and rotor current can be expressed by the stator and rotor flux as is = λ(Lr ψ s − Lm ψ r )
(5)
ir = λ(−Lm ψ s + Ls ψ r ) (6) where λ = 1/ Ls Lr − L2m . The rotor flux can be expressed using the stator flux and the stator current as ψr =
Lr 1 ψs − is . Lm λLm
(7)
B. Effects of Voltage Vectors on Active/Reactive Powers The complex power vector S in the stator side of DFIG can be expressed as [21] S = P + jQ =
3 ∗ i us 2 s
(8)
where “*” is the conjugate operator; P and Q indicates the active power and reactive power, respectively. As the active and reactive powers are to be controlled in the PDPC, it is desirable to deduce the relationship between the complex power and the rotor voltage vector. By differentiating the complex power in (8) with respect to time t, the complex power slope is obtained as dus di∗ dS = 1.5 i∗s + s us . (9) dt dt dt Suppose the supply three-phase voltage are sinusoidal and balanced, i.e., us = |us | ej ω 1 t
(10)
where ω1 is the grid frequency (rad/s). The differentiation of the stator voltage can be obtained from (10) as
II. MODEL OF THE DFIG
us = Rs is +
ψ s = Ls is + Lm ir
(1)
dus = jω1 |us | ej ω 1 t = jω1 us . dt
(11)
The second item in (9) is derived from (1), (2), (5), and (7) as di∗s dψ ∗s dψ ∗r us = λ Lr − Lm us dt dt dt = λLr |us |2 − (λLr Rs + jωr ) (i∗s us )
(2)
− λLm (u∗r − Rr i∗r ) us + jλLr ωr (ψ ∗s us ) . (12)
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Substitute (11) and (12) into (9), the complex power slope can be obtained as [29] dus di∗ dS di∗ = 1.5 i∗s + s us = 1.5 i∗s jω1 us + s us dt dt dt dt = − (λLr Rs − jωsl ) S + 1.5(λLr |us |2 − λLm (u∗r − Rr i∗r ) us + jλLr ωr (ψ ∗s us ))
(13)
where ωsl = ω1 − ωr is the slip speed. If the stator resistance is neglected, the relationship between the stator voltage and the stator flux at steady state can be obtained from (1) as us = jω1 ψ s .
(14)
Substitute (14) into (13), a simplified complex power slope is obtained as dS = − (λLr Rs − jωsl ) S + 1.5(sλLr |us |2 dt − λLm (u∗r − Rr i∗r ) us )
(15)
where s = ωsl /ω1 is the slip. Decomposing the real and imaginary components of (15), the active and reactive power slopes can be expressed as dP = −λLr Rs P − ωsl Q + 1.5sλLr |us |2 dt − 1.5λLm · Re ((u∗r − Rr i∗r ) us )
(16)
dQ = −λLr Rs Q + ωsl P − 1.5λLm dt · Im ((u∗r − Rr i∗r ) us ) .
(17)
Equations (16) and (17) provide practical methods to calculate the slopes of active and reactive powers and avoid the calculation of inverse tangent function in [21]. It should be noted that although (16) and (17) are derived in stationary frame, they have the same forms in other frames, because the power relationship is independent of specific frame. The only thing to be noted is that all the variables should be expressed in the same frame. C. Principle of Basic DPC Based on (16) and (17), the influences of various rotor voltage vectors on both active and reactive powers can be easily obtained by applying various rotor voltage vectors from twolevel inverter. Fig. 1 graphically illustrate the evolutions of active and reactive power slopes versus rotor flux position at the sub-synchronous speed and hyper-synchronous speed for a specific 15-kW DFIG operating at 15-kW active power and zero reactive power. The machine parameters are the same as those introduced in [21]. It should be noted that at the hypersynchronous speed, the stator/rotor flux is rotating in an opposite direction viewed from rotor frame. As a result, the sector sequence is different for the case at sub-synchronous speed. It is seen that the influences of zero vectors on the active power are different at the sub-synchronous speed and hyper-synchronous
Fig. 1. Active and reactive power slopes versus rotor flux position for various rotor voltage vectors: (a) at sub-synchronous speed and (b) at hyper-synchronous speed.
dP speed, i.e., dP dt > 0 for the sub-synchronous speed and dt < 0 for the hyper-synchronous speed irrespective of the motor or generating mode. On the contrary, the influence of the zero vector on reactive power is dependent on both rotor speed and running mode (motor or generator). To be more accurate, dQ dt is positive at the sub-synchronous speed of the motor mode or hyper-synchronous speed of the generator mode, otherwise dQ dt < 0 at the sub-synchronous speed of the generator mode or hyper-synchronous speed of the motor mode [5]. Summarizing the results in Fig. 1, the vector table for the active power and reactive power regulation can be constructed, as shown in Table I [6], [21], where k is the sector number obtained from the rotor flux position in rotor frame. For example, if the rotor flux is located in the first sector, the desired rotor voltage vectors for increasing/decreasing of active and reactive powers are vividly illustrated in Fig. 2. The zero vector is not used in Table I because it will incur the distinguishing between the
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TABLE I VECTOR TABLE FOR ACTIVE POWER AND REACTIVE POWER REGULATION
Fig. 3.
Fig. 2.
Sector division for DPC of the DFIG.
TABLE II SWITCHING TABLE FOR THE NOVEL PDPC
TABLE III SWITCHING TABLE AT STEADY STATE FOR THE PRIOR ART IN [21]
sub-synchronous speed and hyper-synchronous speed, which unnecessarily increases the complexity of system. It should be noted that although Fig. 1 is derived from a specific operating point for a 15-kW DFIG, the conclusion is universal and can be applied to other operating condition. In fact, Table I is the same as [21, Table I] and equivalent to [6, Tables II and III]. The theoretical analysis and physical interpretation are employed in [6] and [21] to describe the influences of various rotor voltage vectors on active/reactive power, while in this paper illustrated figures are presented (although for a specific operating point) to vividly show the relationship between rotor voltage vectors and power slopes, which may be easier to follow. Furthermore, from Fig. 1 it is clearly seen that the influence of zero voltage vectors on active/reactive power is much smaller than other nonzero vectors. This fact is also the theoretical foundation of the proposed three-vectors-based PDPC strategy. III. NOVEL THREE-VECTORS-BASED PDPC STRATEGY Although conventional DPC does not employ the zero vector to avoid possible increased complexity in the switching table,
Control diagram of the proposed three-vectors-based PDPC.
they fail to make full use of the ability of zero vector in reducing ripples of active and reactive powers. As shown in Fig. 1, the zero rotor voltage vector produces very small changes in the active and reactive power slopes, so it is possible to use the zero vector to improve the steady-state performance of conventional DPC. The method of using one active vector and one zero vector during one control period for DTC/DPC has been reported in the past years for induction motors [11], [13], [15], [18], PMSMs [16], [17], and DFIGs [19]. Although a good performance was obtained using two-vectors-based direct control methods, generally they only consider the ripple reduction of one variable and the switching frequencies are usually a few kilohertz. For high-power applications requiring a low switching frequency, using three vectors rather than two vectors during one control period is a better choice [21]. This paper will propose a novel three-vectors-based PDPC, which is more effective than prior art in [21] by using only one switching table and much simpler formula to obtain the vector durations. The issues of switching frequency reduction and digital delay compensation are also addressed. A. Overall Control Diagram Fig. 3 illustrates the overall control diagram of the novel three-vectors-based control strategy. The stator side active and reactive powers are the variables to be directly controlled. The active/reactive powers and slopes are predicted from the model of DFIG, as shown at the bottom of Fig. 3. The errors between the reference value and feedback value of active/reactive powers are transferred into two logical signals through the hysteresis comparators, and then sent to the vector selection block. It should be noted that the switching table used in the vector selection block is different from the classical switching table in [6] and the ones in [21]. Knowing the rotor flux position along with the power error signals, two active vectors and one zero vector will be selected. The duration of each vector are implemented in the vector durations calculation block. The vectors and their durations are sent to the pulse generation block to obtain the triggering signals for the rotor-connected converter. Classical voltage oriented control introduced in [30] is employed for the
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control of the ac/dc converter. The details regarding the several blocks in Fig. 3 will be detailed in the following section.
B. Voltage Vector Selection The proposed voltage vector selection strategy is shown in Table II. In fact, only the active power error is involved in the vector selection regardless of the error sign of reactive power. The first two active vectors produce the same signs on the active power slope while their influences on the reactive power slopes are opposite. For example, if the estimated active power is lower than reference value, V k −1 and V k −2 will be selected to reduce this error. The zero vector V 0,7 is subsequently applied to reduce the active power ripple. The reactive power ripple reduction is achieved by adjusting the durations of V k −1 and V k −2 , because they produce slopes with opposite signs of reactive power. As a comparison, the switching table at steady state for the prior art in [21] is shown in Table III. There are two notable differences between the proposed voltage vector selection strategy and the prior method in [21]. First, the vector selection in Table II is decided according to the sign of the active power error. On the contrary, in Table III, the vector selection is distinguished by the synchronism status of the DFIG, because the active power slopes caused by the zero vectors are different at subsynchronism and hypersynchronism (as shown in Fig. 1). The requirement of speed information makes the method in [21] sensitive to the speed variation. Second, Table III in [21] was only used at steady state for the aim of power ripple reduction and another switching table (see Table I) is still needed for the dynamic process. Only one unified switching table (see Table II) is required in the proposed method. The difference between the proposed Table II and the prior method in Table III are vividly illustrated in Fig. 4, which show the typical steady-state waveforms of active and reactive powers during one control period for the three-vectors-based DPC strategy at hyper-synchronous speed. It should be noted that Fig. 4 is only for illustrative aim and the slopes of active and reactive powers may be not very accurate. The rotor flux is assumed to be located at the third sector (k = 3) and the initial active power is supposed to be greater than the commanding value. The active power slopes for the first two active vectors and the zero vector are indicated by s1 , s2 , and s0 , respectively, and the reactive power slopes by s11 , s22 , and s00 , respectively. According to Table II, V 4 (011) and V 5 (001) will be selected as the first two active vectors followed by the appropriate zero vector V 0 (000) with minimal number of commutations. On the contrary, according to Table III (hypersynchronism status), the prior art in [21] will select V 2 (110) and V 1 (100) as the first two active vectors to increase the active power even if the active power is well above the reference value, because the following zero vector V 0 (000) will reduce the active power at hypersynchronism. Table III simply relies on the zero vector to achieve opposite control on the active power and fails to consider the sign of active power error, which is sometime not very appropriate, especially at the hyper-synchronous speed, as shown in Fig. 4(b).
Fig. 4. Typical steady-state active and reactive power waveforms for threevectors-based DPC strategy at hypersynchronous speed: (a) proposed method and (b) prior method in [21].
C. Determination of Voltage Vectors Durations For the three-vectors-based DPC, it is possible to reduce both active and reactive power ripples at the same time while achieving almost constant switching frequency. The key point is how to decide the duration of each vector during one control period. In prior art [21], the well-known ripple RMS minimization strategy is employed and will be briefly revisited here before introducing our novel method. The duration of the first active vector can be obtained based on reactive power ripple minimization principle as [21] d1 tsp
2 Qref − Qk − s22 d2 tsp = . 2s11 − s22
(18)
In a similar way, the sum of the durations of the first two active vectors is obtained based on the principle of active power ripple minimization as d2 tsp =
2 P ref − P k − s0 tsp 2s12 − s0
(19)
where s12 is the equivalent active power slope of the first two active vectors and it is expressed as s12 =
s1 d1 + s2 (d2 − d1 ) . d2
(20)
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Solving (18)–(20), the durations of the first two active vectors can be obtained as 2s22 P ref − P k + (2s0 − 4s2 ) Qref − Qk d1 tsp = 2s22 s1 − 4s11 s2 + 2s11 s0 − s22 s0 −s22 s0 tsp + (21) 2s22 s1 − 4s11 s2 + 2s11 s0 − s22 s0 (2s22 − 4s11 ) P ref − P k d2 tsp = 2s22 s1 − 4s11 s2 + 2s11 s0 − s22 s0 (4s1 − 4s2 ) Qref − Qk + (2s11 − s22 )s0 tsp + 2s22 s1 − 4s11 s2 + 2s11 s0 − s22 s0 (22) k
k
where Q and P are the reactive and active powers at the kth instant. It is seen that the expressions of durations in (21) and (22) for prior art [21] are complicated and relies much on the DFIG parameters. The complexity is mainly caused by the calculation of the equivalent active power slope of the first two active vectors, as shown in (20). Both active and reactive power slopes of the three vectors (except s00 ) are needed to obtain the final durations. To reduce the complexity in calculating the duration of each vector and improve the parameter robustness, this paper will adopt a simple method requiring no machine parameters, which is expressed as ref P − P k Qref − Qk + (23) d2 = CP CQ where CP and CQ are two positive constants. Similar equation has been proposed in [16] for the torque and flux control of motor drives. In [16], one active vector and one null vector are employed, while for the proposed PDPC, two active vectors and one null vector are employed. However, as the two active vectors obtained from Table II produce the same sign of active power slope, it is feasible to obtain the sum of their durations in a similar way as that in [16]. The feasibility of the simple expression in (23) can be roughly explained by comparing it with (19). As seen from Fig. 1, the active power slope s0 caused by the zero vector is relatively small compared to the slope of active vectors. As a result, if s0 is neglected in (19), the duty ratio d2 will be proportional to the active power error. Similarly, if reactive power (rather than active power) RMS ripple minimization is employed in (19) and neglecting the reactive power slope caused by zero vector, the duty ratio d2 will be proportional to the reactive power error. As both active power and reactive power are of concern, the final expression for d2 can be obtained as (23). Assuming constants in (23) is due to the fact that there are various expressions for d2 . For example, if the principle of deadbeat control is employed, i.e., P ref = P k +1 , the duty ratio d2 can be obtained as d2 tsp =
P ref − P k − s0 tsp . s12 − s0
(24)
The denominators in (19) and (24) are different, which enlightens us that using constant denominators can be seen as a mixture of various duty determination methods.
The criteria for choosing CP and CQ is a tradeoff between the dynamic response and steady-state performance. Generally, larger value of CP and CQ will produce less steady ripples, but the dynamic performance is degraded. The extensive simulation results indicate that the rated VA value of the DFIG is a good starting point for CP and CQ to achieve good compromise between steady and dynamic performance. In other word, the duty ratio d2 can be seen as the sum of the per unit deviations in the active and reactive power. It should be noted that the rated value of the DFIG is just a quick starting point for CP and CQ , and the designer can tune them according to their specific requirements. Nevertheless, it is found that the variations of CP and CQ do not cause significant difference in the system performance, as shown in Section IV. As d2 can be easily obtained from (23), it is not difficult to further obtain the duration of the first active vector from (18) under the principle of ripple RMS minimization. However, in this paper, the duration of the first active vector will be obtained in a deadbeat fashion as introduced in [16], which is expressed as
d1 tsp =
Qref − Qk − s22 d2 tsp . s11 − s22
(25)
The expression of (25) is only related to the reactive power slope calculation, which can be easily obtained from (17). Compared to the prior duty determination methods in (21) and (22), the proposed equations in (23) and (25) are much simpler and have less parameter dependence. Although the five basic machine parameters (Rs , Rr , Ls , Lr , and Lm ) are needed in (17), it is found that the parameter mismatch up to 50% does not cause much influence on the system performance, as shown in Section IV-C. Remark 1: In normal steady-state operation, the following relationship is true: 0 ≤ d1 ≤ d2 ≤ 1. However, during some dynamic process, such as stepped changes in the commanding value, the duty cycle calculated using (21), (22) or (23), (25) may be out of the range of (0, 1) and the value of d1 may be bigger than d2 . In [21], such unusual cases are considered to belong to the dynamic process, so only the active vector selected from Table I will be applied during the whole period. However, this treatment is not very good because it will switch to the conventional switching-table-based DPC whenever d1 and d2 are outside the range of (0, 1) or d1 is greater than d2 , which fails to regulate the active and reactive powers moderately and accurately. In this paper, a more appropriate treatment is proposed to impose saturation limitation to both d1 and d2 , i.e., d1 is limited to the range of (0, 1) and d2 is limited to the range of (d1 , 1). This special treatment only works during some dynamic process for the protection of d1 and d2 and is no longer required in steadystate operation. The proposed saturation limitation strategy is simple but very effective in avoiding possible spikes in the prior art [21]. As a result, smooth and moderate regulation of both active and reactive powers is achieved.
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D. Switching Frequency Reduction For high-power DFIG-based WECS, it is essential to restrict the average switching frequency in the level of a few hundred hertz to reduce the switching losses, hence improving the system efficiency. In this paper, a switching frequency reduction strategy without affecting the system performance is proposed, which is implemented from two aspects. First, an appropriate zero vector following the first two active vectors should be selected. This has been introduced in the voltage vector selection part. Second, the smooth connection between the adjacent vector sequences should be considered to further reduce the switching frequency. Take the typical waveforms in Fig. 4(a) as an example, the selected voltage vector sequence is “011-001-000”. If we considered all possible combination of the three vectors, there are other three kinds of feasible sequences, i.e., “001-011111,” “000-001-011,” and “111-011-001.” The final selection of vector sequence is dependent on the vector sequence applied in the last control period. If the output vector sequence in the last control cycle is “100-110-111” with “111” as the ending vector, the sequence “111-011-001” will be selected, because there are no jumps between the two adjacent vector sequences. The corresponding duration of each vector should also be changed accordingly. It will be shown in the simulation results that this switching frequency reduction strategy is very effective and can reduce the switching frequency up to 48.6%. E. Compensation of Digital Delay It is well known that there is one-step delay in the digital implementation, which means that the voltage vector decided at the kth instant is not applied until the (k + 1)th instant [25], [26], [28]. To eliminate this delay, the feedback active and reactive powers should use the value at (k + 1)th instant, as shown in Fig. 3. The prediction of P k +1 and Qk +1 are obtained from (16) and (17) using the value at kth instant. The rotor voltage ur is reconstructed from the applied voltage vectors as ur = V I d1 + V II (d2 − d1 )
Fig. 5. Transient response of the prior art in [21] for a 15-kW DFIG: (a) without delay compensation and (b) with delay compensation.
(26)
where V 1 , V II are the first two active vectors. The meanings of d1 and d2 are shown in Fig. 4. The influence of one-step delay cannot be neglected for the three-vectors-based DPC approach with low control/sampling frequency, especially when the final commanding voltage vector is directly obtained based on analytical principles, such as the ripple RMS minimization in [21] or the deadbeat method in [28]. On the contrary, the influence of one-step delay is not significant for the proposed PDPC, because the duration determination method for the novel PDPC has less parameter dependence than that in [21]. To illustrate the influence of one-step delay, Figs. 5 and 6 show the responses of a 15-kW-DFIG system controlled by the method introduced in [21] and the proposed PDPC, respectively. For the aim of comparison, the test condition and machine parameters are the same as those introduced in Fig. 11 of [21]. The responses without delay compensation in Fig. 5 are similar to the experimental results shown in [21, Fig. 11], which means that the one-step delay was not considered in [21]. It is clearly seen that the active power ripple is greatly reduced when
the one-step delay is compensated. For the proposed PDPC in Fig. 6, the transient responses are acceptable even the control delay is not compensated, which proves its robustness against control delay. During the simulations, some active power ripples can be observed in Figs. 5(a) and 6(a), and there are even some small steady-state errors in the active power of Fig. 5(a), which are mainly caused by nonmodeled factor [28], [29]. Although these do not affect the system stability, if one-step delay is compensated, the power ripples can be significantly reduced, as shown in Figs. 5(b) and 6(b). To eliminate the steady-state power error or alleviate the power ripples, additional effort, e.g., integral controller, can be added in the control system, as introduced in [28]. F. Operation Under Unbalanced Grid Voltage Control of the DFIG-based WECS under the unbalanced grid voltage is an important topic, especially considering that the power generated by the wind is increasing significantly. For
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Fig. 7.
Schematic diagram of the simulated system.
TABLE IV PARAMETERS OF THE SIMULATED DFIG SYSTEM
IV. SIMULATION RESULTS
Fig. 6. Transient response of the proposed PDPC for a 15-kW DFIG: (a) without delay compensation and (b) with delay compensation.
the DPC-based control methods, it is feasible to achieve good dynamic response while controlling the current distortion and power/torque oscillations without making significant modifications on the established control structure. The key lies in the power reference generation strategy. According to specific aims, various power references can be obtained from the unbalanced grid voltages and currents (generally requiring sequence extraction). This strategy only modifies the power references in the outer loop and can accompany various DPC-based methods [31], [32]. To validate the effectiveness of power reference generation strategy, the simple switching-table-based DPC is employed in [31] and [32], and the experimental results confirm its good performance. It is also possible to extend the power reference generation strategy proposed in [31] and [32] to the proposed PDPC in this paper without modifying the control structure in Fig. 3. However, thorough discussion on operation under unbalanced grid voltage is out of scope of this paper and is not further expanded.
To verify the effectiveness of the proposed method, simulations are carried out in the environment of MATLAB/Simulink for a 2-MW-DFIG system. Fig. 7 shows the schematic diagram of the implemented system. The DFIG is rated at 2 MW and its parameters are listed in Table IV. The nominal converter dc-link voltage is set at 1200 V and the dc capacitor is 16 000 μF. As shown in Fig. 7, a high-frequency ac filter with inductance and resistance connected in parallel is connected to the stator side to absorb the switching harmonics generated by the two converters. One of the main objectives of the grid-side converter is to maintain a constant dc-link voltage and it is controlled by the classical voltage oriented control. The default sampling frequency for the simulations is 2 kHz and the simulation time step is 5 μs. The two constants in (23) are set to be CP = CQ = 2 MVA. During the simulation, the grid side converter is first enabled. After the dc-link voltage is established, the rotor side converter begins to work with the rotor speed rotating at a fixed speed. When the excited stator voltage matches the grid voltage, the stator of the DFIG is connected into the grid and operates under loaded condition. This process is not shown in the following results. A. Comparative Studies To verify the effectiveness of the novel three-vectors-based PDPC, it is compared with the prior art in [21] by G. Abad et al. under the same condition and they are referred as “proposed method” and “Abad’s method” in the following results. The one-step digital delay is compensated for both methods, as introduced in Section III-E. In this paper, the motor convention
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Fig. 9. Zoomed responses of Abad’s method and the proposed method with step change of reactive powers.
Fig. 8. Simulation results of (a) Abad’s method and (b) the proposed method, with a fixed rotor speed under the condition of step changes in active and reactive powers (ω m = 0.8 p.u.).
is assumed, so the generated power is indicated by minus sign. Fig. 8 presents the simulation results of Abad’s method and the proposed method, under the condition of stepped changes in both active power and reactive power. The rotor speed is externally controlled and fixed at 80% synchronous speed. From top to bottom, the curves shown in Fig. 8 are the reactive power, active power, three-phase stator currents, three-phase rotor currents, and the duty ratios of the first two active vectors. The commanding value of the active and reactive power are also shown in Fig. 8 and marked in red color. The reactive power
commanding steps from zero to 0.5 MVar at t = 0.4 s and then steps to −0.5 MVar at t = 0.6 s. The active power commanding steps from zero to −2 MW at t = 0.3 s and then steps to −1 MW at t = 0.5 s. It seen that the reactive power follows the commanding value quickly and accurately. There is small steadystate error in the active power of Abad’s method, as shown in Fig. 8(a), which is mainly caused by the one-step delay compensation, as introduced in Section III-E. On the contrary, this steady-state error does not exist in the proposed method due to the use of CP and Cq , which provides more flexibility to cope with various condition and is more robust against one-step delay, as shown in [16]. The short-time responses of the reactive power for both methods are shown in Fig. 9 and it is seen that the proposed method has quicker dynamic response than that of the prior art [21]. The results of Fig. 8 are obtained under the condition of a fixed rotor speed. Fig. 10 further shows the results with variable rotor speed under the condition of stepped changes in active/reactive powers for Abad’s method and the proposed method. It is seen from Fig. 10(a) that there are large overshoot in reactive power and sag in active power along with large stator currents for Abad’s method when the commanding reactive power steps from zero to 0.5 MVar. Some oscillations can also be observed in both active and reactive powers in the vicinity of the synchronous speed (t = 0.45 s). On the contrary, the responses of active and reactive powers for the proposed method are very smooth, as shown in Fig. 10(b). The phenomenons of large stator currents and oscillations in active/reactive powers in the vicinity of the synchronous speed do not exist in the proposed method, showing strong robustness against the rotor speed variation. It should be noted that during the stepped changes of active/reactive power, some oscillations appear in the active/reactive power and stator/rotor current. This imperfect decoupling in the active power and reactive power is mainly caused by the abrupt change in the vector durations. However, this oscillation has negligible influence on the system performance and does not affect the system stability, because it only lasts for a very short time of one sampling period. Increasing the constants CP and CQ in (23) can alleviate the abrupt change in the vector durations, hence reducing the oscillation, as shown in Fig. 11. To further compare the steady performances of the two PDPC methods, Figs. 12 and 13 present the harmonic spectra of the stator current and rotor current at steady state of −2-MW active power and 0.5-MVar reactive power under the condition of 80% synchronous speed, respectively. It is seen that the THDs of
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Fig. 11. Simulation results of the proposed method with C P = C Q = 3 MVA, with variable rotor speed under the condition of step changes in active and reactive powers.
TABLE V QUANTITATIVE COMPARISON OF THE TWO PDPC METHODS
Fig. 10. Simulation results of (a) Abad’s method and (b) the proposed method with C P = C Q = 2 MVA, with variable rotor speed under the condition of step changes in active and reactive powers.
the stator current and rotor current of the proposed method are only 3.47% and 3.71% (calculated up to 6000 Hz harmonics), better than the results of 3.64% and 3.92% of Abad’s method. The main harmonics are concentrated at 2 kHz and its multiples, which is in accordance with the 2-kHz control frequency of both methods. The average switching frequencies of both methods are illustrated in Fig. 14 and there are insignificant differences between them. The average commutation frequency fav is calculated by counting the total commutation instants of the six legs of twolevel inverter during a fixed period, e.g., 0.05 s in this paper. The quantitative comparisons of the two PDPC methods at steady state of P = −2 MW and Q = 0.5 MVar are summarized in Table V, including average switching frequency fav , active power ripple Trip , reactive power ripple ψrip , THDs of the stator current is and rotor current ir . The active and reactive power ripples are calculated using standard deviations of them. It is seen from Table V that the proposed method has better overall performance than that of prior art [21], except that its active power ripple and switching frequency are slightly
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Fig. 12. Stator current harmonic spectra of the PDPC methods: (a) Abad’s method and (b) the proposed method.
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Fig. 13. Rotor current harmonic spectra of the two PDPC methods: (a) Abad’s method and (b) the proposed method.
bigger. However, there is some steady error in the active power of Abad’s method, as shown in Fig. 8(a), which does not exist in the proposed method. B. Tracking Behavior For wind energy applications, the DFIG is required to operate at variable rotor speed and the torque reference may also be variable, so it is necessary to test the performance of the proposed PDPC under this circumstance. Fig. 15 shows the result of the proposed method operating with variable speed from 0.8 to 1.2 p.u. The active power reference is generated by a 5-Hz sinusoidal waveform with an amplitude of 2 MW. The reactive power reference is stepped from zero to −0.5 MVar at t = 0.4 s and then to 0.5 MVar at t = 0.6 s. It is seen that both the active and reactive powers track the commanding value well under the condition of variable rotor speed, validating the tracking capability of the proposed method.
Fig. 14.
Average switching frequencies of the two PDPC methods.
C. Robustness Test In this part, the robustness against machine parameter variations is tested for the proposed PDPC method. On one hand, the actual stator resistance and rotor resistance may differ from the value used in the control system due to the influence of temperature, skin effect, etc. On the other hand, the mutual inductance
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Fig. 17. Simulation results of the proposed PDPC with switching frequency reduction.
Fig. 15. Tracking behavior of the proposed PDPC with variable rotor speed under the condition of sinusoidal active power reference and stepped reactive power reference. Fig. 18. Average switching frequencies of the proposed PDPC with and without switching frequency reduction strategy.
result shown in Fig. 8(a), exhibiting strong robustness to the parameter variations. D. Switching Frequency Reduction
Fig. 16. Simulation results of the novel PDPC with machine parameter mismatches (R s = 1.5R s , R r = 1.5R r , and L m = 1.5L m ).
may also be variable due to the magnetic saturation. The leakage inductance of the stator and the rotor can be considered constant because their variations during operation are insignificant [28]. Fig. 16 shows the result of the proposed PDPC with parameter mismatches. The test condition is the same as that in Fig. 8(a), except that the stator resistance, rotor resistance, and mutual inductance used in the control system are all increased to 150% of their actual value. It is seen that there is very insignificant difference in the steady and dynamic responses compared to the
In this part, the switching frequency reduction strategy introduced in Section III-D will be tested. Prior simulation results do not consider the switching frequency reduction and the resulting switching frequency is slightly higher than 1 kHz. It is very desirable to reduce the switching frequency to the level of a few hundred hertz, which is attractive for high-power wind energy applications. Fig. 17 presents the simulation results of the proposed PDPC with switching frequency reduction. The sampling frequency is still 2 kHz and the test condition is the same as that in Fig. 8(a). It is seen from Fig. 17 that, compared to Fig. 8(a), the steady-state performances are deteriorated and more ripples in the reactive power can be observed. However, the steadystate performance of Fig. 17 is still acceptable considering the low switching frequency. Fig. 18 further illustrates the average switching frequencies during the time of 0.35 to 0.65s for Figs. 8(a) and 17. It is seen that the average switching frequency is reduced up to 48.6%, from 1.29 kHz in Fig. 8(a) to 663 Hz in Fig. 17, by adopting the switching frequency reduction strategy in this paper.
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TABLE VI QUANTITATIVE COMPARISON OF THE TWO PDPC METHODS CONSIDERING SWITCHING FREQUENCY REDUCTION
Fig. 19. Simulation results of a complete DFIG-based wind generation system with step changes in the wind speed.
The switching frequency reduction strategy can also be applied to the prior art in [21]. Table VI summarizes the quantitative results of the proposed PDPC and Abad’s method at steady state of P = −2 MW and Q = 0.5 MVar. The conclusion is similar to that drawn from Table V, where the switching frequency reduction is not considered. E. Responses Under Torque Control In the prior simulations, the DFIG was assumed to be in speed control, i.e., with the rotor speed set externally, since in a practical system, the wind turbine’s large inertia results in slow rotor speed change [33]. In this part, a complete wind generation system is simulated, which includes a typical 2-MW wind turbine and the DFIG. The DFIG is set in torque control, which means that the rotor speed is the result of both stator/rotor voltage/current and the mechanical torque driven by the wind turbine. To shorten the simulation time, the inertia is set to a small value. Fig. 19 shows the simulation results when the wind speed steps from 6 to 12 m/s at t = 0.3 s and then to 8.5 m/s at t = 1.3 s. The active power reference value is obtained from the maximum power-tracking curve [34]. The reactive power is step changed from zero to −0.4 MVar at t = 0.6 s and then to −0.2 MVar at t = 1.5 s. It is seen from Fig. 19 that the simulation results are satisfactory and maximum power tracking is achieved when the wind speed varies.
Fig. 20. Experimental results of steady-state performance: (a) Abad’s method, left: without delay compensation, right: with delay compensation and (b) the proposed PDPC, left: without delay compensation, right: with delay compensation.
V. EXPERIMENTAL TESTS Apart from the simulation validation from a 2-MW-DFIG system, a scaled-down 20-kW experimental prototype is also constructed to test the proposed PDPC. The test setup is the same as that introduced in [22], except the sampling frequency is 2 kHz in this paper. A dSpace 1104 control board was employed to implement the real-time algorithm coding using C language. All the experimental results are recorded using the ControlDesk interfaced with DS1104 and PC. The DFIG is driven by a dc motor at 1200 r/min (sub-synchronous speed) during the tests, unless explicitly indicated. The steady-state response of P and Q for both methods are shown in Fig. 20. The results obtained without delay compensation are shown in the left and the results with delay compensation are shown in the right. It is clearly seen that, without one-step delay compensation, the ripples of both P and Q are much bigger. The proposed PDPC presents lower power ripples than Abad’s method. A quantitative comparison for both methods is illustrated in Table VII. When the control delay is compensated, the active and reactive power ripples are reduced up to 42.2% and 58.6%, respectively, for Abad’s method. Similar reduction in power ripples can be observed for the proposed PDPC with better steady-state performance. The harmonic performance of stator
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TABLE VII QUANTITATIVE COMPARISON OF STEADY STATE AT 1200 R/MIN, * DONATES ONE-STEP DELAY COMPENSATION
Fig. 22. Rotor current harmonic spectra: (a) Abad’s method, THD = 10.39% and (b) the proposed PDPC, THD = 7.43%.
Fig. 21. Stator current harmonic spectra: (a) Abad’s method, THD = 8.13% and (b) the proposed PDPC, THD = 5.98%).
and rotor currents for the proposed PDPC is also better than that of Abad’s method, as shown in Figs. 21 and 22. Apart from the steady-state performance comparison, the dynamic responses of both methods under the condition of stepped change in P from 0 to −0.75 p.u. are shown in Fig. 23, where the reactive power is fixed at Q = −0.5 p.u. It is seen that both methods achieve similar dynamic performance, while much lower active power ripple and less current harmonics can be observed in the proposed PDPC, which validates the superiority of the proposed PDPC. Fig. 24 further presents the dynamic responses of both methods when the reactive power steps from 0 to −0.75 p.u., where the active power is fixed at −0.5 p.u. and the rotor speed is at hyper-synchronous speed of 1800 r/min. It is seen that the proposed PDPC exhibits much lower harmonics in both stator and rotor currents. There are less reactive power ripples in the proposed method than that of Abad’s method, but the difference is
Fig. 23. Experimental results of dynamic response with Q = −0.5 p.u. and P steps down to −0.75 p.u. at 0.1 s: (a) Abad’s method and (b) the proposed PDPC.
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ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped to improve the quality of this paper. REFERENCES
Fig. 24. Experimental results of dynamic response with P = −0.5 p.u. and Q steps from 0 p.u. down to −0.75 p.u. at 0.1 s when the rotor speed is at hypersynchronous speed of 1800 r/min: (a) Abad’s method and (b) the proposed PDPC.
not as significant as that in Fig. 23. The dynamic performances of both methods are similar. VI. CONCLUSION This paper proposes an improved three-vectors-based PDPC to achieve both active power and reactive power ripple reduction with simple calculation. Compared to prior art using two switching tables, this paper employs only one switching table to select three appropriate voltage vectors. The duration of each vector is obtained in a much simpler way, which brings the benefits of simplicity and robustness against machine parameters variations and control delay. By dynamically selecting the most appropriate candidate vector sequence, the switching frequency can be reduced up to 48.6%, which is useful for high-power wind energy applications. The effectiveness of the proposed PDPC is validated by a series of simulation results from a 2-MW-DFIG system. The experimental results from a scaled-down laboratory setup are also presented to confirm the theoretical analysis of the proposed PDPC.
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Yongchang Zhang (M’10) received the B.S. degree from Chongqing University, Chongqing, China, in 2004, and the Ph.D. degree from Tsinghua University, Beijing, China, in 2009, both in electrical engineering. From August 2009 to August 2011, he was a Postdoctoral Fellow at the University of Technology Sydney, where he was engaged in PMSM drives for PHEV and control of doubly fed induction generator for wind energy applications. Since August 2011, he has been with the North China University of Technology and is currently an Associate Professor in electrical engineering. His current research interest is model predictive control for power converters and motor drives.
Jiefeng Hu (S’12) received the B.E. and M.E. degrees from the Beijing University of Aeronautics and Astronautics, Beijing, China, in 2007 and 2009, respectively, both in electrical engineering. He is currently working toward the Ph.D. degree at the University of Technology, Sydney (UTS), Ultimo, N.S.W., Australia. From January 2011 to March 2013, he was involved in the research of the minigrid within CSIRO’s Energy Transformed Flagship, Australia. He is currently a Research Associate in UTS. His research interests include electric drives, control of power converters, and microgrids. Mr. Hu received the NSW/ACT Postgraduate Student Energy Awards by the Australian Institute of Energy, in October 2010.
Jianguo Zhu (S’93–M’96–SM’03) received the B.E. degree from the Jiangsu Institute of Technology, Nanjing, China, in 1982, the M.E. degree from the Shanghai University of Technology, Shanghai, China, in 1987, and the Ph.D. degree from the University of Technology, Sydney (UTS), Ultimo, N.S.W., Australia, in 1995, all in electrical engineering. He is currently a Professor of Electrical Engineering and the Head for School of Electrical, Mechanical and Mechatronic Systems at UTS. His current research interests include electromagnetics, magnetic properties of materials, electrical machines and drives, power electronics, and green energy systems.
