Comparison of Six Digital Current Control Techniques for Three-Phase Voltage-Fed PWM Converters Connected to the Utility Grid Robinson F. de Camargo
Humberto Pinheiro
Power Electronics and Control Research Group – GEPOC Federal University of Santa Maria - UFSM ZIP CODE: 97105-900 – Santa Maria, RS – Brazil
[email protected] [email protected]
I. INTRODUCTION More and more voltage-fed three-phase converters have been used to provide improvements in the current drawn equipments connected to the grid. In three-phase three-wire systems applications stand out the three-phase PWM rectifiers, shown in Fig. 1, which offer a better alternative with respect THD of the input currents, displacement power factor and low output voltage ripple if compared with traditional uncontrolled or phase controlled rectifiers. In addition, PWM rectifiers present some features that actually are required in the market today that are: near unit power factor, bidirectional power flow and fast dynamic responses [1]. Regarding the control, the use of digital techniques introduces advantages as flexibility and fixed switching frequency if compared with analogue counterparts [2]. Digital techniques [2-14, 17-29, 32] can be classified in linear and nonlinear [3]. Among the nonlinear ones stand out the Variable Structure Controllers (VSC). This is a broad class of nonlinear controllers that may include slide mode, hysteresis and delta modulation controllers [2]. Usually, they result in robust systems with good transient responses. However, they often merge the controller with the modulation circuit making it difficult the test and debugging. In addition, they usually operate with variable frequency or require complex control laws. Other nonlinear controllers such as feedback linearization [11, 15], passivity [32] and Lyapunov-based control [10] also are considered in the literature for PWM converters. The DC and AC side dynamics are usually considered in the design. The additional complexity of these approaches may only be justified in systems where the DC side filter is small. Other nonlinear controllers such as the ones based on artificial neural networks [22] and fuzzy logic [2] are not strong candidates since the PWM converter models are well known and its nonlinearity is not severe. Although nonlinear controllers have good performance, linear controllers are usually
0-7803-9033-4/05/$20.00 ©2005 IEEE.
adopted since its design and implementation are simple and they are well known. Linear controller for the AC currents of the three-phase PWM rectifiers can be implemented in stationary and synchronous frames. Among them are: stationary [2, 3, 8, 12, 18] and synchronous PI [2, 3, 8, 16], state feedback [30, 31], predictive [2], deadbeat [30] and repetitive [24] controllers. In an isolated manner, the performance of linear current controllers for three-phase voltage-fed PWM rectifiers [3, 8, 12, 16, 18, 24, 30, 31] has been investigated in the literature. Some papers make a comparative study of current controllers [22, 25, 26, 31], however, they do not investigate the performance of most popular linear current controller under unbalance and harmonics in the grid voltages. This paper proposes a comparative analysis for six linear digital current control techniques applied to three-phase voltage-fed PWM rectifiers in stationary frames (STF) and synchronous frames (SYF), where the performance of these controllers are investigated under unbalance and harmonics in the grid voltages. Moreover, basic requirements are established and comparison criteria are used, which justified the current controllers selected. Also, abacuses, tables and experimental results are presented to demonstrate the similarity and differences of these controllers. The comparison contributes as a tool for the designers to select the best current control technique for a given application. PCC OTHERS 3 LOADS
Abstract – This paper presents a comparative analysis of six digital current control techniques applied to three-phase voltage-fed PWM converters connected to the utility grid. Basic requirements are established and comparison criteria are used to justify the selected current control technique. The comparison contributes for the selection of the best current control technique for a given application. Finally, abacuses, tables and experimental results are shown in order to present the performance of these controllers under unbalances and harmonics in the grid voltages.
3φ AC MAIN Lf
+ v - dc
LOADS
Three-phase three-wire PWM Rectifier
Fig. 1 – Basic scheme of the three-phase three-wire systems in the presence of a three-phase PWM rectifier.
II. BASIC REQUIREMENTS FOR SELECT CURRENT CONTROL TECHNIQUES There are many discrete current control techniques in the literature as mentioned in the Introduction. In order to restrict the comparative analysis basic requirements are established, as follow:
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(i) Current control techniques are designed considering the digital implementation. Due the flexibility, because the controllers can be modified easily through of software; (ii) Simplicity, in terms of computational effort, in order to make it possible implementation in a fixed-point and low cost DSP controllers; (iii) Grid voltages are measurement. Although the use of voltage estimation technique reduces the number of sensors [17], they usually result in a reduced stability margin [12].
IV. CURRENT CONTROL TECHNIQUES By taking into account the basic requirements in the Section II, six current control techniques have been selected. The basic structure of these controllers is given by the block diagram in the Fig. 2. The main characteristics of these current controllers are presented bellow. Three-phase PWM Rectifier
Lf
+ -
III. COMPARISON CRITERIA SVM
The key point in a comparative analysis is the definition of suitable comparison criteria, as well as selection of realistic test situations. The following comparison criteria have been selected to carry out the comparative analysis on the AC side PWM rectifiers.
ux
αβ or dq Transformation
DIGITAL CURRENT CONTROLLER
A. Power Factor (PF)
iy
This is defined as the ratio of the active power (W) of the fundamental wave, to the apparent power (VA) of the fundamental wave, according to IEEE Std. 100-1996, that is, P DF = . (1) S B. Displacement Power Factor (DPF) This is defined as the ratio of the active power (W) of the fundamental wave, to the apparent power (VA) of the fundamental wave, according to IEEE Std. 100-1996, that is, V I cos ( θ1 − φ1 ) DPF = 1 1 = cos ( θ1 − φ1 ) . (2) V1 I1 C. Unbalance Factor (UF)
ix
-
+
-
refy +
refx
vdc VOLTAGE CONTROLLER + vdc*
Fig. 2 – Block diagram of algorithms and controllers for three-phase PWM rectifiers, where x and y represent αβ or dq coordinates.
A. Deadbeat Current Control in Stationary Frame (DCC) This technique stand out for its simplicity [4], as follow in Fig. 3. However, it has stability problems under parameter uncertainties, as demonstrated in [12]. Moreover, the deadbeat response is not significant feature for PWM rectifiers. As a result of the large bandwidth of this controller, the output currents are sensitive to noises and THD of the input currents are larger than the other techniques. Grid
Unbalance factor in the input currents of the rectifier can be defined as the maximum deviation from the average of the three-phase current, divided by average of the three-phase current, as definition in the IEEE Std.1159-1995, that is, ⎛ I frms − I avg UFi = ⎜ ⎜ I avg ⎝
uy
max
⎞ ⎟ 100 ; ⎟ ⎠
wαβ(k) 2z-1
Fαβ
refαβ(k) + _
eαβ(k)
z-1
kbase
+
uαβ(k)
+
_ z-1
(3)
Hαβ
+ +
z-1I +
Cαβ
i( k )
Gαβ
Robust Deadbeat Plant in STF
Fig. 3 – Basic scheme of the deadbeat current controllers in STF.
D. Total Harmonic Distortion (THD) Total harmonic distortion in the input currents of the rectifier: for this the THD of currents are used. It is defined in the IEEE Std. 519-1992 or IEC 61000-2-2, as follow: ⎛ ∞ ⎞ THDi = ⎜ ∑ I h2 I1 ⎟ 100 ; (4) ⎜ h=2 ⎟ ⎝ ⎠ E. Computational Effort (CE) The differences among the techniques with respect their computational effort can be measured through number of instructions per sampling cycle implemented in a DSP controller.
The main equations of the controller in discrete domain are: u αβ ( k ) = kbaseeαβ ( k − 1) + 2w αβ ( k − 1) − u αβ ( k − 1) , (5) where: eαβ ( k ) = ⎡⎣refαβ ( k ) − i αβ ( k ) ⎤⎦ ; kbase = Lm ( Z baseTs ) . (6) Current references of α and β axes at the internal current loop respectively are: ref α (k ) = ucc ( k ) vα ( k − 1) , (7) , refβ (k ) = ucc ( k ) vβ ( k − 1) (8) where, ucc(k) is the control action of voltage loop, which can be understood as the equivalent conductance of the load. The
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ucc(k) will be assumed constant, since the objective is to compare the performance of the current controllers. B. Resonant Current Regulators in Stationary Frame (RCR) This current regulator is very simple, as the previous controller. This is based in a resonant regulator that ensures zero steady-state error for sinusoidal references and disturbances, similar to the repetitive technique [24]. The RCR is implemented through a bandpass filter [18, 22] in the discrete-time, as shown in the Fig.4. However, if the grid frequency varies, it is necessary to implement a grid frequency adaptation algorithm. This algorithm increases the complexity, which offsets the main feature of this controller, that is, its simplicity.
ref d (k ) = ucc ( k ) , ref q (k ) = 0 ,
(17) where, ucc(k) is the control action of voltage loop, in all cases ucc(k) is considered constant. Grid
wdq(k) Fdq
refdq(k) + e dq(k) -
refαβ(k) + _
eαβ(k)
b2
z-2 b1
+ +
_
a2
udq(k)
H dq
+
z-1I
+
ψ(k)
Cdq
i(k)
+ Gdq
Fig. 5 – Basic scheme of the PI current controllers in SYF.
D. Reactive and Active Power Control in Synchronous Frames (RAPC)
Fαβ z-2
uαβ(k)
_ a1
PI
Plant in SYF
Grid
wαβ(k)
(16)
z-1
Hαβ
+ +
z-1I +
Cαβ
i( k )
Gαβ
Bandpass filter Plant in STF
Fig. 4 – Basic scheme of the resonant current regulators in STF.
The main equations of the controller in discrete domain are: uαβ ( k ) = b1eαβ ( k ) + b2eαβ ( k − 2 ) − a1uαβ ( k − 1) − a2uαβ ( k − 2 ) , (9) where: eαβ ( k ) = ⎡⎣refαβ ( k ) − i αβ ( k ) ⎤⎦ , (10) and b1, b2, a1 and a2 are the designed coefficients of the bandpass filters [19]. Current references of α and β axes at the internal current loop respectively are: ref α (k ) = ucc ( k ) vα ( k ) , (11) refβ (k ) = ucc ( k ) vβ ( k ) , (12)
Several techniques are proposed to control directly reactive and active power [13, 14, 21, 26]. The power controllers present in [14, 21, 26] used different concepts to calculate the reactive and active power. The first reference is more complex because it uses positive and negative sequence controllers while the last two use the dc-link voltage to calculate the power. The technique proposed in [13] is considered here due its simplicity. The instantaneous reactive and active power are indirectly controlled, as shown in the Fig. 6. It is assumed that the grid voltage is constant in dq axis, however under large voltage unbalance the THD in the input current may be significant. Rede
wdq(k) Fdq
refdq(k) + e dq(k) -
C. Synchronous Frame PI Current Regulator (SFPI) This is the most popular controllers used in SYF [2, 3, 8, 16, 22]. It uses two PI compensators, one in the d and other in q axis. This controller guarantees zero error in steady state which is a superior feature if compared with the PI controller in STF [18]. However, this technique usually does not takes into account the coupling between the d and q axis, under reference steps the dynamics of the d axis affects the q axis current and vice versa. Many synchronization methods can be used, here for simplicity purpose the MSRF [31] have been adopted. The main equations of the controller in discrete domain are: u dq ( k ) = K1e dq ( k ) − K 2e dq ( k − 1) + u dq ( k − 1) , (13) where: edq ( k ) = ⎡⎣ref dq ( k ) − i dq ( k ) ⎤⎦ , (14) and the gains are: K1 = K PZ + K IZ e K 2 = K PZ , (15) The current references of the d and q axis at the internal current loop respectively are:
PI
udq(k)
Hdq
+ +
z-1I
ψ(k)
Cdq
i(k)
. ;x
P (k); Q (k)
+ Gdq
Plant in SYF
Fig. 6 – Basic scheme of the PI power controllers in SYF.
The main equations of the controller in discrete domain are: u PQ ( k ) = K1e PQ ( k ) − K 2e PQ ( k − 1) + u PQ ( k − 1) , (18) where: eP ( k ) = [ ref P ( k ) − P ( k )] , (19) eQ ( k ) = ⎡⎣ refQ ( k ) − Q ( k ) ⎤⎦ and the gains are: K1 = K PZ + K IZ e K 2 = K PZ , (20) The power references of the d and q axis at the internal current loop respectively are: ref d (k ) = ucc ( k ) , (21) ref q (k ) = 0 , (22)
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E. Decoupling by State Feedback Plus dq Axis Servo Controllers (DSFdq) The theory of the decoupled by state feedback very well discuss in [20, 28-30]. This is based in the obtaining of decoupling matrixes Mdes and Kdes, so that three-phase PWM rectifier can be represented as two SISO, which simplify the design of controllers in the dq axis, as shown in the Fig.7. However, this technique used a dq servo controller to guaranteed zero error in steady state, what increase more complexity to the implementation algorithms, if compare to the others current controllers.
vq ( k ) = vq ( k − 1) + ref q (k ) − iq ( k ) .
(32) where, the reference of the q axis at the internal current loop respectively is: ref d (k ) = ucc ( k ) , (33) ref q (k ) = 0 , (34)
refdq(k)
Grid wdq(k)
Feedforward of the Grid
Grid
wdq(k )
wdq(k)
Rp
Fdq
udq_des(k)
+ Mdes
+
Fd q
vdq(k)
refdq(k)+ -
+
+
vdq(k -1)
z-1I
K1
+ -
+ Mdes
udq(k) H dq
+
++ +
z-1I
ψ(k)
Cdq
+
z-1I
ψ(k)
+
Cdq
i(k)
Gdq
+
Plant in SYF
Decoupled Controller By State Feedback
Plant in SYF Decoupled controller by State Feedback
+
i(k)
Gd q
servo controller
H dq
Kdes
Fig. 8 Basic scheme of the decoupling by state feedback plus q servo controllers in SYF.
Kdes
K22
Fig. 7 – Basic scheme of the decoupling by state feedback plus d and q servo controllers in SYF.
V. COMPARISON ANALYSIS
This controller is similar to the previous subsection [28, 29], with the advantage that, it is not necessary an additional servo controller in the d axis, besides to guarantee zero error, only design of Rp matrix [28]. However, this controller does not take into account unbalance and harmonics in the grid voltages. Moreover, as a result of the large bandwidth in the d axis there is an increase of THD in the input currents. The main equations of the controller in discrete domain are: T udq _ des ( k ) = Mdes ⎡⎣ud ( k −1) uq ( k −1) ⎤⎦ + Kdesψdq ( k ) + Rpwdq ( k ) , (29) where: T (30) ψ ( k ) = ⎡⎣i dq (k − 1) u dq _ des ( k − 1) ⎤⎦ , and the equation of servo controllers is: T (31) uq ( k ) = k1q vq ( k ) − k2 q ⎣⎡iq (k ) uq (k − 1) ⎦⎤ , where, the subscript x represents d or q axis. State equation of the integrator, given by:
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15 12 9
.
F. Decoupling by State Feedback plus q Axis Servo Control (DSFq)
This section presents a comparative analysis of the six digital control techniques as described in the previous section. Fig. 9 presents the effects of grid voltage unbalances on the unbalance of the currents drawn by the PWM rectifier. It is possible to observe that the best performances in terms of the current unbalance are obtained with the RAPC controller. The dq current controllers have similar performances and the DCC and RCR techniques present the worst performance. Fig. 10 shows the effects of grid voltage unbalances on the THD in the currents drawn by the PWM rectifier. It is possible to see for the DCC and the RCR the voltage unbalance does not affect the input current THD. Due to the frequency response of the RCR controller, it presents the best result in this case. The dq current control techniques present a good performance for grid voltage with unbalance lower than 10%. The RAPC present a good performance when the grid voltage unbalance is small, but its performance degrades as the unbalance of the grid voltage increase.
UFi (%)
The main equations of the controller in discrete domain are: T u dq _ des ( k ) = M des ⎣⎡ud ( k − 1) uq ( k − 1) ⎦⎤ + K des ψ dq ( k ) , (23) where: T (24) ψ ( k ) = ⎡⎣i dq (k − 1) u dq _ des ( k − 1) ⎤⎦ , Equation of servo controllers is: T u x ( k ) = k1x vx ( k ) − k2 x [ix (k ) ux (k − 1)] , (25) where, the subscript x represents d or q axis. State equation of the integrator, given by: vx ( k ) = vx ( k − 1) + ref ix (k ) − ix (k ) . (26) where, the references of the d and q axis at the internal current loop respectively are: ref d (k ) = ucc ( k ) , (27) ref q (k ) = 0 , (28)
6 3 0
0
5
10
15
20
25
% Grid Voltage Unbalance Fig. 9 – Grid voltage unbalances versus UF in the PWM rectifier currents drawn.
Fig. 11 presents the input current THD under harmonics in the grid voltages. The 5th, 7 th and 11 th harmonics have been considered in the grid voltages with the same amplitude. The all cases THD of the input currents increases with THD of the grid voltages. Fig. 12 shows the comparison of the number of instructions per sampling cycle to the six current control techniques. The two αβ STF present the best performance and dq SYF needs more computational effort, because the dq transformation and synchronization method required.
design parameters in Table I. The DSP TMS320F241 was used to the implementation of the control algorithms, mainly due it low cost and satisfactory performance for this application. Figs. 13-18 show experimental results of the three-phase PWM rectifier. The grid voltages present THDv=2.5 % and UFv=5.8 %. All techniques present similar DPF and PF under these disturbances. The DSFdq presents the lower THDi, the SFPI and DSFq exhibit a good performance and the DCC and RCR presented the worst performance under these conditions.
15
TABLE I. PARAMETERS TO THREE-PHASE PWM RECTIFIER. Parameters Quantities Grid frequency 60 Hz Input filter (L) 2,5 mH Sampling frequency in the current loop 10 kHz Switching frequency 10 kHz DSP TMS320F241
THDi (%)
12 9 6 3 0
0
5
10
15
20
25
% Grid Voltage Unbalance Fig. 10 – Grid voltage unbalances versus THD in the currents drawn.
25
Fig. 13. Experimental results using DCC. Three-phase input currents of the PWM rectifiers and phase-to-neutral grid voltage. Horizontal scale: 5 ms/div. Vertical scale of currents: 2 A/div. Vertical scale of voltage: 10 V/div. DPF=0.998, PF=0.996, THDi=5.8 %, UFi=1.4%.
THDi (%)
20 15 10 5 0
0
5
10
15
20
25
Fig. 14. Experimental results using RCR. Three-phase input currents of the PWM rectifiers and phase-to-neutral grid voltage. Horizontal scale: 5 ms/div. Vertical scale of currents: 5 A/div. Vertical scale of voltage: 10 V/div. DPF=0.999, PF=0.998, THDi=4.2 %, UFi=0.92%.
% Grid Voltage Harmonics Fig. 11 – Grid voltage unbalances versus THD in the currents drawn.
Number of instrutions
300 250 200 150 100 50 0
Fig. 15. Experimental results using SFPI . Three-phase input currents of the PWM rectifiers and phase-to-neutral grid voltage. Horizontal scale: 5 ms/div. Vertical scale of currents: 5 A/div. Vertical scale of voltage: 10 V/div. DPF=0.999, PF=0.999, THDi=3.5 %, UFi=2.1%.
Fig. 12 – Number of instructions of each current control technique.
VI. EXPERIMENTAL RESULTS This section presents experimental results for the threephase PWM rectifier with current control strategies presented previously. The experimental setup operates according to the
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VII. CONCLUSION
Fig. 16. Experimental results using RAPC. Three-phase input currents of the PWM rectifiers and phase-to-neutral grid voltage. Horizontal scale: 5 ms/div. Vertical scale of currents: 5 A/div. Vertical scale of voltage: 10 V/div. DPF=0.999, PF=0.998, THDi= 5.3 %, UFi=4 %.
Fig. 17. Experimental results of DSFdq. Three-phase input currents of the PWM rectifiers and phase-to-neutral grid voltage. Horizontal scale: 5 ms/div. Vertical scale of currents: 5 A/div. Vertical scale of voltage: 10 V/div. DPF=0.999, PF=0.999, THDi= 2.7 %, UFi=2.6 %.
In this paper a comparative study of digital current control are presented. The basic requirements and suitable criteria are used to compare these techniques. Six digital current controllers have been select emphasize their different characteristics. Then, a comparative analysis has been carried out a three-phase PWM rectifier setup with the same parameters as follow in Table I. In this analysis was considered unbalance and harmonics in the grid voltages. Table II summarizes the controller performances, where each current control has been availed according to three different indexes, which emphasize the main advantages or disadvantages for each technique. This comparison contribute as a important tool for the designers in the selection of the digital current control, where it was considered real conditions as unbalance and harmonics in the grid voltages. It is important emphasize that the MSRF method is used for simplicity, however it present distortion in the synchronism signals. This problem can be reducing using another synchronization method that considers these disturbances. ACKNOWLEDGMENT The authors would like to thank the CEEE, CAPES, CNPq and FAPERGS for the financial support.
Fig. 18. Experimental results of DSFq. Three-phase input currents of the PWM rectifiers and phase-to-neutral grid voltage. Horizontal scale: 5 ms/div. Vertical scale of currents: 5 A/div. Vertical scale of voltage: 10 V/div. DPF=0.999, PF=0.998, THDi=3.6 %, UFi=3.2 %.
TABLE II – COMPARATIVE ANALYSIS TO THE FOUR CONTROL TECHNIQUES THDi of the input current under: Unbalance current Displacement Power under unbalance Grid voltage Harmonics in the Unbalance plus Power Factor Factor voltage main Unbalance grid voltages harmonics grid Control Techniques voltages DCC 1º 1º 3º 2º (>10%) 3º 3º RCR 1º 1º 3º 1º (>2%) 1º 3º SFPI 1° 1° 2º 2º (6%) 3º 3º DSFdq 1° 1° 2º 2º (
