4. CONTROL OF ZERO-SEQUENCE CURRENT IN PARALLEL THREE-PHASE CURRENT-BIDIRECTIONAL CONVERTERS

Transcription

1 4. CONRO OF ZERO-SEQUENCE CURREN IN PARAE HREE-PHASE CURREN-BIDIRECIONA CONVERERS 4. A NOVE ZERO-SEQUENCE CURREN CONRO 4.. Zero-Sequence Dynamics he parallel boost rectifier moel in Figure.4 an the parallel voltage source inverter in Figure.8 show that the zero-sequence ynamics are governe by their z channels. It is interesting to note that both moels have the same z channel equivalent circuit except for the current irection. Since v = v + v + v ) ( v + v + v ), (4.) z c ( a b c a b c the zero-sequence current is etermine by the ifference in their common-moe voltages For a single converter, the common-moe voltage oes not cause any zero-sequence current because physically there is no such current path. he z channel is actually an open circuit. Besies, the common-moe voltage oes not affect the converter control objectives, such as voltage regulation an current control. herefore, the z channel is normally not consiere in the control esign for a single converter. When the two converters are in parallel, the zero-sequence current path is forme. A small ifference between the two common-moe voltages may cause a large zerosequence circulating current, because the z channel is an unampe circuit with only inuctors, an their ESRs in practical cases. Figure 4. shows the z channel moel of the parallel boost rectifier system. he current irection is counter clockwise for the parallel voltage source inverter system. 77

2 + i z v z c Figure 4. Zero-sequence ynamics moel of parallel boost rectifiers. 4.. A New Control Variable In the moulation of the switching network for the three-phase converters, a spacevector moulation (SV) technique is commonly use, for example, [], [4], [57]. Figure 4. shows that the reference vector is in sector I. It can be synthesize by the vectors pnn an ppn, an their uty cycles an can be obtaine by projecting the reference vector onto the two vectors, as escribe in Figure 4.. he uty cycle of the zero vector is: 0 =. (4.) npn ppn npp III IV nnp II V ref I VI pnp pnn Figure 4. Space-vector moulation in three-phase current-biirectional converters. o optimize switching losses, total harmonics istortion an maximum moulation inex, ifferent SV schemes prouce ifferent triple harmonics by istributing 0 ifferently [54]-[6], thus result in ifferent z. For example, Figures 4.3 shows one typical PW pattern in one switching cycle. he PW pattern is so-calle minimalloss PW, in that only two phase legs switch in one cycle. At every one-sixth AC line 78

3 cycle (/60Hz, for example), one of the phase legs that carries the highest current oes not switch. a s a b s b c s c (pnn) (ppn) Figure 4.3 inimal-loss SV. 0 (ppp) With this PW pattern, the z-channel uty cycle can be calculate: z a + b + c = + ( + 0) + 0 = + + =. (4.3) With this scheme, the phase-leg uty cycles an the uty cycle z are shown in Figures 4.4 an 4.5, respectively. 0.0 (-) : t(s) c (-) (-) : t(s) b (-) (-) : t(s) a (-) t(s) Figure 4.4 Phase-leg uty cycles a, b an c with minimal-loss SV. 79

4 .0 (-) : t(s) ((a+b)+c).5 (-) t(s) Figure 4.5 Duty cycle z with minimal-loss SV. Figure 4.6 shows another commonly use PW pattern with alternating zero vectors. a s a b s b c s c (nnn) (pnn) (ppn) (ppp) Figure 4.6 Alternating zero-vectors SV. With this PW pattern, the z-channel uty cycle can be calculate: z = a + b + c = ) + ( ) = ( 0. (4.4) With this scheme, the phase-leg uty cycles an the uty cycle z are shown in Figures 4.7 an 4.8, respectively. 80

5 Figure 4.7 Phase-leg uty cycles a, b an c with alternating zero-vectors SV. Figure 4.8 Duty cycle z with alternating zero-vectors SV. Although ifferent schemes have ifferent phase-leg uty cycles, they always have the same phase-to-phase uty cycles, as shown in Figure 4.9. he phase-to-phase uty cycles are use in the phase-to-phase averaging [53]. 8

6 Figure 4.9 Phase-to-phase uty cycles ab, bc an ca. he istribution of the zero vectors can vary without affecting the phase-to-phase uty cycles an the control objectives, such as the input AC currents an the output DC voltage. his inicates that z can be controlle by controlling the istribution of 0. Base on this fact, a new control variable k is introuce as follows: k = ppp, (4.5) where ppp is the uty cycle of the zero vector ppp, as illustrate in Figure 4.0. a s a b s b c s c ( 0 k) (nnn) (pnn) (ppn) k (ppp) Figure 4.0 New control variable k. Usually, for the scheme with alternating zero vectors, k=0.5 0, as shown in Figure 4.6. With the efinition in (4.5), z becomes z a b c ( + 3 = + + = + + k) + ( + k) + k = + k. (4.6) 8

7 herefore, z v c = (( + + 3k ) ( + + 3k )) v = 3( k k) v c c, (4.7) assuming both converters have the same reference vector, thus the same an. As a result, the new average moel of the zero-sequence ynamic with the new control variable k is shown in Figure i z 3 ( k k ) v c Figure 4. Zero-sequence ynamics moel with new control variable k. he efinition of the control variable is base on the SV scheme with alternating zero vectors. he scheme in Figure 4.3 cannot be use for the irectly parallel threephase converters. First of all, the k is always equal to 0 an not variable. Seconly, the scheme has a iscontinuous average z-channel uty cycle. he iscontinuity causes a strong zero-sequence isturbance to the parallel system [4]. herefore, the scheme is practically not usable for the irectly parallel three-phase converters, although the scheme has less switching losses Implementation Since the z-channel is basically a first-orer system, the control banwith of the zero-sequence current loop can be esigne to be very high, an a strong current loop that suppresses the zero-sequence current can be achieve. In the esign of the single converter with current loops, normally only two current sensors are neee, because the sum of the three phases currents is always zero. With two converters in parallel, three current sensors are neee in orer to measure the zerosequence current. Figures 4. an 4.3 show the implementation of the zero-sequence current control for a two-parallel three-phase boost rectifier system an a two-parallel voltage source inverter system. In a two-parallel converter system, it is sufficient to 83

8 control the zero-sequence current in one of the two converters since there is only one zero-sequence current. v A i a i p v c v B v C i b i c v a v b v c C R / abc qz i z 0 H z k 0 k 0 k ref (0) v c v cref i i q SV Current Controller q i ref i qref = 0 Voltage Controller i p i a v i a b v b i c v c abc q i i q SV Current Controller q i ref i qref = 0 Figure 4. Control implementation for parallel boost rectifiers. 84

9 v c i p v b v a i a i b v B v A v c i c v C v C abc qz abc q C R/ ref (0) k i z v v q k 0 k 0 H z 0 i q i Voltage Controller v ref v qref q SV Current Controller i ref i qref i p i a v b v a i b v c i c abc q i q i q SV Current Controller i ref i qref Figure 4.3 Control implementation for parallel voltage-source inverters. he converter in Figures 4. an 4.3 has the same control esign as that use for a single converter, which normally has an q channels current loops. Besies the regular an q channel control, the converter in Figures 4. an 4.3 has a zerosequence current controller shae with a rectangle. he zero-sequence current control part is ae onto the regular control for a single converter. Since it is implemente 85

10 within the iniviual converter an oes not nee any aitional interconnecte circuitry, it allows moular esign. 4. SIUAION AND EXPERIENA RESUS o emonstrate the concept presente in the previous sections, a parallel three-phase rectifier system is chosen as an example. Both the average an switching moels were built in SABER simulator. he average moel is not only use for the control esign, but it is also much more efficient in simulation than the switching moel, which is sometime ifficult to converge for such a iscrete high-orer system. A breaboar system of a two-parallel three-phase boost rectifier system was built. A preliminary experiment was conucte to valiate the zero-sequence control concept. 4.. Simulation Results he parameters of the simulation moel are escribe below: V 0 V; ω = π 60 ra/s; V = 400V; P = 0 ~ 5kW ; = 50 H ; rms( a, b, c) = ESR = 45mΩ,, = 00µ F H i, = +, π 60 s c First of all, the system was simulate without the zero-sequence current control. It was emonstrate that any iscrepancies between the two parallel converters (for example, ifferent switching frequencies, ifferent power stage parameters or ifferent switching eatime, all of these cases were simulate) may cause a large circulating current. A previous research propose to use an SV without zero vectors to avoi the zerosequence current [4]. However, since it is not a feeback control, the zero-sequence current can be avoie only if the two converters are ientical. If there is even a small ifference, a large zero-sequence current still coul occur. Figure 4.4 illustrates the example that both converters use the SV without zero vectors. he only ifference between the two converters is that there is a small eatime ifference (one is 0.µs, an 86 o, µ C, ESR C, = 50mΩ, f 3 Hz sw = k, f sw = 6kHz, H iq, = +, π 60 s H v = +. π 50 s

11 the other is 0.µs). he eatime ifference causes a DC offset in their common-moe voltage v z c. As a result, a zero-sequence current with a DC offset exists in the parallel system. Figure 4.4(a) shows the zero-sequence current. As a consequence, Figure 4.4(b) shows that the phase currents also have a DC offset. i z i z (a) Zero-sequence currents. i a i a (b) Input phase currents. Figure 4.4 Average moel simulation using SV without zero-vectors. 87

12 Figure 4.5 shows another case, in which the two rectifiers have ifferent switching frequencies (one is 3 khz an the other is 6 khz). Figure 4.5(a) shows that a significant low-frequency circulating current exists in the system. he currents i z an i z are zero-sequence currents for the boost rectifiers an, respectively. he circulating current causes istorte input line currents i a an i a, as shown in Figure 4.5(b). he circulating current has both low- an high-frequency components. he high-frequency one is basically a beat-frequency component cause by the two switching frequencies 6kHz an 3kHz. 5A/iv i z i z (a) Zero-sequence currents. 5A/iv i a i a (b) Input phase currents. Figure 4.5 Average moel simulation without zero-sequence current control (fsw=3 khz, fsw=6 khz). 88

13 By applying the zero-sequence current control, the waveforms in Figure 4.6 show that the circulating current is almost gone. Only high-frequency current ripples still exist, an they can be attenuate by filters. 5A/iv i z i z (a) Zero-sequence currents. 5A/iv i a i a (b) Input phase currents. Figure 4.6 Average moel simulation with zero-sequence current control (fsw=3 khz, fsw=6 khz). Figure 4.7 shows the simulation results using the switching moel without the zerosequence current control. Figure 4.8 shows the simulation results using the switching moel with the zero-sequence current control. 89

14 i z i z i a i a Figure 4.7 Switch moel simulation without zero-sequence current control. i z i z i + i a a i a i a Figure 4.8 Switch moel simulation with zero-sequence current control. 4.. Experimental Results A two-parallel, three-phase boost rectifier breaboar system was built an teste. he experiment setup can be referre in [4], [8]. Some parameters are shown below: 90

15 Input voltage: 3Φ, V l-l =08V; Output voltage: V c =400V; Rate power: 5kW/converter; Switching frequency: f sw =3.kHz, f sw =5.6kHz; Input inuctor: 56µH; Output capacitor: 00µF; IGB moules: OSHIBA G 50JYS50; DSP: ADSP0. In orer to implement the zero-sequence control, the PW part of the DSP coe was moifie. Instea of a fixe uty cycle for zero-vector ppp, a variable uty cycle is applie base on measure zero-sequence current. Even when the same PW applies to the two converters shown in Figure 4.9, a strong zero-sequence interaction can still be observe, as shown in Figure 4.0, if without zero-sequence control. (a) PW for the converter. (b) PW for the converter. Figure 4.9 Experimental waveforms PW applie to the parallel boost rectifiers. 9

16 5A/iv i z 5A/iv i a i z i a (a) Zero-sequence currents. (b) Input phase currents. Figure 4.0 Experimental waveforms parallel interactions with open-loop operation. he experimental results with current-loop close are shown in Figures 4. an 4.. he output DC voltage was not regulate. Because of this, the testing power is very low in orer to avoi high DC output voltage. Figure 4. shows the zero-sequence currents an the line currents without zerosequence current control, whereas Figure 4. shows the waveforms with control. Due to non-uniform practical conitions, such as ifferent elays in the control circuits an sensor loops, the experimental waveforms are less uniform than simulation waveforms. herefore, the beat-frequency component as appeare in simulation oes not pronounce in the experiment. hree noticeable ripples in Figure 4.(a) are ue to istortion introuce by zero-crossing etection. 9

17 5A/iv i z i z (a) Zero-sequence currents. 5A/iv i a i a (b) Input phase currents. Figure 4. Experimental waveforms without zero-sequence current control (fsw=3.khz, fsw=5.6khz, unsynchronize). 93

18 5A/iv i z i z (a) Zero-sequence currents. 5A/iv i a i a (b) Input phase currents. Figure 4. Experimental waveforms with zero-sequence current control (fsw=3.khz, fsw=5.6khz, unsynchronize). 94

19 4.3 GENERAIZAION O PARAE N-NUBER OF -PHASE CURREN- BIDIRECIONA CONVERERS his section generalizes the zero-sequence moeling an control concept to parallel N-number of -phase current-biirectional converters, such as full-brige, three-phase three-leg an four-leg rectifiers an inverters, as shown in Figure 4.3 with two converters in parallel, for example. oa/ Source v a v b oa/ Source v a v b v c i 0 Source /oa i 0 Source /oa v a vb v a vb vc (a) Parallel full-brige rectifier/inverter. (b) Parallel three-phase three-leg rectifier/inverter. oa/ Source v a v b v c v n i 0 Source /oa v a vb vc v n (c) Parallel three-phase four-leg rectifier/inverter. Figure 4.3 Parallel multi-phase current-biirectional converters. 95

20 4.3. Zero-Sequence Current Dynamics oel Base on the phase-leg averaging presente in Chapter, an average moel of a multi-phase converter can be constructe by simply connecting multiple phase-leg moels with the rest of the circuit. Figure 4.4 shows an average moel of a generalize -phase converter. he subscript represents the number of phases of the converter. Referring to Figure 4.3, a full-brige converter has = an φ =a, φ =b; a three-phase, three-leg converter has =3 an φ =a, φ =b, φ 3 =c; an a three-phase, four-leg converter has =4 an φ =a, φ =b, φ 3 =c, φ 4 =n, assuming the phase n has the same inuctance as the other phases. v c i p v φ i φ i φ Source /oa v φ oa/ Source C φ i φ iφ vc φ φ φ v c 0(ref) Figure 4.4 ulti-phase converter s average moel. When N-number of -phase converters are in parallel, multiple zero-sequence current paths are forme, as shown in Figure 4.5. he zero-sequence current is efine as the sum of all phases currents: i i z K, zn = i = i φ + i φ + i + K+ i φ + K+ i φ N. (4.8) he circuit of Figure 4.5 has equations: φ K, φ v c v c i φ t i φ t = φ = v φ c v c i φ t i t = = φ = = v c v c N i t N i t. (4.9) 96

21 In orer to simplify the equations an to extract the zero-sequence components, one can obtain the following equations by summing up (4.9). ( ( φ = + + φ + + φ + + ) v c ) v ( i c N φ ( i + i φ t + i + + i t φ ) + + i =. (4.0) ) v c i p v φ v φ i φ i φ Source /oa oa/ Source C φ iφ v φ i φ i φ c z φ v c 0(ref) vφ N iφ N i p v i i i v φn φ N i φ N c zn φ N v c Figure 4.5 N-parallel -phase converters average moel. A zero-sequence uty cycle is efine as the sum of the uty cycle of all phase legs in each converter. z K, zn = = φ + φ φ + + φ N. (4.) With the efinition of zero-sequence currents in (4.8) an zero-sequence uty cycles in (4.), the moel of Figure 4.5 can be simplifie as follows: 97

22 z zn v = c v c iz = t N i t zn. (4.) Equation (4.) escribes the ynamics of the zero-sequence currents. Figure 4.6 shows the equivalent circuit of the ynamics escribe in equation (4.). N i z i zn z v c zn v c Figure 4.6 Zero-sequence ynamics moel of N-parallel -phase converters. A control variable k is efine as the uration for which all top switches (usually efine as p switch, while the bottom switches are efine as n switch) are close: k = p p p. (4.3) φ Figure 4.7 epicts the k in one switching pattern of an -phase converter; as an example, k is the uration of the zero-vector ppp in a three-phase three-leg converter. φ φ k Sφ p S φ Figure 4.7 Control variable k in N-parallel -phase converters. Using the efinition in (4.3), the zero-sequence uty cycles in (4.) can be rewritten as follows: p 98

23 z K, zn = = φ + φ φ + + φ = k N + = k N + active activen, (4.4) where active is the total uty cycle of the active switching vectors. For example, in a three-phase space-vector moulate converter, = + active. Substituting (4.4) into (4.), a new zero-sequence average moel with the control variable k is obtaine: iz ( k + active ) vc = t =. (4.5) izn ( kn + activen ) vc N t he equivalent circuit is shown in Figure 4.8. As a result, the zero-sequence current can be controlle by controlling k, k,, an k N. N i z i zn ( k + active ) v c ( kn + activen ) vc Figure 4.8 Zero-sequence ynamics moel with new control variable k in N-parallel -phase converters Zero-Sequence Current Control For the N-parallele converter system, there are N- inepenent zero sequence currents because i i + K + i 0. (4.6) z + z zn = herefore, only N- converters are neee to have a zero-sequence current control. For example, if k N is set to a constant value K, then the steay state has a unique solution: 99

24 k, k k = K + N N = K + = K. activen activen active, activen (4.7), If all N converters are controlle, the resulting equivalent circuit is N current sources in parallel, which is not practical. First, the N current sources will have interactions because of the constraint in (4.6). Secon, assuming that in steay state all currents are zero, the number of solutions is infinite because there are N- inepenent equations as in (4.8), but N unknowns (k, k,, k N ): k k = k N active active activen = =. (4.8) herefore, it is necessary to have N- converters controlle in an N-parallel converter system. Figure 4.9 shows the zero-sequence control in an N-parallel -phase converter system. he controllers are esigne within iniviual converters. he converter N is not controlle an has a constant K. he preceing analysis has been valiate by simulation on a three-parallel, threephase converter system constructe in Saber. he system is a phase-leg average converter moel as shown in Figure

25 oa/ Source φ φ k k PI i z 0 Source /oa φ N φ N i zn- k N k N PI 0 φ N φ N K Figure 4.9 Zero-sequence current control implementation for N-parallel -phase converters. Figure 4.30 shows the zero-sequence current without control. he three converters have ifferent switching frequencies (f sw =0kHz, f sw =7kHz, f sw3 =5kHz) an shifte switching clocks (shift clock =0µs, shift clock =0µs, shift clock3 =0µs). It can be seen that a strong zero-sequence interaction exists in the parallel system. By applying the control to two of the parallel converters, the zero-sequence currents are well controlle, as shown in Figure

26 i z3 i z i z Figure 4.30 hree-parallel three-phase boost rectifiers without zero-sequence current control. i z i z i z3 Figure 4.3 hree-parallel three-phase boost rectifiers with zero-sequence current control. 0