IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013
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Origin of Cross-Coupling Effects in Distributed DC–DC Converters in Photovoltaic Applications Juha Huusari, Member, IEEE, and Teuvo Suntio, Senior Member, IEEE
Abstract—Long strings of photovoltaic (PV) modules are found to be vulnerable to shading effects, causing significant reduction in the system power output. To overcome this, distributed maximum power point-tracking (DMPPT) schemes have been proposed, in which individual dc–dc converters are connected to each PV module to enable module-wise maximum power extraction. There are two main concepts to implement DMMPT systems: series and parallel configuration, describing the connection of the output terminals of the converters. Both systems are studied intensively, with innovative solutions to encountered operational challenges and novel control methods. However, a comprehensive dynamic model for neither system has been presented so far. This paper fills the gap by presenting small-signal models for both configurations, explaining the observed operational peculiarities. The analytical claims are verified with a practical system comprising two maximum power point-tracking buck–boost converters.
Fig. 1.
PV-cell equivalent circuit.
Fig. 2.
PV-cell current-voltage characteristics.
Index Terms—Cascaded converters, dc–dc power conversion, photovoltaic (PV) power systems.
I. INTRODUCTION HE importance of providing a reliable and sustainable source of energy and electricity has become a major concern over the past years. Conventional fossil-based sources of energy are widely seen as nonsustainable and polluting, thus forming need for green alternatives. Moreover, political decisions may further narrow down the number of energy sources to be utilized. Most notably, these include the announcement to run down the nuclear power plants in Germany by 2022 [1]. Among the most studied alternative sources is the photovoltaic (PV) electricity [2], [3], in which the irradiation coming from the Sun is transferred into electrical energy via the PV effect [4]. Successful utilization of PV electricity requires expertise ranging from semiconductor physics and material science to power electronics. At present, technical difficulties encountered with the practical implementation of power electronic converters can be seen as one of the key obstacles [5], [6]. This paper continues the authors’ previous work by supplementing the analytical predictions given in [7] with experimental results and a more detailed analysis. The analysis explicitly explains the observed cross-coupling effects and, further, explains
T
their disappearance due to applied input-voltage feedback loop. Furthermore, this paper presents experimental evidence supporting both the analytical claims presented by author as well as the simulated results presented by Petrone et al. in [8]. Series and parallel system configurations are discussed and it is shown, that the parallel configuration is virtually free of cross-coupling effects, thus providing significantly better performance in distributed maximum power-point tracking (DMPPT) applications. The rest of the paper is organized as follows: Brief introduction to PV systems is given in Section II. Section III discusses the modeling of interfacing buck–boost converters and presents small-signal models for the cascaded structures. Measurement results obtained from experimental cascaded system of two buck–boost converters are presented in Section IV. Finally, the conclusions are drawn in Section V. II. PV SYSTEMS
Manuscript received August 27, 2012; revised October 30, 2012; accepted December 10, 2012. Date of current version March 15, 2013. The research work for this manuscript was carried out at Tampere University of Technology. Recommended for publication by Associate Editor J. A. Cobos. J. Huusari is with the ABB Corporate Research, CH-5405 Baden-D¨attwil, Switzerland (e-mail:
[email protected]). T. Suntio is with the Department of Electrical Energy Engineering, Tampere University of Technology, FI-33101 Tampere, Finland (e-mail: teuvo.
[email protected]). Digital Object Identifier 10.1109/TPEL.2012.2235860
The simplified electrical equivalent model of a PV cell comprises a photocurrent source and a parallel connected diode (see Fig. 1). The PV cell itself behaves as a highly nonlinear current source with limited output voltage as depicted in Fig. 2. It is known on the basis of the PV-cell voltage-current characteristics that the generated power reaches its maximum only under specific
0885-8993/$31.00 © 2012 IEEE
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loading. This maximum power point (MPP) separates the constant current region (CC), where the PV-cell current stays relatively constant from the voltage region (CV), where the PV-cell voltage remains relatively constant. Because the PV-cell terminal voltage and current are both proportional to ambient operating conditions, the loading must be adjusted to provide maximum power under all operating conditions. This is known as MPP tracking. A practical method to implement MPP tracking is to utilize a switching converter between the PV generator and the system load. The converter provides an interface for transferring power between the PV generator and the load [9], [10] as well as galvanic isolation if an isolated converter is used [10], [11]. A converter capable of transferring power from low to high voltage is required when interfacing an individual PV module, e.g., to an inverter, because the voltage of the PV module is usually insufficient for the proper operation of the inverter—a typical PV module comprising 54 cells generates approximately 26 V at MPP. Furthermore, a practical PV system frequently experiences nonuniform illumination especially in built environment, leading to partially shaded individual PV modules whose global MPP may occur at a significantly lower voltage than under uniform illumination [12]. Conventionally, large-scale PV electricity systems are comprised of long strings of PV modules, which are interfaced to the utility grid by means of a VSI-type inverter [10], [13]. Each string contains a number of PV modules connected in series, thus increasing the string voltage high enough for the inverter. These strings have been found to be vulnerable to shading effects, in which the generated power of the string is severely limited by modules that are shaded, e.g., by clouds or shadows caused by nearby objects [12], [14]. Due to the series connection, each module has to carry equal current, which may force the operating point of some modules away from the MPP [10], [14], [15]. To overcome this, DMPPT systems have been proposed, in which each PV module has a dedicated interfacing converter [11], [16]–[18]. DMPPT converters are the first part in a two-stage conversion chain, where the dc power produced by the PV modules is interfaced into the ac utility grid by means of an inverter. The two-stage structure contains, therefore, a high-voltage dc link between the dc–dc converters and the inverter. Typically, there are a number of individual converters transferring power into the common dc link [18]–[22]. The general structures of DMPPT systems are presented in Figs. 3 and 4, depicting the series configuration and the parallel configuration, respectively. In the series configuration (see Fig. 3), the outputs of individual interfacing converters are connected in series. Thus, the dc-link voltage is distributed between the converter output terminals according to power levels [23]. In the parallel configuration (see Fig. 4), the output terminals of the interfacing converters are connected directly to the dc link (i.e., the input of a inverter). Each converter has to endure full dc-link voltage at the output terminal, which requires specific solutions to meet the high conversion ratio. The DMPPT concepts have been under intensive study, with numerous publications discussing the implementation of MPP
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013
Fig. 3.
Series DMPPT configuration.
Fig. 4.
Parallel DMPPT configuration.
tracking and the design process of interfacing dc–dc converters (e.g., [8], [24]). Despite the research activity, there are hardly any papers presenting a clear and comprehensive dynamic analysis (i.e., small-signal model) for DMPPT systems. In [25] Siri et al. discuss MPP tracking in parallel configuration, but the dynamic considerations are based only on a single converter. The most comprehensive dynamic considerations have been presented by Femia et al. [19], in which a system of two series-connected converters is analyzed and a small-signal model is presented applying Middlebrook’s extra element method. However, the presented results have only limited predicting power, possibly due to difficult way of presentation and focus on the stability issues. For instance, the model given by Femia et al. is unable to predict the cross-coupling effects, observed in simulations by Petrone et al. [8]. The cross-coupling effects are an undesired property of cascaded converters, causing disturbances in the converter operation [8]. III. MODELING In order to conveniently model and analyze the operation of a switched-mode converter, a linear model for the converter is required. Conventionally, the state-space averaging approach [26] is used to obtain a linear small-signal model describing the circuit operation in frequency domain. Frequency-domain analysis is mandatory to correctly predict the operation of the circuit in time domain, i.e., to guarantee stable and controlled
HUUSARI AND SUNTIO: ORIGIN OF CROSS-COUPLING EFFECTS IN DISTRIBUTED DC–DC CONVERTERS IN PHOTOVOLTAIC APPLICATIONS
power processing as well as to predict the circuit response to changes in operating conditions. Moreover, the control of a switched-mode converter has to be designed and verified with frequency-domain methods. In state-space averaging, the operational subcircuits defined by switching action are analyzed separately and corresponding state-space equations are developed according to well-known Kirchhoff’s circuit laws. In a state-space model, the system output variables and the derivatives of the system state variables are given as a function of the input and the state variables. An averaged state-space model is obtained, when the equations describing the subcircuits are averaged over one switching period, according to the durations that each subcircuit is active. The averaged model, therefore, presents the averaged, time-invariant behavior of the circuit but is nonlinear by nature [27]. A linear model is finally obtained, when the averaged equations are linearized around a specific operating point, i.e., partial derivatives of each variable are developed from the state equations. When the averaged state-space consisting of time-domain differential equations is linearized around a certain operating point, the resulting linearized time-domain state space, i.e., the ˆ (t), small-signal model [28] can be expressed as in (1), where x ˆ (t), and y ˆ (t) are vectors containing the state variables, input u variables, and output variables, respectively. The matrices A, B, C, and D contain the effects of the parasitic elements within the circuit, such as parasitic resistances, as well as the effects of the inductances and capacitances in the system. Additionally, depending on the converter topology, these matrices contain typically some average values of the output, input or state variables, such as the average output or input voltage d ˆ (t) = Aˆ x x(t) + Bˆ u(t) dt ˆ (t) = Cˆ y x(t) + Dˆ u(t).
(1)
To obtain the frequency-domain small-signal model, Laplace transformation is applied to (1), yielding the following frequency-domain equations, where s = jω denotes the Laplace variable sX(s) = AX(s) + BU(s) Y(s) = CX(s) + DU(s).
(2)
The output variables Y(s) can be solved from (2), yielding Y(s) = [C(sI − A)−1 B + D]U(s) = G(s)U(s).
(3)
The matrix G(s) contains six transfer functions, describing the transfer functions between input variables U and output variables Y. In this paper, the H-parameter scheme is used to describe an individual dc–dc converter in a distributed PV system, because an H-parameter network model has a current-type source and a voltage-type load. Using matrix notation, the transfer functions can be presented as in ⎡ ⎤ iˆin Zin Toi Gci u ˆin ⎣u = (4) ˆo ⎦ . Gio −Yo Gco iˆo cˆ
Fig. 5.
H-parameter network.
Fig. 6.
Schematic diagram of a current-fed buck–boost converter.
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Characteristic to H-parameter scheme is that a CC source is connected to the input terminal and a constant-voltage-type load at the converter output terminal. Therefore, the input variables ˆo cˆ]T and the output variables of the system are U = [iˆin u T Y = [ˆ uin iˆo ] , respectively. According to control engineering principles, only the system output variables are controllable by means of a feedback network, allowing only the PV terminal voltage (i.e., converter input voltage) to be controlled. A network model for the H-parameter scheme is shown in Fig. 5, equaling (4). It should be noted, that contrary to the conventional two-port network model, the positive output current is now defined to flow out of the output terminal, to comply with presentation in (4). A. Model for a Buck–Boost Converter The power stage of a current-fed buck–boost converter is presented in Fig. 6, with an additional output-side CL-type filter (C2 , L2 ) included, to provide continuous output current. The input variables are, therefore, u ˆo , iˆin , the state variables ˆC1 , u ˆC2 and the output variables u ˆin , iˆo . The buck– iˆL1 , iˆL2 , u boost converter operates as follows: During the on time (ton ), the diodes conduct and during the off time (toff ), respectively, the switches conduct, yielding two operational subcircuits. This inversion of control signals is required to obtain a stable operation with input-voltage control [29], [30]. The same result can be achieved if the feedback loop is multiplied with −1, as shown, e.g., in [19]. Combining the equations depicting the on time and off time subcircuits, the averaged model (5) is obtained, where the parasitic elements include the inductor resistances rL1 , rL2 , the switch channel resistances rds1 , rds2 , the capacitor resistances rC1 , rC2 , and the diode resistances and the forward voltage drops rD1 , rD2 and UD1 , UD2 , respectively
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013
d iL 1 dt =
rL 1 + d(rD 1 + rD 2 + rC 2 ) + d (rC 1 + rd s1 + rd s2 ) iL 1 L1 +
d d d rC 1 drC 2 iL 2 + uC 1 − uC 2 + iin L1 L1 L1 L1
−
d(UD 1 + UD 2 ) L1
d d rC 2 rC 2 + rL 2 1 1 iL 2 = iL 1 − iL 2 + uC 2 − uo dt L2 L2 L2 L2
Fig. 7.
H-parameter network with nonideal source.
d 1 d uC 1 = − iL 1 + iin dt C1 C1 d d 1 uC 2 = iL 1 − iL 2 dt C2 C2 uin = −d rC 1 iL 1 + uC 1 + rC 1 iin io = iL 2 .
(5)
By linearizing the averaged model, the resulting small-signal state space can be given by dˆ R1 DrC2 ˆ D D u ˆC1 − u ˆC2 iL2 + iL1 = − iˆL1 + dt L1 L1 L1 L1 +
D rC1 ˆ U1 ˆ d iin − L1 L1
dˆ DrC2 ˆ rL2 + rC2 ˆ 1 1 u ˆC2 − u ˆo iL1 − iL2 + iL2 = dt L2 L2 L2 L2
Fig. 8. Network model for series-connected H-parameter networks with nonideal sources.
rC2 IL1 ˆ + d L2 d D 1 ˆ IL1 ˆ u ˆC1 = − iˆL1 + d iin + dt C1 C1 C1 d D ˆ 1 ˆ IL1 ˆ u ˆC2 = d iL1 − iL2 + dt C2 C2 C2 u ˆin = −D rC1 iˆL1 + u ˆC1 + rC1 iˆin + rC1 IL1 dˆ iˆo = iˆL2
(6)
The transfer functions describing the input and output dynamics of the source-affected H-parameter network can be presented as in (9). Manipulating the transfer functions to share a common denominator results in two special transfer functions: The input impedance at open-circuited output terminal Zin−o co and the ideal input impedance Zin−∞ [31]. The special transfer functions are discussed more in detail in, e.g., [32] ⎤ ⎡ T G Z in
where the additional variables R1 and U1 are defined as follows: R1 = rL1 + D(rD2 + rD2 + rC2 ) + D (rds1 + rds2 + rC1 ) U1 = (rD1 + rD2 + rC2 − rds1 − rds2 − rC2 )IL1 − rC2 IL2 + UC1 + UC2 + UD1 + UD2 − rC1 Iin .
(7)
The ideal small-signal model can be expanded to contain the effects of a nonideal source, described by an arbitrary source admittance YS , yielding the network model shown in Fig. 7. The actual input current iˆin is computed from Fig. 7 and substituted in (4) as follows: ˆin . iˆin = iˆinS − YS u
(8)
Thus, the input variable iˆin is replaced by the current of the actual source iinS .
⎢ 1 + Zin YS ⎢ ⎣ G io
1 + Zin YS
oi
1 + Zin YS −Yo
1 + Zin−o co YS 1 + Zin YS
ci
⎥ ⎥. 1 + Zin−∞ YS ⎦ Gco 1 + Zin YS (9) 1 + Zin YS
B. Models for Cascaded Configurations To model the small-signal behavior of cascaded (i.e., either series connected or parallel connected) converters in DMPPT applications, two system models representing both alternatives are formed using the H-parameter scheme. In both models, the individual H-parameter networks have nonideal current sources (iinS1 , iinS2 ) at the input terminals and a constant-voltage load (uo ) at the system output. A network model for the complete system is formed by merging two individual network models together, yielding the series-connected system model (see Fig. 8) and the parallel-connected system model (see Fig. 9), respectively.
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Fig. 9. Network model for parallel-connected H-parameter networks with nonideal sources.
By computing the transfer functions from the series configuration, one obtains the matrix expression as shown in (10), where the superscript “S” denotes source interactions and “c” cross-coupling effects ⎤ ⎡ˆ iinS1 ⎤ ⎡ S,c S,c S,c S,c ⎥ ⎢ˆ ⎡ ⎤ Tcr1 Toi1 Gci1 Gcr1 Zin1 u ˆin1 ⎢ iinS2 ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ S,c S,c S,c S,c ⎥ ⎢ ˆin2 ⎦ = ⎢ u ˆ ⎥. ⎢ T Z T G G ⎣u o oi2 cr2 in2 ci2 ⎦ ⎢ ⎣ cr2 ⎥ ⎥ ⎢ S,c S,c S,c S,c c ˆ iˆo ⎣ 1 ⎦ Gio1 Gio2 −Ytot Gco1 Gco2 cˆ2 (10) Similar analysis can be performed for the parallel configuration, resulting in the transfer functions shown in ⎤ ⎡ˆ iinS1 ⎥ ⎡ ⎤ ⎡ S ⎤⎢ˆ S u ˆin1 0 Toi1 GSci1 0 Zin1 ⎢ iinS2 ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥⎢ S S Zin2 ˆin2 ⎦ = ⎣ 0 Toi2 0 GSci2 ⎦ ⎢ u ˆo ⎥ . ⎣u ⎥ ⎢ ⎥ ⎢ S GSio1 GSio2 −Ytot GSco1 GSco2 ⎣ cˆ1 ⎦ iˆo cˆ2 (11) According to (10), there are cross couplings between the two input sources, both from the converter 1 control signal to converter 2 input voltage (i.e., u ˆin2 /ˆ c1 , Gccr2 ), from converter 1 input c ), and vice current to converter 2 input voltage (ˆ uin2 /iˆin1 , Tcr2 versa. This would imply that the series connection might suffer from effects caused by the cross couplings. On the other hand, there are no cross-coupling effects between parallel-connected converters in the presented system configuration (i.e., with an ideal load) as shown in (11), indicating superior performance over the series configuration [7]. The cross-coupling effects can be explained intuitively as well, by examining the system structures in Figs. 8 and 9. In parallel configuration, for each network there are two constant control engineering input variables, namely, the input current and the output voltage. In series configuration, however, only the input current is such constant input variable. Because the
Fig. 10. Network model for parallel-connected H-parameter networks with nonideal load.
input variable related to the output terminal is not constant, the output variable related to the input terminal (in this application, the input voltage) is a product of two nonconstant variables: the modulo m(d), defining the conversion ratio of the converter, and the output voltage uo . In parallel configuration, the input voltage for each converter is a product of the modulo m(d) and the constant output voltage uo . Thus, the output voltage for each converter is not affected by neighboring converter, which explains the zeros in transfer function matrix in (11). C. Nonideal Load in Parallel Configuration By adding an arbitrary series impedance to the load, the effects of load-imposed cross couplings in parallel configuration can be studied. The series configuration with a nonideal load is not examined, as the existence of cross couplings in series configuration is already verified, as shown in (10). The network model for parallel-connected H-parameter networks with common, nonideal load is presented in Fig. 10. The corresponding transfer functions can be presented in matrix form as in (12), where subscript “L” denotes load effect and additional subscript “c” cross-coupling effect ⎡ˆ ⎤ iin1 ⎤ ⎡ L ⎤ ⎡ L,c L,c ⎢ L Toi1 GLci1 Gcr1 ⎢ iˆin2 ⎥ Zin1 Tcr1 u ˆin1 ⎥ ⎥ ⎥ ⎢ L,c ⎥⎢ ⎢ L,c ⎢ L L L u ˆ u ˆ = ⎣ in2 ⎦ ⎣ Tcr2 Zin2 Toi2 Gcr2 Gci2 ⎦ ⎢ oL ⎥ ⎥. ⎥ ⎢ L iˆo GLio1 GLio2 −Ytot GLco1 GLco2 ⎣ cˆ1 ⎦ cˆ2 (12) Thus, in the parallel configuration neither the system nor nonideal sources introduce cross-coupling effects, that would interfere with the system operation, whereas in series configuration the cross couplings emerge already from the system configuration. The actual effect on the system performance is dictated by the magnitude of these nonidealities (i.e., by the magnitude of corresponding impedances), as indicated by the corresponding equations. In the following section, the relevant open-loop
transfer functions for both system configurations are discussed more in detail, revealing conditions in which there might emerge cross couplings in parallel-connected system, for instance. D. Open-Loop Transfer Functions for Cascaded Configurations For the series-connected system, the transfer functions become rather complex due to cross couplings and inclusion of source nonidealities. If source nonidealities are neglected, the cross-coupling-affected transfer functions related to input port 1 [i.e., the first row in the matrix in (10), without (Toi can be given as in (13)]. Only these transfer functions are examined, as they are the most important considering the scope of this paper
Yto t Yo 1 Yo 2
Gcc i1 = Gc i1 − To i1 Gc o 1 Gcc r1 = −To i1 Gc o 2
Yo 2 Yo21
Yto t Yo 1 Yo 2
(13)
c where Tcr1 represents the input cross-coupling transfer function and Gccr1 the control cross-coupling transfer function. The open-loop transfer functions for parallel-connected system with nonideal sources in (11) are equal to the corresponding sourceaffected transfer functions for a single converter, except the system output admittance, which can be expressed by S Ytot = Yo1 + Yo2 +
Gio1 Toi1 YS1 Gio2 Toi2 YS2 + . 1 + Zin1 YS1 1 + Zin2 YS2
(14)
The open-loop transfer functions for the input channel 1 in parallel configuration with a nonideal load can be given as in (III-C), corresponding to the first row in the matrix in (12)
ZL To 1 Gio 2 1 + ZL Yto t
ToLi1 = To i1
1 . 1 + ZL Yto t
GLc i1, c = Gc i1 ,c GLc r1 =
1 + ZL Yo 1 −∞ 1 + ZL Yto t
10
180 90 0 −90 −180 −270 1 10
10
2
2
3
10 Frequency (Hz)
3
10 Frequency (Hz)
4
10
4
10
5
10
5
10
ZL To 1 Gc o 2 1 + ZL Yto t
(15)
According to (13), the greater the output admittance in a single converter (the smaller the output impedance), the weaker are the cross-coupling effects. In other words, the performance of series configuration is improved, if the individual converters operate as voltage sources having small output impedance. Respectively, a current source behavior increases the cross-coupling effects. Considering nonideal load in parallel configuration, the smaller the load impedance, the weaker the cross-coupling effects. In PV applications, the load for DMPPT converters is the input terminal of the inverter, which acts as a voltage-type load due to inverter control structure, maintaining the input voltage at a constant value. This would imply that the inverter input impedance is low and the load-imposed cross couplings would be negligible. Using the analytical model presented earlier, predictions were calculated for the most important transfer functions. As shown in the following figures, these are the open-loop input cross coupling (Tcr−o , Fig. 11), open-loop control cross coupling (Gcr−o , Fig. 12), and the input voltage loop gain (L, Fig. 13).
20 0 −20 −40 −60 1 10
10
180 90 0 −90 −180 −270 1 10
10
2
2
3
10 Frequency (Hz)
3
10 Frequency (Hz)
4
10
4
10
5
10
5
10
Fig. 12. Predicted Gcr-o in series configuration: CC (solid line), MPP (dashed line), and CV (dash-dotted line).
Magnitude (dB)
TcLr1, c =
1 + ZL Yo 1 −sc i 1 + ZL Yto t
Phase (deg)
L Zin 1 = Zin 1
20 0 −20 −40 −60 1 10
Fig. 11. Predicted Tcr-o in series configuration: CC (solid line), MPP (dashed line), and CV (dash-dotted line).
Magnitude (dB)
Tccr1 = −To i1 Gio 2
Yo 2 Yo21
Phase (deg)
c Zin 1 = Zin 1 − To i1 Gio 1
Magnitude (dB)
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Phase (deg)
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40 20 0 −20 −40 −60 1 10
10
0 −45 −90 −135 −180 1 10
10
2
2
3
10 Frequency (Hz)
3
10 Frequency (Hz)
4
10
4
10
5
10
5
10
Fig. 13. Designed loop gain: CC (solid line), MPP (dashed line), and CV (dash-dotted line).
E. Closed-Loop Transfer Functions for Cascaded Configurations The closed-loop transfer functions in series configuration can be computed most conveniently by the corresponding control engineering block diagram, shown in Fig. 14, based on Fig. 8. Fig. 14 represents a general closed-loop model for H-parameter
HUUSARI AND SUNTIO: ORIGIN OF CROSS-COUPLING EFFECTS IN DISTRIBUTED DC–DC CONVERTERS IN PHOTOVOLTAIC APPLICATIONS
T
iˆ
uˆ
Y
++ +
++ +
G ++ +
iˆ
T
Z
G
uˆ
G
Gse1 cˆ
G
G
eˆ1 − +
+ −+
uˆ iˆ
cˆ
++ G +
G
iˆ
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G
G
+ ++
Z
G
T
+ −
uˆ
G
+ ++
uˆ iˆ
T
Fig. 14. Closed-loop block diagram for series configuration: Source nonidealities included.
network with input voltage control and, thus, is valid in ideal and nonideal case. In later analysis, we treat the nonideal case in which the input currents iin are replaced by iinS , and the transfer functions with respective nonideal ones. Because the series configuration is mostly plagued by inputside cross-coupling effects, the block diagram in Fig. 14 can be used to examine the closed-loop cross-coupling transfer functions, corresponding to the matrix equation (10). For input channel 1, these are the input cross-coupling transfer funcS,c =u ˆin1 /iˆinS2 and the control cross-coupling transfer tion Tcr1−c S,c ˆin1 /ˆ uref2 . For clarity, the superscript “S,c” function Gcr1−c = u has been left out of the transfer functions in the following discussion. The transfer functions of the input voltage loops are defined as L1 = Gse1 Gc1 Gci2−o ,
L2 = Gse2 Gc2 Gci2−o
(16)
where Gse represents voltage sensing gain and Gc a lumped transfer function, containing the effects of A/D converter and modulator. To examine Tcr1−c , for example, all other input variables are set to zero and the expressions for the output variables are written. Manipulation of these equations yield expression for Tcr1−c as shown in
Tcr1−c
1 = 1 + L1
Gcr1−o L2 Gc2 1 + L2 . (17) Gcr2−o L1 L2 Gci1−o 1 + L1 1 + L2
Tcr1−o − Zin2−o 1−
Gcr1−o Gci2−o
Considering the low-frequency operation, where the magnitude of both input-voltage loop gains L1 and L2 is large (therefore, the ratio L2 /(1 + L2 ) equals unity and 1 + L1 ≈ L1 ), the corresponding transfer function can be approximated by
Tcr1−c
1 ≈ L1
Tcr1−o − Zin2−o 1−
Gcr1−o Gc2
Gcr1−o Gcr2−o Gci2−o Gci1−o
≈ 0.
(18)
Fig. 15. Closed-loop block diagram for parallel configuration: Source nonidealities included.
Respectively, the control cross-coupling transfer function Gcr1−c can be presented by
L2 Gcr1−o Gc2 1 − 1 1 + L2 Gcr1−c = (19) G L1 L2 G 1 + L1 cr1−o cr2−o 1− Gci2−o Gci1−o 1 + L1 1 + L2 and the corresponding low-frequency approximation by Gcr1−c ≈
1 Gcr1−o Gc2 (1 − 1) = 0. Gcr1−o Gcr2−o L1 1− Gci2−o Gci1−o
(20)
According to (17) and (19), the loop gain L1 dominates the low-frequency value of the transfer function, effectively forcing the magnitudes |Tcr1−c | to a very low value and canceling |Gcr1−c | out. Therefore, the closed-loop operation acts as to reduce the input cross-coupling effects, thereby improving the system performance. Considering the parallel configuration with nonideal sources, the block diagram for input-voltage controlled system can be expressed according to (11) as shown in Fig. 15. If the load nonidealities are included, the system block diagram equals that presented in Fig. 14, where the individual transfer functions contain the load effects. Similar analysis reveals, that although there are load-imposed cross couplings in parallel configuration, the input-voltage control loop effectively cancels them out. F. Observed Cross-Coupling Effects An excellent example on actual effect of the discussed cross couplings is presented in [8]: A system of three series-connected converters was studied by introducing an irradiance sequence into PV modules attached to each converter. The converters were operated with an open-loop MPPT mode, in which the MPPT block gave duty cycle to converters directly and in closed-loop
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013
TABLE I DETAILS FOR SAS CONFIGURATIONS
Nominal ’Shaded’
Isc (A)
Uoc (V)
Impp (A)
Umpp (V)
1.00 0.50
19.00 18.00
0.90 0.45
16.00 15.50
Open-loop operation of the system [8].
Phase (deg)
Fig. 16.
Magnitude (dB)
TABLE II COMPONENT DETAILS IN BUCK–BOOST CONVERTERS Component
Value
Component
Details
L1 L2 C1 C2
120 µH 100 nH 440 µF 33 µF
D1 , D2 S1 , S2
SK56C IPD200N15N3
20 0 −20 −40 −60 1 10
10
180 90 0 −90 −180 1 10
10
2
2
3
10 Frequency (Hz)
3
10 Frequency (Hz)
4
10
4
10
5
10
5
10
Fig. 18. Measured Tcr-o in series configuration: CC (solid line), MPP (dashed line), and CV (dash-dotted line).
Fig. 17.
Closed-loop operation of the system [8].
MPPT mode, in which the MPP tracker gave reference value for input-voltage feedback loop. The resulting response in the input voltages of the converters at open loop are shown in Fig. 16, revealing the disturbance caused by change in one PV module irradiance. As a result, the MPP operation is lost for all modules. Respectively, under closed-loop operation (see Fig. 17), there is no visible disturbance. Thus, the closed-loop operation clearly improves the energy yield of the system. IV. EXPERIMENTAL VERIFICATION The input source for cascaded buck–boost converters was realized by configuring isolated solar array simulator (Agilent E4360A, “SAS”) modules according to values shown in Table I. Both buck–boost converters had identical circuit structure, with power stages comprising components shown in Table II. Both converters operated at 100 kHz and were controlled by individual DSCs (TMS320F28335). The system had a constantvoltage load of 50 V, realized with an electronic load (EA EL3400-25). The converters were connected in series configuration
and in parallel configuration and for both setups, the input cross coupling and the control cross-coupling transfer functions were measured using Venable Instruments’ Model 3120 frequency response analyzer. A. Open-Loop Measurements The measured input cross-coupling transfer functions at open ˆin1 /iˆinS2 ) for series and parallel configurations loop (Tcr−o = u are presented in Figs. 18 and 19, respectively. According to Figs. 18 and 19, the analytical predictions presented in Section III are proven valid as the series configuration contains strong cross coupling at low frequencies. On the basis of Fig. 18, the magnitude of Tcr−o is significant at low frequencies. Thus, the input voltage uin1 is strongly affected by changes in input current iinS2 . On the other hand, as indicated by analytical study, the parallel configuration is virtually free of such cross-coupling effects. The measured control cross-coupling transfer functions at ˆin1 /ˆ c2 ) for both system configurations open loop (Gcr−o = u are shown in Figs. 20 and 21. For series configuration the Gcr−o , although weaker than Tcr−o , is still significant at low frequencies and, thus, contributes to experienced disturbances. Respectively, the parallel configuration does not suffer from this cross coupling. The only visible
Magnitude (dB)
0 −20 −40 −60 1 10 360 180 0 −180 −360 1 10
2
10
3
10 Frequency (Hz)
4
10
5
10
2
10
3
10 Frequency (Hz)
4
10
20 0 −20 −40 2
10
3
10 Frequency (Hz)
4
10
5
10
Phase (deg)
Phase (deg)
90 0 −90 2
10
3
10 Frequency (Hz)
4
10
Magnitude (dB)
2
10
3
10 Frequency (Hz)
4
10
5
10
−180 −360 2
10
3
10 Frequency (Hz)
4
10
5
10
0 −20 −40 −60 1 10
2
10
3
10 Frequency (Hz)
4
10
5
10
5
−180 −360 −540 1 10
10
Fig. 20. Measured Gcr-o in series configuration: CC (solid line), MPP (dashed line), and CV (dash-dotted line).
Phase (deg)
−60 1 10
0
180
−180 1 10
−40
Fig. 22. Measured Tcr-c in series configuration: CC (solid line), MPP (dashed line), and CV (dash-dotted line).
Magnitude (dB)
Magnitude (dB)
Fig. 19. Measured Tcr-o in parallel configuration: CC (solid line), MPP (dashed line), and CV (dash-dotted line).
−60 1 10
−20
−540 1 10
5
10
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0
0 Phase (deg)
Phase (deg)
Magnitude (dB)
HUUSARI AND SUNTIO: ORIGIN OF CROSS-COUPLING EFFECTS IN DISTRIBUTED DC–DC CONVERTERS IN PHOTOVOLTAIC APPLICATIONS
2
10
3
10 Frequency (Hz)
4
10
5
10
Fig. 23. Measured Gcr-c in series configuration: CC (solid line), MPP (dashed line), and CV (dash-dotted line).
in parasitic elements and actual component values between the model and the prototype.
0 −20
B. Closed-Loop Measurements
−40 −60 1 10 360 180 0 −180 −360 1 10
2
10
3
10 Frequency (Hz)
4
10
5
10
To investigate the effect of closed-loop control on the performance of series configuration, an input-voltage control loop was designed for the buck–boost converters with a PID controller presented in Gcc (z) =
2
10
3
10 Frequency (Hz)
4
10
5
10
Fig. 21. Measured Gcr-o in parallel configuration: CC (solid line), MPP (dashed line), and CV (dash-dotted line).
effect shown between 100 Hz and 1 kHz is due to resonance caused by the components in the power stage, namely C1 and L1 . Yet, the magnitude remains below −20 dB, indicating very weak cross coupling at worst case. Comparison with Figs. 11– 13 shows that practical measurements support the analytical predictions well. Minor differences are caused by mismatches
4.981z −3 − 4.374z −2 − 4.963z −1 + 4.393 . (21) −0.786z −3 − 0.370z −2 + 0.157z −1
The measured closed-loop cross-coupling transfer functions are presented in Figs. 22 and 23. Thus, the input-voltage control loop in series configuration effectively eliminates the cross couplings introduced by changes in the input current (see Fig. 22) or in the reference (see Fig. 23) of the neighboring converter, thereby improving the system performance significantly. C. Time-Domain Operation With MPP Tracker Individual MPP-tracking algorithms were implemented in series configuration, using perturb and observe method. To examine the system operation under changing environmental
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Fig. 24.
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013
Open-loop MPP-tracking: Transients SAS module 2 (2.0 s/div).
Fig. 26. Closed-loop MPP-tracking: Sudden increase in the output power of SAS module 2 (0.2 s/div).
V. CONCLUSION
Fig. 25. Closed-loop MPP-tracking: Sudden decrease in the output power of SAS module 2 (0.2 s/div).
conditions, a transient was introduced in the other SAS module, so that the module output varied between two different IV-curves with a period of 10 s. The transient modeled a sudden drop in the irradiance, resulting in the “shaded” curve described in Table I. Fig. 24 shows the system behavior under transient, when both converters operated at open loop, with the MPP block giving the duty cycle directly to the converters. The examined variables were the input voltages of the converters (i.e., the SAS module 1 and 2 output voltages), the output voltage SAS module 2 (experiencing the transient) and, finally, the current delivered to the voltage-type load. Next, the MPP algorithm was added to the closed-loop control scheme, where the MPP block gave reference value for the input voltage control. Similar transient was introduced to the other SAS module and the system operation was verified as previously described. The results are shown in Figs. 25 and 26, respectively. The presented results validate the simulated predictions given by Petrone et al. [8] and the analytical claims presented in this paper. It can be concluded, that input-voltage control is mandatory to obtain a high-performance system with seriesconnected DMPPT converters.
This paper discussed the dynamic properties of distributed MPP-tracking dc–dc converters in cascaded configurations. Comprehensive small-signal analysis was presented with main focus on interaction between two individual converters. It was shown, that in series configuration the system structure itself introduces cross-coupling effects that disturb the intended system operation. Similar cross-coupling effects were found in parallel configuration only after including nonideal properties to the system load. However, it was concluded that due to properties of the system load (i.e., a grid-connected inverter), the parallel configuration does not actually suffer from cross-coupling effects. The observations were used to explain the observed behavior of series-connected converters, most notably by explicitly showing that input-voltage control has to be applied to remove the cross-coupling effects. Further, time-domain measurements performed with prototypes verified the simulated predictions presented by other authors. This paper clearly pointed out the strength of frequency-domain analysis and its importance in research around PV interfacing dc–dc converter systems. REFERENCES [1] The Federal Government of Germany. (2011, Jun.). “Switching to the electricity of the future (accessed 14.5.2012),” [Online]. Available: http://www.bundesregierung.de/Content/EN/Artikel/_2011 /06 /2011-0609-r egierungserklaerung_en.html?nn=454766 [2] B. Kroposki, R. Margolis, and D. Ton, “Harnessing the sun: An overview of solar technologies,” IEEE Power Energy Mag., vol. 7, no. 3, pp. 22–33, May/Jun. 2009. [3] D. Abbott, “Keeping the energy debate clean: How do we supply the world’s energy needs?” Proc. IEEE, vol. 98, no. 1, pp. 42–66, Jan. 2010. [4] T. M. Razykov, C. S. Ferekides, D. Morel, E. Stefanekos, H. S. Ullal, and H. M. Upadhaya, “Solar photovoltaic electricity: Current status and future prospects,” Solar Energy, vol. 85, no. 8, pp. 1580–1608, Aug. 2011. [5] G. Petrone, G. Spagnuolo, R. Teodorescu, M. Veerachary, and M. Vitelli, “Reliability issues in photovoltaic power processing systems,” IEEE Trans. Ind. Electron., vol. 55, no. 7, pp. 2569–2580, Jul. 2008. [6] M. A. Eltawil and Z. Zhao, “Grid-connected photovoltaic power systems: Technical and potential problems—a review,” Renewable Sustainable Energy Rev., vol. 14, no. 1, pp. 112–129, Jan. 2010. [7] J. Huusari and T. Suntio, “Distributed MPP-tracking: Cross-coupling effects in series and parallel connected dc/dc converters,” in Proc. Photovolt. Spec. Conf., 2012, pp. 3103–3109.
HUUSARI AND SUNTIO: ORIGIN OF CROSS-COUPLING EFFECTS IN DISTRIBUTED DC–DC CONVERTERS IN PHOTOVOLTAIC APPLICATIONS
[8] G. Petrone, C. A. Ramos-Paja, G. Spagnuolo, and M. Vitelli, “Granular control of photovoltaic arrays by means of a multi-output maximum power point tracking algorithm,” Prog. Photovoltaic: Res. Appl., DOI: 10.1002/pip.2179. [9] W. Li and X. He, “Review of nonisolated high-step-up dc/dc converters in photovoltaic grid-connected applications,” IEEE Trans. Ind. Electron., vol. 58, no. 4, pp. 1239–1250, Apr. 2011. [10] S. Kjaer, J. Pedersen, and F. Blaabjerg, “A review of single-phase gridconnected inverters for photovoltaic modules,” IEEE Trans. Ind. Electron., vol. 41, no. 5, pp. 1292–1306, Sep./Oct. 2005. [11] Q. Li and P. Wolfs, “A review of the single phase photovoltaic module integrated converter topologies with three different dc link configurations,” IEEE Trans. Power Electron., vol. 23, no. 3, pp. 1320–1333, May 2008. [12] A. M¨aki, S. Valkealahti, “Power losses in long string and parallelconnected short strings of series-connected silicon-based photovoltaic modules due to partial shading conditions,” IEEE Trans. Energy Convers., vol. 27, no. 1, pp. 173–183, Mar. 2012. [13] J. Myrzik and M. Calais, “String and module integrated inverters for single-phase grid connected phovoltaic systems—A review,” in Proc. IEEE Power Tech Conf., 2003 [14] B. Liu, S. Duan, and T. Cai, “Photovoltaic dc-building-module-based BIPV system—concept and design considerations,” IEEE Trans. Power Electron., vol. 26, no. 5, pp. 1418–1429, May 2011. [15] J. Imhoff, J. R. Pinheiro, J. L. Russi, D. Brum, R. Gules, and H. L. Hey, “Dc-dc converters in a multi-string configuration for stand-alone photovoltaic systems,” in Proc. IEEE Power Electron. Spec. Conf., 1976, pp. 2806–2812. [16] K. Kim and P. Krein, “Photovoltaic converter module configurations for maximum power point operation,” in Proc. Power Energy Conf. Illinois, 2010, pp. 77–82. [17] H. J. Bergveld, D. B¨uthker, C. Castello, T. S. Doorn, A. de Jong, R. van Otten, and K. de Waal, “Module-level dc/dc conversion for photovoltaic systems,” in Proc. IEEE Int. Telecommun. Energy Conf., 2011, pp. 1–9. [18] G. Walker and P. Sernia, “Cascaded dc-dc converter connection of photovoltaic modules,” IEEE Trans. Power Electron., vol. 19, no. 4, pp. 1130– 1139, Jul. 2004. [19] N. Femia, G. Lisi, G. Petrone, G. Spagnuolo, and M. Vitelli, “Distributed maximum power point tracking of photovoltaic arrays: Novel approach and system analysis,” IEEE Trans. Ind. Electron., vol. 55, no. 7, pp. 2610– 2621, Jul. 2008. [20] C. Deline, B. Marion, J. Granata, and S. Gonzalez. (2011, Jan.). “A performance and economic analysis of distributed power electronics in photovoltaic systems (accessed 14.5.2012),” pp. 1–15, [Online]. Available:http: //www.nrel.gov/docs/fy11osti/50003.pdf [21] L. Linares, R. W. Erickson, S. MacAlpine, and M. Brandemuehl, “Improved energy capture in series string photovoltaics via smart distributed power electronics,” in Proc. IEEE Appl. Power Electron. Conf., 2009, pp. 904–910. [22] L. Chang, K. Sun, Y. Xing, L. Feng, and H. Ge, “A modular grid-connected photovoltaic generation system based on dc bus,” IEEE Trans. Power Electron., vol. 26, no. 2, pp. 523–531, Feb. 2011. [23] A. I. Bratcu, I. Munteanu, S. Bacha, D. Picault, and B. Raison, “Cascaded dc-dc converter photovoltaic systems: Power optimization issues,” IEEE Trans. Ind. Electron., vol. 58, no. 2, pp. 403–411, Feb. 2011. [24] K. Siri and M. Willhoff, “Optimum power tracking among seriesconnected power sources with uniform voltage distribution,” in Proc. IEEE Aerosp. Conf., 2011, pp. 1–11. [25] K. Siri, V. A. Caliskan, and C. Q. Lee, “Peak power tracking in parallel connected convertors,” IEE Proc. Circuits, Devices Syst. G, vol. 140, no. 2, pp. 106–116, Apr. 1993. ´ [26] R. D. Middlebrook and S. Cuk, “A general unified approach to modelling switching-converter power stages,” in Proc. IEEE Power Electron. Spec. Conf., 1976, pp. 18–34. [27] D. Maksimovi´c, A. M. Stankovi´c, V. J. Thottuvelli, and G. C. Verghese, “Modeling and simulation of power electronic converters,” Proc. IEEE, vol. 89, no. 6, pp. 898–912, Jun. 2001. [28] R. D. Middlebrook, “Small-signal modeling of pulse-width modulated switched-mode power converters,” Proc. IEEE, vol. 76, no. 4, pp. 343– 354, Apr. 1988.
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[29] J. Lepp¨aaho, T. Suntio, “Solar-generator-interfacing with a current-fed superbuck converter implemented by duality-transformation methods,” in Proc. Int. Power Electron. Conf., 2010, pp. 680–687. [30] J. Lepp¨aaho, L. Nousiainen, J. Puukko, J. Huusari, and T. Suntio, “Implementing current-fed converters by adding an input capacitor at the input of voltage-fed converter for interfacing solar generator,” in Proc. Int. Power Electron. Motion Control Conf., 2010, pp. 81–88. [31] J. Huusari and T. Suntio, “Current-fed quadratic full-bridge buck converter for pv systems interfacing: Dynamic characterization,” in Proc. IEEE Energy Convers. Congr. Expo., 2011, pp. 487–494. [32] T. Suntio, M. Hankaniemi, and M. Karppanen, “Analysing the dynamics of regulated converters,” IEE Proc. Electric Power Appl., vol. 153, no. 6, pp. 905–910, 2006.
Juha Huusari (S’09–M’12) received the M.Sc. (Tech.) and D.Sc. (Tech.) degrees (the latter with distinction) in electrical engineering from Tampere University of Technology, Tampere, Finland, in 2009 and 2012, respectively. Since August 2012, he has been with ABB Corporate Research, Baden-D¨attwil, Switzerland. He has authored and coauthored 12 conference publications and eight journal publications. He also has one international patent application. His current research interests include analysis and design of photovoltaic inverters as well as dc/dc converters in photovoltaic power systems. Dr. Huusari is a Member of the IEEE Power Electronics, the IEEE Industrial Electronics, and the IEEE Power Engineering Societies.
Teuvo Suntio (M’98–SM’08) received the M.Sc. (Tech.) and D.Sc. (Tech.) degrees in electrical engineering from Helsinki University of Technology, Espoo, Finland, in 1981 and 1992, respectively. From 1977 to 1991, he was with Fiskars Power Systems as a Design Engineer and R&D Manager. From 1991 to 1992, he was at Ascom Eergy Systems Oy as an R&D Manager. From 1992 to 1994, he was an Entrepreneur in power electronics design consultancy, and from 1994 to 1998 he was at Efore Oyj as a Consultant and Project Manager. Since 1998, he has been a Professor specializing in switched-mode power converter technologies first at the University of Oulu, Electronics Laboratory, and from August 2004 at the Tampere University of Technology, Department of Electrical Energy Engineering. His current research interests include dynamic modeling and control design of switched-mode power converters in dc–dc systems as well as in renewable energy applications. He holds several international patents and has authored more than 180 international scientific journal and conference papers, the book Dynamic Profile of Switched-Mode Converter—Modeling, Analysis and Control, Wiley-VCH, Weinhein, Germany, 2009 as well as two book chapters. Prof. Suntio is a Member of the IEEE Power Electronics, IEEE Industrial Electronics, IEEE Circuits and Systems, and IEEE Power and Energy Societies as well as a member of European Power Electronics and Drives Association. From the beginning of 2010, he has served as an Associate Editor for the IEEE TRANSACTION OF POWER ELECTRONICS. He served also as a quest Editor-inChief of the Special Issue on Power Electronics in Photovoltaic Applications of the IEEE TRANSACTIONS ON POWER ELECTRONICS in 2011–2012.
