Abstract— Multi-port DC-DC converters are very attractive
because they have the potential to combine different energy
sources and management strategies. Their utilization is not only
reducing the cost and size of the conventional energy sources
but also facilitating the entrance of alternative energy sources on
the market. In this paper, the modeling and analysis of such
power electronic devices with N ports are discussed. A new
representation of the average current flowing in and out of the
terminals of an N-port converter is presented. This new average
current, named as the cyclic average current (CAC), has shown
advantages for the modeling, optimization, and control of the N-
port converter. The large-signal model (LSM) and small-signal
state-space model (SSM) of an N-port converter are derived using
CAC and compared to the LSM and SSM derived using the
average current (OAC) in the literature. The modeling procedure
is demonstrated for a 5-port DC-DC converter. Simulation is then
used to compare the LSM and SSM models and observe their
performance with respect to a detailed model (DM).
I. INTRODUCTION
The costs of conventional forms of energy such aspetroleum, coal, and natural gas are continuously rising dueto their increasing demands, the environmental changes, andtheir scarcity in certain regions. Their devastating pollutioneffects on the atmosphere are also becoming a worldwide con-cern. Advancements in the exploitation of alternative energysources such as photovoltaics, fuel cells, wind turbines, alongwith storage devices such as batteries and supercapacitorshave also been observed during the past decade. Renewableenergy sources do not create pollution or at least to smallamounts. Nowadays, both developed and developing countriesare acknowledging the fact that some of the prevailing energyproblems can be tackled by incorporating renewable energies.Hence, combining the most available forms of energy withthe means of a power converter as one and managing it moreefficiently could be a key to offset the crisis of pollution andproblematic energy sources. The multi-port DC-DC converteris a system that addresses this necessity. It finds its efficacyin many applications such as micro grid applications orhouseholds that are seeking to harness the wind or solar energyto reduce their utility bills.
Various topologies of the multi-port DC-DC converter havebeen proposed in the literature. It starts with the Dual BridgeDC-DC converter (DAB), which is identified as the build-ing block of the multi-port DC-DC converter [1]–[4]. The
DAB then evolves to the three-port bidirectional converter(TAB) [5]–[7]. In [8], the three-port converter was effectivelyextended to a four-port converter using a high-frequencytransformer as the energy-coupling device. An extension of thethree-port converter to an N-port is suggested in [9]. Differenttopologies of the N-port converter are also presented in [1]and [10].
An example of a multi-port DC-DC converter with a 5-windings transformer serving as a common denominator tocombine the different energy sources is shown in Figure 1.Each sink/source is connected to its respective winding viaan H-bridge, which attributes the bidirectional power flowproperty to the terminal. The zone of study of the multi-portDC-DC converter only encloses the transformer, the H-bridges,and their filtering elements. Modeling an N-port converter is anon-trivial task since the modeling difficulty increases with thenumber of ports and the method adopted. An efficient approachto overcome this difficulty is by extending the averaging wave-forms method carried out in [3], [11] for a 3-windings deltatransformer to an N-winding transformer with a generalizeddelta (G-delta) circuit as shown in Figure 2. Knowing theaveraging waveforms of the converter, the average power atits terminals is estimated. The average current is then obtainedfrom the equation of the average power. The resulting averageexpressions for power and current were also used in [5], [6],[7], [8], [12].
This paper first presents a modified version of the averagecurrent presented in [3], [11]. For distinction purposes, theaverage current in [3], [11] is named as the original averagecurrent (OAC) and the modified version is named as the cyclicaverage current (CAC). Second, the two average currents,which are the mathematical representations of the converter,are used to obtain its equivalent large-signal model (LSM).The implementation of four different types of sources/sinks isalso discussed. The small-signal state space model (SSM) isthen obtained from the large-signal model. Lastly, a simulativecomparison of the LSMs and SSMs is performed on the 5-portconverter and the results are discussed.
II. AVERAGE CURRENT AND POWER FLOW
The averaging waveforms method is applied to the G-deltacircuit. The OAC at the ith port of the converter is given in (1).
Fig. 2. Equivalent G-delta Circuit of an N-port converter.
Ii =N
j=1,j =i
iij =N
j=1,j =i
VjYijf1(φij) (1)
where Yij =1
ωLij, ω is the switching frequency, Vi is the
voltage of port i, φij is the phase shift between bridge i andbridge j, φij ∈ [−π,+π] and
f1(φij) = φij
1− |φij |
π
(2)
The turns ratio and the prime sign usually included in (1)to identify Vj as a primary referred variable are omittedbecause the equations herein are per-unitized [13]. Studying amulti-port converter indicates the possibility of dealing with
N ports, each of which may have different voltage, current,and power levels. In order to make the subsequent analysesmore convenient, the equations of the multi-port converter areper-unitized.
From a practical point of view, the phase shift between twobridges is limited as given in (3). The expression of f1(φ),which regroups all the φ terms and dictates the behavior of (1),is plotted in Figure 4 for large values of the phase shift.Assuming that the voltages and admittances in (1) are fixed,the maximum average current is reached at π
2 . Beyond π2 ,
the average current is decreasing and then hits zero at π. Fora phase shift greater than |π| the average current convergestoward infinity. The delay in phase shift cannot exceed πbecause it automatically becomes an advance in phase shiftand f1(φ) does not portray this behavior.
φij ≤ |π| (3)
One of our future objectives is to perform a power op-timization for the converter. In order to achieve this objec-tive, a search algorithm (e.g. Newtons algorithm, Broydensalgorithm, Evolutionary algorithm etc.) with some imposedequality and inequality constraints is required. Equation (3)would be included as part of the inequality constraints. It iseasily observed from the plot of f1(φ) that failing to constrainthe phase shifts using (3) would permit the algorithm to searchfor feasible solution on the extreme curves, which convergetoward ±∞. So it would be impossible to verify whether thesolution generated by the algorithm is practical.
It is useful to consider the initial condition of a four-portconverter shown in Figure 3. Each waveform is tagged by anarrowhead symbol. The phase shift of each port is defined asthe gap from the initial condition to the new position of thearrowhead as depicted in Figure 3b. Equation (4), which allowsall the phase shifts to be expressed in term of the phase shiftsbetween any port and the reference port, is always true withthis definition of phase shift. However, since the phase shift isconstrained by (3), a periodicity is observed over a 2π cycle(i.e. φ14 in Figure 3b is the same as φ
14 ). Therefore, if anequation for the average current that considers the periodicityis derived, the phase shifts of the N-port converter could beallowed to take on any value from −∞ to +∞.
φij = φ1j − φ1i (4)
To illustrate the periodicity of the average current, it is derivedtwice more for phase shifts between [π, 2π] and [2π, 3π]. Byinduction, the average current at the terminal of an N-portDC-DC converter is then generalized as in (5) for phase shiftsbetween [−∞, +∞]. It is important to note that when τij is anodd number the average current is negative, which is identicalto the average current obtained when Port j is advanced withrespect to Port i (−π ≤ φij < 0). Conversely, when τij iseven, the average current is positive, which is identical to theaverage current obtained when Port j is delayed with respectto Port i (0 < φij ≤ π). Additionally, when τij is equal to
864
Fig. 3. New phase shifts definition a.) Initial condition b.) After phase shift.
zero, (5) becomes (1).
Ii =N
j=1,j =i
VjYijf2(φij) (CAC) (5)
where τij = floor
φij
π
, φij ∈ [−∞,+∞] and
f2(φij) = (−1)τij (φij − τijπ)
1− (φij − τijπ)π
(6)
The plot of f2(φ) is also shown in Figure 4. it is still boundedwithin the minimum and maximum values of −π
4 and π4 ,
respectively. It is important to note that (6) does not requirethe absolute value anymore as in (2). It also complies withthe relationship between phase shifts of (4). It is expected thatafter linearization of the converter using (5), the subsequentsteady-state equations that will be implemented into the searchalgorithm would naturally force the algorithm to look for thephase shifts within the desired range of (3). Moreover, most ofthe search algorithms are very dependent on the initial guessand with a solution space as delicate as the one depicted byf1(φ) , the initial guess of the phase shifts has to considertheir signs very carefully or the algorithm would convergetoward a wrong solution. On the other hand, the solutionspace depicted by f2(φ) allows a random initial guess andits cyclic characteristic would eventually force the algorithmto converge to a true solution. No matter what the solutionturns out to be, the equivalent solution that complies with (3)can be determined using (7). Therefore, when the algorithmruns for a large number of iterations without locking on anysolution, it would simply mean that the objective functionhas no practical solution for the specific transformer setup(i.e. selected voltages and maximum power available at eachsource). Another advantage of (5) is that the use of limitersfor the phase shifts during the controller design is eliminated.
φ
ij = φij −τij +
sinτijπ
2
π (7)
III. MODELING THE N-PORT DC-DC CONVERTER
A strategy is developed in [14] for appropriately selectingthe state variables, independent and dependent port variables,and control variables of a multi-port converter. The lineariza-tion of the large-signal model of the converter to obtain its
−10 −5 0 5 10−20
−15
−10
−5
0
5
10
15
20
X: 1.575Y: 0.7854
X: 3.175Y: −0.03327
Phase rad
f 1(φ) a
nd f 2(φ
)
f1(φ)f2(φ)
Fig. 4. Plots of f1(φ) and f2(φ) for large value of the phase shift.
small-signal model is then discussed. The modeling strategyin [14] is simple to follow only if the designer is ableto derive all the differential equations describing the multi-port converter. A simple modeling approach based on twofundamental elements, which are the average currents in (1)and (5), is presented in this paper. It is important to notethat (1) and (5) do not account for the resistances included inthe per-unitized G-delta circuit. Hence, the natural dampingof the system is not modeled as a first approximation.
A. Detailed Model (DM) and Large-signal Model (LSM)
The DM is modeled using the Kirchhoff’s Current Laws(KCL) and Kirchhoff’s Voltage Laws (KVL) formulas ob-tained from the equivalent G-delta circuit. Conceptual circuitdiagrams of the LSM similar to the ones in [15] are presentedfor the four types of components in Table I. The LSM forthe ith port is obtained by connecting a controlled currentsource of magnitude given in (1) or (5) to the ports inputL-C filter [12]. KCL and KVL are then applied to obtain themodeling equations. Since the port is bi-directional, the inputfilter can also be called an output filter. The port configurationwhen the input filter is connected to a dependent source withconstant supplied power Pi representing the WT or the PVsystem is shown in Figure 5(a). In order to determine theaverage current of each port, the voltage and the phase angleat each port must be known.
The utility grid (UG) is assumed to have unbounded power,which means that if the load is requesting power that is wellbeyond the availability in the renewable resources and the bat-tery, the utility grid is able to compensate the deficiency. Theport configuration of the utility grid is shown in Figure 5(b).The current ii drained from the utility is dependent of theconverter but the voltage vini is independent of the converterand is established by the circuit connected to the input ofthe converter. Hence, the voltage vini is known and usedas an input to the multi-port DC-DC converter. This circuitconfiguration is also applicable to the battery bank (BB). It isimportant to note that the voltages of BB and UG are morelikely to be constant but the circuit model is applicable to
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variable voltage sources.The load is assumed to be purely resistive. The correspond-
ing circuit configuration for the load is shown in Figure 5(c).
B. Small-signal Model (SSM)
The SSM of the N-port converter is obtained by linearizingthe equivalent circuits in Figure 5 using the demonstrationin [16], [17] . The state space average model is then realizedusing the SSM. The capacitor voltage and inductor current areused as the states for each port. Hence, there are a total of 10states for a 5-port DC-DC converter. After linearization aroundthe steady state points (Ii, Vj , φij), the linearized and steadystate equations at the three different types of ports are givenin (8) and (9). To reduce the number of phase angles in theseequations, the phase angles ϕij(t) are written in term of thephase angles ϕ1j(t) and ϕ1i(t).
Linearized equations (8)
Cidvi(t)
dt= −ii(t)−
N
j=1,j =i
Kij vj(t)
−j<i
j=2,j =i
Tijϕ1j(t) +N
j=1,j =i,i =1
Tijϕ1i(t)
−N
j>i
Tijϕ1j(t) (All Ports)
Lidii(t)
dt= vi(t)−
V 2i
Piii(t) (WT )− (PV )
Lidii(t)
dt= vi(t)− vini(t) (UG)− (BB)
Lidii(t)
dt= vi(t)−Riii(t) (LD)
Steady state equations (9)
−Ii −N
j=1,j =i
KijVj = 0 (All Ports)
Pi = −ViIi (WT )− (PV )Vini = Vi (UG)− (BB)
Vi = RiIi (LD)
where Kij = (−1)τij Yij(φij − τijπ)
1− (φij − τijπ)π
and Tij = (−1)τij YijVj
1− 2(φij − τijπ)
π
for CAC, and
Kij = Yijφij
1− |φij |
π
and Tij = YijVj
1− 2φij
π
for
OAC.The states and input variables are defined as follows:
x =v1(t) v2(t) .. v5(t) i1(t) .. i5(t)
(10)
u =ϕ12(t) ϕ13(t) .. ϕ15(t) vin1 vin2
(11)
The state space average model of the 5-port converter can bederived using (8).
(a) Connected to a dependent source.
(b) Connected to a constant voltage source.
(c) Connected to a load.
Fig. 5. Large-signal model of port i with filter.
TABLE ILIST OF ENERGY SOURCES CONNECTED TO THE 5 PORTS
The information of the energy sources connected to the 5-port converter is given in Table I. The switching frequencyis chosen to be fs = 20 kHz. It is inferred from thetable that the net power rating of the transformer is 1 kW(sum of power ratings of individual ports). From here on,the angles are in radians and the rest of the parameters inpu unless specified. The turns ratios for the 5 bridges areN1 = 1, N2 = V2/V1, N3 = V3/V1, N4 = V4/V1 andN5 = V5/V1, respectively. These turns ratios are selected in away that the voltage at each bridge represents the base voltagefor per unitization at the matching bridge. The battery bankis chosen to be the reference bridge so that the base power isPb = 75 W. Since the lowest power is selected as the basepower, the per unit values at the 4 other ports are expected tobe greater than 1. The per unit specifications of the converterwith a G-delta circuit connected to the 5 sources are shownin Table II. The filter components of each port are selected tominimize the ripples in the currents and voltages. Their valuesare shown in Table III.
The DM, LSM and SSM of the 5-port converter for bothOAC and CAC are implemented in Matlab. The results shown
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TABLE II5-PORT CONVERTER WITH G-DELTA CIRCUIT
Port Current Power Parallel Parallel(pu) Rating Resistance Reactance
in this section aim to provide a relative comparison in theperformance of the LSM and SSM when the variations inthe phase shifts and input voltages assume small values, largevalues, or when the phase shifts are assigned values outsidethe restricted interval of (3). The simulation process of theDM and LSM is straightforward. The converter is modeled asexplained above and the desired phase angles are applied inSimulink using step functions. The simulation is then run tillit reaches the steady state. The DM and LSM simulates theactual voltages of the capacitors and currents of the inductors.Conversely, the SSM simulates the variations in the voltagesof the capacitors and currents of the inductors from a quiescentpoint after the inputs are perturbed. In order to compare theLSM to the SSM, the steady state voltages of the capacitorsobtained from the LSM for a selected and practicable quiescentpoint need to be fed to the SSM. A non-practical point wouldbe a situation where the load transfers power to the other ports.The voltages of the capacitors are the only unknowns to deter-mine the state space matrices of the SSM. First, the quiescentpoint is selected: (Vin1 = 1, Vin2 = 1, V3 = 0.3463, V4 =0.2915, V5 = 0.9313, φ12 = −0.5650, φ13 = −0.7469, φ14 =−0.8435, φ15 = −0.1768). Second, the variations in thephase angles and the input voltages are chosen. The smallvariations are chosen to be: (vin1 = 0.02, vin2 = 0.01, ϕ12 =0.0313, ϕ13 = −0.0149, ϕ14 = 0.0152, ϕ15 = −0.0513).The large variations are chosen to be: (vin1 = 0.12, vin2 =0.23, ϕ12 = −0.5650, ϕ13 = 0.2450, ϕ14 = −0.2988, ϕ15 =0.7136). Third, a new quiescent point is chosen outside (3) andsmall input variations are performed to analyze the response
of the models. The new quiescent point chosen outside of (3)is obtained by adding 2π to the previous one.
For the DM and LSMs, the inputs are immediately steppedto (φ + ϕ, Vin + vin) and the voltages of the capacitors andcurrents of the inductors are initialized at the quiescent point.Consequently, the voltages of the capacitors and currents of theinductors, which are the modeled variables, should vary fromthe quiescent point to the new steady-state point. In the case ofthe SSMs, the quiescent point is used to determine the statespace matrices then the inputs are stepped to (ϕ, vin). Thenew steady-state voltages of the capacitors and new steady-state currents of the inductors are obtained by adding theirvalues at the quiescent point to their variations obtained fromthe simulation. The load is selected to be R5 = 0.15 (320 Ω).
The voltages of the capacitors and currents of the inductorsof each port, when small variations are performed on the inputsare shown in Figure 6. Even though the LSMs and SSMslack the modeling of the winding resistances and that theSSMs are a linearization of the LSMs, they all represent agood approximation of the dynamics of DM when the inputvariations are small. The SSMOAC actually shows results veryclose to the DM ones. However, this is a coincidence related tothe selection of the quiescent point. When the quiescent pointis changed, SSMOAC most likely yields the worst results. Itis observed that the smaller the variations the more accuratethe SSMs are. The BB and UG voltages are forced to 1, sono offset is expected.
The voltages of the capacitors and currents of the inductorsof each port, when large variations are performed on the inputsare shown in Figure 7. The results of LSMCAC and LSMOACalways agree as expected. Now the deviations of the SSMsfrom the LSMs are more apparent. Compared to the SSMs, theresults of the LSMs are much closer to the results of the DM.The SSMCAC performs better than SSMOAC as the inputvariations are increased.
The voltages of the capacitors and currents of the inductorsof each port, when the phase shifts are selected outside of (3)are shown in Figures 8. One of the advantages of modelingthe converter using CAC instead of OAC is now clear. TheLSM and SSM based on CAC maintain the same results asshown in Figures 6, while the models based on OAC yieldedunpredictable results. SSMOAC is unable to predict the initialvalues of the currents because its steady-state equations areonly valid in the range of (3). Additionally, SSMOAC andLSMOAC do not agree on the same results for all the ports.
V. CONCLUSION
The modeling of the multi-port DC-DC converter with Nports is presented in this paper. A new representation ofthe converter average current flow is presented. This newaverage current has shown to have many advantages whenit comes to the modeling, optimization, and control of theN-port converter. The G-delta circuit can be analyzed usingthe validated SSMs or LSMs. Given the derived state spacemodel of the N-port converter, our future work is to develop
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a power optimization algorithm and design control strategiesfor different applications.
REFERENCES
[1] H. Tao, A. Kotsopoulos, J. L. Duarte, and M.A.M. Hendrix, “Family ofmultiport bidirectional DC-DC converters,” IEE Proceedings on ElectricPower Applications, vol. 153, no. 3, pp. 451-458, May 2006.
[2] C. Iannello, S. Luo, and I. Batarseh, “Full bridge ZCS PWM converterfor high-voltage high-power applications,” IEEE Trans. on Aerospaceand Electronic Systems, vol. 38, no. 2, pp. 515-526, April 2002.
[3] R. W. De Doncker, D. M. Divan, and M. H. Kheraluwala, “A three-phase soft-switched high-power-density dc/dc converter for high-powerapplications,” IEEE Trans. on Industry Applications, vol. 27, no. 1, pp.63-73, Jan./Feb. 1991.
[4] D. M. Divan, “Diodes as pseudo active elements in high-frequency dc/dcconverters,” in Proc. IEEE Power Electronics Specialist Conf., pp. 1024-1030, April 1988.
[5] H. Tao, A. Kotsopoulos, J. L. Duarte, and M. A. M. Hendrix, “A soft-switched three-port bidirectional converter for fuel cell and supercapacitorapplications,” in Proc. IEEE Power Electronics Specialists Conf., pp.2487-2493, June 2005.
[6] H. Tao, A. Kotsopoulos, J. L. Duarte, and M. A. M. Hendrix, “Triple-half-bridge bidirectional converter controlled by phase shift and PWM,”in Proc. IEEE Applied Power Electronics Conf. and Exposition, pp. 1256-1262, March 2006.
[7] M. Michon, J. L. Duarte, M. Hendrix, and M. G. Simoes, “A three-portbi-directional converter for hybrid fuel cell systems,” in Proc. IEEE PowerElectronics Specialists Conf., pp. 4736-4742, June 2004.
[8] M. Qiang, X. Zhen-lin, and W. Wei-yang, “A novel multi-port DC-DCconverter for hybrid renewable energy distributed generation systemsconnected to power grid,” in Proc. IEEE International Conference onIndustrial Technology, pp. 1-5, April 2008.
[9] H. Tao, A. Kotsopoulos, J. L. Duarte, and M. A. M. Hendrix, “Multi-input bidirectional DC-DC converter combining DC-link and magnetic-coupling for fuel cell systems,” in Proc. IEEE Industry ApplicationsConference, pp. 2021-2028, October 2005.
[10] Z. Qian, O. Abdel-Rahman, M. Pepper, and I. Batarseh, “A zero-voltage switching four-port integrated DC/DC converter,” in Proc. IEEETelecommunications Energy Conference, pp.1-8, June 2010.
[11] D. Liu and H. Li, “A novel multiple-input ZVS bidirectional DC-DC converter,” in Proc. IEEE 31st Annual Conference of IndustrialElectronics Society, pp. 579-584, November 2005.
[12] H. K. Krishnamurthy and R. Ayyanar, “Building block converter modulefor universal (AC-DC, DC-AC, DC-DC) fully modular power conversionarchitecture,” in Proc. IEEE Power Electronics Specialists Conference,pp. 483-489, June 2007.
[13] J. J. Grainger and W. D. Stevenson, JR., Power System Analysis, 1st Ed,1994.
[14] D. C. Hamill, “Generalized small-signal dynamical modeling of multi-port DC-DC converters,” in Proc. 28th Annual IEEE Conference onPower Electronics Specialists, pp. 421-427, June 1997.
[15] Y. M. Chen and Y. C. Liu, “Development of multi-port converters forhybrid wind-photovoltaic power system,” in Proc. of IEEE Region 10International Conference on Electrical and Electronic Technology, pp.804-808, August 2001.
[16] M. Santhi, “Dynamic analysis of multi-output push-pull ZCS-QUASIresonant converter,”in Proc. of IET-UK International Conference onInformation and Communication Technology in Electrical Sciences, pp.229-237, December 2007.
[17] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics,2nd Ed, 2004.
0 0.5 1 1.5 2 2.5 30.8
1
1.2
V 1 pu
Battery Voltage
0 0.5 1 1.5 2 2.5 30.9
1
1.1
V 2 pu
Utility grid Voltage
0 0.5 1 1.5 2 2.5 3
0.35
0.4
0.45
0.5
V 3 pu
Wind Turbine Voltage
0 0.5 1 1.5 2 2.5 30.2
0.3
0.4
0.5
V 4 pu
PV panel Voltage
0 0.5 1 1.5 2 2.5 30.7
0.8
0.9
1
V 5 pu
Time in s
Load Voltage
DMLSMCACLSMOACSSMCACSSMOAC
(a) Capacitors’ voltages.
0 0.5 1 1.5 2 2.5 31.5
2
2.5
3
I 1 pu
Battery Terminal Current
0 0.5 1 1.5 2 2.5 3−5
−4
−3
−2
I 2 pu
Utility grid Terminal Current
0 0.5 1 1.5 2 2.5 3−5
−4.5
−4
−3.5
I 3 pu
Wind Turbine Terminal Current
0 0.5 1 1.5 2 2.5 3−5
−4.5
−4
−3.5
I 4 pu
PV panel Terminal Current
0 0.5 1 1.5 2 2.5 34.5
5
5.5
6
6.5
I 5 pu
Time in s
Load Terminal Current
DMLSMCACLSMOACSSMCACSSMOAC
(b) Inductors’ currents.
Fig. 6. Effect of small input perturbations.
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(a) Capacitors’ voltages.
(b) Inductors’ currents.
Fig. 7. Effect of large input perturbations.
0 0.5 1 1.5 2 2.5 3−5
0
5
10
V 1 pu
Battery Voltage
0 0.5 1 1.5 2 2.5 3−2
0
2
4
V 2 pu
Utility grid Voltage
0 0.5 1 1.5 2 2.5 3−5
0
5
V 3 pu
Wind Turbine Voltage
0 0.5 1 1.5 2 2.5 3−5
0
5
V 4 pu
PV panel Voltage
0 0.5 1 1.5 2 2.5 3−2
−1
0
1
V 5 pu
Time in s
Load Voltage
DMLSMCACLSMOACSSMCACSSMOAC
(a) Capacitors’ voltages.
0 0.5 1 1.5 2 2.5 3−20
0
20
40
I 1 pu
Battery Terminal Current
0 0.5 1 1.5 2 2.5 3−40
−20
0
20
I 2 pu
Utility grid Terminal Current
0 0.5 1 1.5 2 2.5 3−20
−10
0
I 3 pu
Wind Turbine Terminal Current
0 0.5 1 1.5 2 2.5 3−15
−10
−5
0
I 4 pu
PV panel Terminal Current
0 0.5 1 1.5 2 2.5 3−30
−20
−10
0
10
I 5 pu
Time in s
Load Terminal Current
DMLSMCACLSMOACSSMCACSSMOAC
(b) Inductors’ currents.
Fig. 8. Effect of phase shifts chosen outside restricted interval.
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