2013 XXIV International Conference on Information, Communication and Automation Technologies (ICAT)October 30 – November 01, 2013, Sarajevo, Bosnia and Herzegovina
Abstract—Photovoltaic (PV) systems that inject energy di-rectly to the grid have attracted much attention over the lastyears due to their lower cost per watt with respect to otherphotovoltaic applications and the incentives that governmentsoffer for such systems. In a grid-connected PV system a powerinverter is required to optimize the energy transfer from thephotovoltaic modules to the power grid. Considering the non-linear time-varying nature of grid-connected PV systems allowsto obtain well-defined mathematical description of the problemthat can be useful for the design of control schemes. Nevertheless,control structures that explicitly take into account the non-linearelectrical model of the PV modules usually depend on parametersthat are unknown and/or difficult to measure. Consequently, suchcontrollers should normally be used with estimator schemes. Inthe present paper a control and estimator scheme for a full-bridge central PV inverter which is valid for a wide range of PVtechnologies is presented. As shown with a simulation study, therequired control objectives have been achieved and the unknowntemperature dependent have been correctly estimated.
I. INTRODUCTION
One of the most cost-effective ways of profiting photo-voltaic (PV) energy is by feeding the generated PV energydirectly into the utility grid system. The grid can absorb PVpower that is surplus to current needs, making it available foruse by other customers and reducing the amount of energythat has to be generated by conventional means. At night or oncloudy days, when the output of the PV system is insufficientto the energy needs, the grid will provide the backup energyfrom conventional sources. The lack of a battery subsystemnot only represents a considerable cost and size reduction ofthe whole system but also increases its reliability: while a PVpanel lasts more than twenty years a battery operates for atmost five years and need periodic maintenance, [1].
The main components of a grid-connected PV systemsinclude a series-parallel connection arrangement of the avail-able PV panels and a power conditioning system that reg-ulates the power transfer from the PV panels to the grid.A control strategy for the power conditioning system has tobe designed in order to extract and to properly transfer themaximum available power from the PV panels to the grid.A main difficulty in the design of such control scheme isthe inherent nonlinear time-vayring electrical characteristicsof the grid-connected photovoltaic inverter and its dependanceon unpredictable environmental conditions such as temperatureand solar irradiance.
A common approach to deal with the control of the PVpower conditioning subsystem is to make use of heuristicalgorithms and control design methodologies that do not fullyconsider the non linear electrical model of the PV panel. Theaforementioned approach lacks of a well-defined mathematicalproblem description that might difficult the derivation of anoptimal solution. On the other hand, control structures thatexplicitly take into account the non linear electrical model ofthe PV panels (e.g. [2], [3]) usually depend on parametersof the model that are unknown and/or difficult to measure.Consequently, such controllers should normally be used withestimator schemes.
This paper presents a control and estimator scheme for afull-bridge central PV inverter. The P-passive controller, whichwas introduced in [2] without the estimator, renders the systemglobally stable. The online estimator, introduced in [4] where itwas applied to a PV battery charging system, allows to obtainthe temperature dependent parameters of the PV generator1.
The grid-connected photovoltaic (GPV) system that isconsidered in the present paper is shown in Fig. 1. The systemconsists of an array of PV modules connected to the utility gridby means of a full-bridge inverter. This configuration in whichall the PV panels are linked to a unique power inverter unitis known as “central inverter”. A grid-connected PV system
1A photovoltaic generator (PVG) is considered in this document as a devicethat is designed and constructed to generate electric energy based on thephotovoltaic effect, it is a general term that includes PV cells, PV modulesand PV arrays.
based on a central inverter is one of the most prevailing config-urations since, under uniform irradiance conditions of the PVpanels, it represents a good trade-off between the extractedenergy and the design complexity of the power inverter [5],[6], [7]. Moreover, the full-bridge power inverter stage is atypical output stage present in other GPV configurations suchas the widely used two-stage boost-buck configuration (e.g.,[8], [1]) or the multilevel configuration (e.g., [9]). As statedin [10], this common output stage enables to extrapolate thecentral inverter’s control scheme to other configurations.
The schematic diagram of the full-bridge central inverterconfiguration is shown in Fig. 1. Here x1 and x2 are theaverage values of the input capacitor voltage and the outputinductor current, respectively. The utility grid voltage vg isassumed to be sinusoidal with a constant amplitude A anda constant frequency ω, i.e., vg = A sin(ωt). The PV arrayoutputs a current, ipv(·) depends on the PV array’s voltage,vpv = x1, a set of constant parameters and the PV array’stemperature and incident solar radiation. The full-bridge in-verter consists of four switches controlled by the signals δand δ which take values in the discrete set {0, 1} (i.e., OFFor ON, respectively). The switch control signals are generatedvia a pulse-width modulation (PWM) scheme with a duty ratiofunction u ∈ [−1, 1] by the controller. This means that ifthe switching frequency is sufficiently high, the dynamicalbehavior of the GPV system can be approximated by thefollowing set of differential equations
Cx1 = −ux2 + ipv(·)Lx2 = ux1 − vg.
(1)
The control scheme that generates signal u should be ableto
C1. deliver a sinusoidal current in phase with the utilityvoltage of the power grid, i.e.,
x∗2 = k vg
where k is a positive value that defines the outputcurrent amplitude.
C2. set x1 to a value that assures the maximum powergeneration from the PVG, i.e.,
x∗1 = x1|PPV(x1)=max{PPV}
where PPV = x1iPV and Pmpp = max{PPV} is thePVG’s maximum power value.
There are mainly two control schemes that have been usedto achieve the aforementioned control objectives. The blockdiagram of the these control schemes are depicted in Fig. 2and Fig. 3 and are referred as one-loop controller and two-loopcontroller, respectively.
Controller Inverter
x∗1
x∗2
x1
x2
Fig. 2. Block diagram of the one-loop controller
Innercontrol
loopInverter
x∗1 Outer
controlloop
x1
x2
Fig. 3. Block diagram of the two-loop controller
The control scheme of Fig. 3 is the most common onefound in the literature (e.g., [11], [12], and [13]). It iscomposed of two loops: an inner current control loop thatdetermines the duty ratio for the generation of a sinusoidaloutput current, and an outer loop that determines the outputpower according to the maximum power point (MPP) of thePV array (see Figure 3). Notice that in this case it is onlyneeded to define the desired value of x1.
On the other hand, the scheme shown in Fig. 2 requiresthe definition of the desired values of both x1 and x2. Thiscontrol scheme as reported in [2] and [3] allows to achieveglobal stability and has comparable performance to that of thetwo-loop controller.
Regarding the definition of x∗1 and x∗
2 it should be noticedthat they are related by means of the following power balanceequations obtained from (1)
x∗1Cx∗
1 + x∗2Lx
∗2 = ipv(·)x∗
1 − x∗2vg (2)
If, according to control objective C.1, we make x∗2 = kvg
it can be verified that x∗1 needs to be time varying (and
periodic) to meet (2). Nevertheless, as it has been analyzedand experimentally verified in [3] the time varying componentsof x∗
1 can be neglected without affecting considerably theclosed-loop performance of the system. In [3] the followingexpressions for x∗
1(t) and x∗2(t) have been proposed and
validated
x∗2(t) =kA sinωt
k =2Ppv(x
∗1)
A2
x∗1(t) =x∗
1
If we consider x∗1 given, e.g. by a maximum power point
tracker, we still need to explicitly know ipv(·) in order to obtainx∗2. The expression of ipv(·) is obtained from the electrical
model of the PVG that is explained in the following section.
III. PV ELECTRICAL MODEL
We deal with a level of abstraction that represents thePVG as a two port system in which the current circulatingthrough the device, ipv, and the voltage presented acrossits terminals, vpv, are related by means of a non-linearfunction which depends on the PVG’s temperature, T , theincident solar radiation, S, and a set of constant parametersρ = (ρ1, ρ2, . . . , ρn). The number, n, and the values ofthe aforementioned parameters depend on the materials andmanufacturing process of the PVG. Consequently, the mostsuitable models for the different PVG technologies have notonly different number and values of ρ but also different waysin which T and S affect their electrical behavior ( [14], [15],
[16], [17]). Nevertheless, in almost all cases the relationshipbetween ipv and vpv has the following general form
ipv = Λ(S, T, ρ)− φ(S, T, vpv, ρ) (3)
where Λ(·) ≥ 0 and φ(·) ≥ 0 are two functions that varydepending on the PVG technology. The common factor inthe majority of the cases is that φ(·) is bijective and strictlyincreasing with respect to vpv.
For the vast majority of monocrystalline and polycristallinePV generators equation (3) can be approximated to the follow-ing implicit expression
that is, the electrical model of this class of PV generators al-lows the separation of two independent components dependingonly on the solar incident radiation, S, and the temperature,T .
where Ψ and α depend only on the temperature and ρ, and zis a new defined variable depending on ρ, the PV current, ipv,and the PV voltage, vpv. Thus, it is possible to rewrite (4) ina simpler way, i.e.,
Provided all the necessary values and measurements re-quired to calculate y and z, a simple on-line estimation schemeto obtain θ1 and θ2 can be derived as it will be described inthe following section. For instance, for the typical one-diodeelectrical model of a PV generator, i.e.,
ipv = Ig(S)− Is(T )(expα(T )vpv)
assuming Ig ≫ Is we have that
y = ln (Ig − ipv)
θ1 =α(T )
θ2 = ln(Is)
z =vpv.
IV. ESTIMATOR
As mentioned previously, the estimator used in this paperhas been presented in [4]. For the sake of completeness themain results and analysis found in [4] are repeated next.
Consider (5) rewritten in the following way
y = θTΦ (6)
where Φ = [z 1]T , θ = [θ1 θ2]T and the dependence of θ on
the temperature and the other parameters is omitted from thenotation for simplicity.
Defining θ = [θ1 θ2]T as the estimated values of θ one can
make use of the following estimator
˙θ = ΓΦ
(
y − ΦTθ)
(7)
where Γ is a positive-definite matrix. Defining the param-
eter error θ = θ − θ, (7) can be written in the classical form
˙θ = −ΓΦΦTθ. (8)
It is well–known, e.g. [18], that the zero equilibrium of thisequation is stable with Lyapunov function
V =1
2θTΓ−1θ.
Indeed, the derivative of this function is,
V = −(
θTΦΦTθ)
≤ 0. (9)
Recall that, as discussed previously and according to (2),at steady state vpv is periodic and thus persistently exciting,and thus the dynamical system (8) will reach the desired
stabilization equilibrium, i.e., θ1 = θ2 = 0.
For the sake of illustration and further discussion of thepresented estimator scheme we consider a PV array withthe electrical charactertistics shown in Fig. 4 and a choiceof vpv similar to that present in the normal steady stateoperation of a central inverter, i.e.,
vpv = vpv + 0.03vpv sin(200πt) (10)
where vpv is the mean value of vpv.
Additionally, Γ has been defined as
Γ =
[
γ11 γ12γ21 γ22
]
(11)
where the γij are chosen to tune the dynamics of the es-timator and make Γ positive-definite. For instance choosingγ11 = γ1a + γ1b > 0, γ22 = γ2a + γ2b > 0 and γ12 + γ21 =2√γ1b
√γ2b > 0 yields Γ positive-definite.
Figures 5, 6, 7 and 8 show the dynamic behavior of theestimator using (10) with vpv = 600 V and the electricalcharacteristics of the PV array shown in Fig.4. As it can beseen in the aforementioned figures, we can identify two well-defined behaviors of the estimator dynamics, namely,
1) the dynamics when the trajectory of θ1 and θ2 inthe phase plane has not reached the nullcline. Thisdynamics, which are shown in Figures 5 and 6,corresponds to the linear dynamics defined by (7)which, by means of Γ can be made arrive to thenullcline arbitrary fast.
2) the dynamics when the trajectory of θ1 and θ2 inthe phase plane has reached the nullcline and evolves
towards θ1 = θ1 and θ2 = θ2. When the nullcline isreached the following equation holds
ipv = θ1vpv + θ2 = θ1vpv + θ2. (12)
Notice the estimator’s trajectory takes significantlymuch more time to reach the desired equilibrium thanto reach the nullcline. In this case, the dynamics of
0 100 200 300 400 500 600 700
vpv (V)
0
1
2
3
4
5
6
7
8i p
v(A
)
0 100 200 300 400 500 600 700
vpv (V)
0
500
1000
1500
2000
2500
3000
3500
4000
Ppv
(W)
Fig. 4. i− v and P − v curves of the PV array considered for simulations.The values of the temperature dependent parameters are θ1 = 0.026 andθ2 = −15.818. The red circle marks the maximum power point.
TABLE I. PARAMETERS OF THE ESTIMATOR AND THE SIMULATION.NOTE: THE SIMULATION HAS BEEN DONE USING Python 2.7.3 WITH THE
numpy AND scipy LIBRARIES.
.
Parameter Value
γ11 0.1
γ12 222.5
γ21 -200
γ22 256
θ1(t = 0) 0.029
θ2(t = 0) -17
Absolute tolerance 5 × 10−8
Relative tolerance 5 × 10−8
the trajectory mainly depends on the spectral richnessof vpv and the definition of Γ. This dynamics can beseen in Figures 7 and 8.
The values of the parameters of the simulation are shownTable I.
V. P-PASSIVE CONTROLLER AND ESTIMATOR
In this section the estimator scheme described previously isused with the P-Passive controller described in [2] and [3]. Thiscontroller is a one loop controller with a similar structure tothe control scheme shown in Fig. 2. The estimator is requiredfor the temperature dependent parameters needed to obtain x∗
1and x∗
2.
A. P-Passive controller
The P-Passive controller is discussed in detail in [2]and [3]. The main result obtained in the cited works is thatthe following control signal achieves a globally asymptoticallystable closed-loop system
u =Lx∗
2 + vg
x∗1
−Kp (x∗1x2 − x∗
2x1) (13)
where x1 = x1 − x∗1, x2 = x2 − x∗
2 and Kp is the controllergain that can be tuned to obtain the desired dynamics. Noticethat, as discussed previously, in order to obtain x∗
Fig. 5. Evolution of θ1 and θ2 before the estimator’s trajectory reach thenullcline. The black solid line represents the nullcline, i.e., when (12) holds.
The red circle is the point where θ = θ.
0.024 0.025 0.026 0.027 0.028 0.029 0.030−18.0
−17.5
−17.0
−16.5
−16.0
−15.5
−15.0
−14.5
−14.0
θ1
θ2
Fig. 6. Evolution of θ1 and θ2 in the phase plane before the estimator’strajectory reach the nullcline.
0 20 40 60 80 1000.0245
0.0250
0.0255
0.0260
0.0265
0.0270
0.0275
0.0280
0.0285
0.0290
0 20 40 60 80 100−17.0
−16.5
−16.0
−15.5
−15.0
Time (s)
Time (s)
yy
Fig. 7. Evolution of θ1 and θ2
0.024 0.025 0.026 0.027 0.028 0.029 0.030−18.0
−17.5
−17.0
−16.5
−16.0
−15.5
−15.0
−14.5
−14.0
θ1
θ2
Fig. 8. Evolution of θ1 and θ2 in the phase plane. The black solid linerepresents the nullcline, i.e., when (12) holds. The red circle is the point
where θ = θ.
B. Assumptions
The simulation tests were performed based on the follow-ing assumptions
• the electrical characteristics of the PV array model canbe written in the form of (5) where the only unknownparameters are those depending on the temperature,i.e., θ1 and θ2. More specifically, the electrical char-acteristics of the PV array considered in the simulationis shown in Fig. 4,
• x∗1 is assumed given and the reference value of x2 is
calculated using θ and is denoted as x∗2 = kA sin(ωt).
C. Simulation results and discussion
Figures 9, 10 and 11 show the simulation results of thepresented control and estimator scheme with the assumptionsstated in the previous section and with x∗
1 = 600 V. FromFigure 9 it is shown that the control and estimator schemeachieve both control objectives, i.e., a desired value of x1 andan output current, x2 in phase with the utility grid voltage.
Figures 10 and 11 show the simulation results of theestimator. Figure 10 show the evolution of the estimates of
θ1 and θ2, where it can be seen that they do not reach thecorrect values until at least 200 s. On the other hand, fromFigure 11 it is possible to see that the estimation value of x∗
2,
i.e, x∗2 = kA sinωt, converges in approximately 0.5 s. The
difference on the convergence of the estimations of θ and k isdue to the two different dynamic behaviors that the estimatorexhibits, as discussed in the preceding section. Indeed, noticethat when the estimator achieves the nullcline equation (12)
holds, and thus ipv(x1) = ipv(x1) even though θ 6= θ.
VI. CONCLUSIONS AND FUTURE WORK
In the present paper a control and estimator scheme thatconsiders the non-linear time-varying electrical characteristicsof a PV inverter have been presented. As shown with a simula-tion study, the required control objectives have been achieved
0.0 0.1 0.2 0.3 0.4 0.5450
500
550
600
650
700
0.0 0.1 0.2 0.3 0.4 0.5
−30
−20
−10
0
10
20
30
x1
x2
Time (s)
Time (s)
Fig. 9. Simulation results of the evolution x1 (top) and x2 (bottom) usingthe controller defined in (13) and the estimator (7). In red are the desired valueof x1 (top) and the scaled waveform of the utility grid voltage (0.1vg).
0 50 100 150 2000.018
0.020
0.022
0.024
0.026
0.028
0.030
0 50 100 150 200−17
−16
−15
−14
−13
−12
−11
θ1
θ2
Time (s)
Time (s)
Fig. 10. Simulation results of the evolution of θ1 and θ2 using the controllerdefined in (13) and the estimator (7). The dash lines denote the actual valuesθ1 and θ2.
and the unknown temperature dependent estimated values havebeen obtained. The estimator and controller scheme is valid fora wide range of PV technologies. Future work will focus onthe implementation of the estimator and the definition of anon-heuristic algorithm that is able to obtain the PV array’smaximum power point.
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