Modeling of Steady State Operation of Two-Input DC-DC Converter for Combining of Harvested and Reserve Powers

Modeling of the designed two input DC-DC converter for combining harvested and reserve powers is presented in the paper. Analytical model of the converter in steady state conditions was derived using differential equations averaging method. The results agreement comparing with the converter circuit simulation in LTSPICE software is shown. Analytical expression between control variables (switching duty cycles) and independent quantities (sources’ voltages and load current) is derived. A converter controller algorithm seeking to minimize the power drawn from the reserve source is suggested. DOI: http://dx.doi.org/10.5755/j01.eee.19.9.5649


I. INTRODUCTION
The electric energy generated by renewable sources (photovoltaic, wind, hydro, etc.) most often has to be converted to the electrical energy having voltage and current compliant to the requirements of an electrical load.A variety of electronic power converters (DC-DC, DC-AC, AC-DC) are used for this purpose [1].It became common that these converters are also responsible for combining and shaping energies generated by the several renewable sources and/or energy storage batteries in order to supply the load.In this paper we consider the system configuration shown in Fig. 1.The reserve source U2 supplies the energy from the power grid.This is the energy that a consumer is billed for.We assume that U2is always capable of supplying the power demanded for the normal load operation.The energy generated from the renewable source, for example solar panel, is fluctuating over the day and year season.It is possible that instantaneous power drawn from the source U1 is not sufficient to fully supply the load.The proposed setup Manuscript received January 25, 2013; accepted April 8, 2013. is aimed to minimize energy consumed from the reserve source U2 as a result of some portion of energy taken from the source U1.Such a configuration could be suitable for loads having DC plug for power supply and a solar panel mounted individually at the consumer side.The conversion from DC (from renewable source) to AC, connecting AC to the power grid or wiring new AC lines to the load location are the difficulties that may be avoided using the proposed setup.
In this paper we target some design issues of the two-input DC-DC converter.The input voltages V1 and V2 in general are not equal.Even more voltage V1 is not fixed due to the varying availability of the renewable energy.We also assume that the load voltage V0 is equal to V2, since V2 is the output of the original DC power supply of the load (laptop computer, TV set-top-box, charger, antenna amplifier, etc.).We will present a steady state analysis of the suggested converter.It is expected that the results of this analysis will support the possibility of implementing the DC-DC converter for combining power generated by the renewable and reserve sources.In the future definition of the converter's dynamic characteristics will also be needed for the design of closed-loop control of the converter.Meanwhile, we focus on electronic schematics without a feedback controller.

II. STRUCTURE OF THE CONVERTER
Many publications report investigations of the so called current sharing techniques and structures [2], [3].However, the most often authors deal with the problem of ensuring current balance as in the implementation of redundant power supplies [4].In our case it is the opposite, i.e. consumption of the energy from the renewable source is the priority, while the remaining energy should flow from the reserve source.Some authors have already presented similar approach using conversion to AC and/or using energy storage batteries [5], [6].In order to minimize the cost of the converter we attempt to avoid batteries and conversion to AC.In addition to operating two power supplies in parallel the converter must ensure voltage level conversion.Therefore, we expect that switching DC-DC converter with two inputs should serve for the purpose (Fig. 2

A. Analytical averaged model of the converter
To obtain the steady state values of the output quantities , we will follow the well established methodology of averaging the system of differential equations that describe circuit of the converter dependent on the status of control switches [7]- [9].The application of the method was already demonstrated for multiple-input [10] and multiphase [11] converters.Following the Kirchhoff's laws the systems of differential equations are composed for variables for each circuit shown in Fig. 4. For example, for the first circuit in Fig. 4: In a matrix form the system (1) can be presented [8], [10] by , where ( , ) T LC iu  x is the vector state space quantities, is the vector of independent quantities, and matrixes A1 and B1 are: where Correspondingly for the circuits 2 and 3 in Fig. 4 the matrixes A2, B2 and A3, B3 can be expressed as follows: where where Then the averaged equations system can be written [8] , where averaged matrixes are: In the steady state conditions all derivatives are assumed to be equal to zero, and the solution of ( 9) is obtained by solving the system of linear equations [8]  where: From the waveforms given in Fig. 6 it was obvious that the converter was operating in the continuous current mode (CCM).The used method of differential equations averaging is applicable for converter modeling in CCM [8].Steady state output voltage V0st was estimated at the time moment t = 1.5 ms (see Fig. 6).Comparison of V0st values obtained using LTSPICE simulation and those calculated from (15) have shown very good agreement at several combinations of duty cycles d1 and d2, and load current I0.The mismatch between the values was only at the milivolt level.

C. Relationship between control and independent quantities
In order to approve control of the output voltage, it is of interest to derive the expression relating duty cycles d1 and d2 (control variables), the independent quantities V1, V2 and I0, and the output voltage V0.To achieve this we use the third equation from (15) 0 31 32 13 1 23 2 33 0 , where c13 and c23 are corresponding elements of matrix C, and e13, e23 and e33 are corresponding elements of matrix E.
Expressions for iLst and uCst are obtained substituting (7) and ( 8) to (11).Then from (22) the duty cycle can be expressed The correctness of the expression was verified by numerically solving (15) using Matlab fsolve function.No mismatch was observed.The duty cycle d2 calculated according to (23) must also be constrained by the following inequality 1 0 If calculated d2 does not comply to the (24) that means the converter is not able to ensure the requested output voltage.

IV. A SUGGESTION FOR THE CONTROL ALGORITHM
Usually a controller of a switching converter is implemented as a PID controller.It takes feedback from the output voltage (and current) and seeks to adjust duty ratio such that to keep V0 equal to reference voltage Vref.In our case Vref = V2.Influence of load current and input voltage fluctuations on V0 be suppressed (first criterion) by the controller operating in disturbance rejection mode.In the analyzed two input converter two control variables are available (d1 and d2).It could be many combinations of d1 and d2 that ensure the preset output voltage.Therefore, we suggest that the second criterion for the adaptive control algorithm should be the minimization of the power P2 where i2st can be obtained from (15).In Fig. 7(a) the dependences of P2 vs. d1 at two load currents are shown, assuming that d2 is selected using ( 23) and ( 24) in order to ensure output voltage equal to V2.It can be seen that there exists an optimal value of It must be commented that the drop of the P2 curve in the range of 1 0.85 d  , is due to the fact that it is no longer possible to find a value for d2 such that to keep 02 VV  .It can be seen from Fig. 7(c).Indeed, the d2 was calculated according to (23) but finally was set to 21 1 dd  if the condition 12 1 dd  was not satisfied.Therefore, the range of 1 0.85 d  (when I0 = 5 A) should not be treated as a normal operating region of the converter.When plotting Fig. 7 the voltage of the source V1 was modeled by the function presented in [12], which describes generated voltage of a photovoltaic cell (PV) in respect to its load current 1PV i . In particular the output voltage of the PV cell PV2-70W can be expressed [12] PV The following parameters of the converter (Fig. 2) d , which is yet not investigated.

V. CONCLUSIONS
The results of steady state modeling of the considered two-input DC-DC converter approve its suitability for combining unstable harvested power with reserve source power for DC load supply.The control variables are duty cycles of electronic switches connected in the branches of each power source.By manipulating these duty cycles, the output voltage may be kept constant in the certain range of renewable source power fluctuations and load current demand.
It was demonstrated that there exists an optimal set of control variables (duty cycles) that minimize the power drawn from the reserve source.

Fig. 1 .
Fig. 1.Power supply from two sources via two-input converter.
the duty cycles of PWM control signals of the switches S1 and S2 are: