Modeling, Design Procedureand Control of a Low-Cost High-Gain Multi-Input Step-Up Converter

: The use of several different sources to feed a load jointly is convenient in many applications, in particular those where two or more renewable energy sources are employed. These applications include energy harvesting, hybrid vehicles, and off-grid systems. A multi-input converter able to admit sources of different characteristics and select the output power of each source is necessary in such applications. Several topologies of multi-input converters have been proposed to this aim; however, most of them are controlled by simple strategies based on a small signal model of multi-input converters. In this work, a low cost high gain step-up multi-input converter is analyzed. A nonlinear model is derived. Using this model, a detailed design procedure is proposed. A 500 W converter prototype was constructed to conﬁrm that the model predicted the real behavior of the converter. Using the nonlinear model, indirect voltage control of basic converters was extended to the multi-input converter. The obtained controller had a fast performance, and it was robust under load and input voltage variations. With the obtained model, the proposed design procedure, and the controller, a converter that was initially proposed for photovoltaic applications was enabled to be used in a broader range of applications. The herein exposed ideas for modeling, the design procedure, and control could be also applied to other multi-input converters.


Introduction
An intense research effort is being made to increase the use of renewable energy in all human activities. Techniques to take advantage of solar [1][2][3][4], wind [5][6][7], and hydrogen based [8][9][10], among other kinds of clean energies, are being developed. In some applications, it is necessary to jointly use several of these sources to feed a single load. If several sources collaborate to feed a load, it would be convenient to select the most adequate source at any given time (due to cost, availability, or other parameter) to feed a load. If a single source is not sufficient, a second source could be used; if both are not sufficient, a third can be used, and so on. For this to be possible, a multi-input converter is necessary [11]. In addition to regulating the load voltage and being robust under load variations, such a converter should accept sources with variable voltage and different voltages among them. Furthermore, this converter should be able to extract different energy portions from each source, and the energy that each source provides must be independent of its output voltage. Among the Although in [48], a converter was proposed for photovoltaic applications, in this work, it is shown that with a proper control strategy, the converter can accept sources of different voltages, and each source can provide different amounts of current independent of its voltage. It is also shown that with an adequate control strategy, the converter can be robust to input voltage and load variations.
For the sake of simplicity, in what follows, the two input case is considered. However, once the two input case is understood, it is not difficult to extend the results to the general case. The special case of two inputs is shown in Figure 2. The PWM signals applied to the transistor are shifted a half of period from each other. Depending on the switches' position, the converter has four operation modes shown in Figure 3. Note that Mode 1 and Mode 3 are the same. Capacitor C p is disconnected in these modes, thus during the time the converter is in any of these modes, voltage trough C p is held. The case where both transistors are off is avoided by restricting the duty cycle of both transistors to be larger than 0.5. One of the main advantages of this converter is the possibility to operate each transistor at different duty cycles without having one stage operating in continuous conduction mode and the other in discontinuous conduction mode, in contrast to the common interleaved boost converters [36]. From these operation modes, in the next section, a nonlinear model is obtained.

A Nonlinear Model for the Converter
The state-space standard notation is useful in the modeling and control of dynamical systems. For this reason, hereafter, electrical variables and state variables' notation given by: are used interchangeably. Using the previous notation and applying Kirchhoff laws to every circuit depicted in Figure 3, the following models can be obtained: • Mode 1 and Mode 3: Introducing the notation: Expressions (2)-(4) can be written in a single set of expressions as follows: The expressions (6) model the converter depicted in Figure 2. Variables u 1 and u 2 are regarded as the control inputs of this model. Note that these variables can only have two values because u 1 , u 2 ∈ {0, 1}; thus, Model (6) is a nonlinear switching model.
It has been shown in discontinuous differential equation theory [50] and by the heuristic approach [51] that if switching signals u 1 and u 2 change fast enough, an average model can be obtained. The average model results:˙x wheref means the average of f . Note that the average model looks very similar to the switching model; however, the meaning of the variables is different. Due to the definitions of u 1 and u 2 given by (5), the relations between transistors' duty cycles d 1 , d 2 , andũ 1 ,ũ 2 are: Useful information can be obtained from Models (6) and (7). From the average Model (7), the conditions for the existence of an equilibrium (stationary) state can be found. Switching Model (6), on the other hand, gives detailed information about what happens within a switching period. From this information, appropriate components' values can be found to keep signals' ripples lower than predefined values.
From Model (7), stationary values can be found. In the stationary state, the derivative of average signals is zero, that isẋ 1 =ẋ 2 =ẋ 3 =ẋ 4 = 0. Hence, equaling the right hand side of Expressions (7) to zero, the following is obtained: wheref means the stationary value of f . Expression (9b) shows that the output voltage is equal to the sum of two voltages as if there were two independent boost converters. Hence, it is convenient to define the voltage gain for each source as: Note that the output current is given by: Thus, from Equations (9c), (9d) and (11): Expression (12) means different duty cycles yield different stationary inductor currents. To further make stationary current values precise, it is necessary to specify the amount of power that each source contributes to total power delivered to the load. To this aim, note that in the ideal case: Suppose that the contribution rate of V in 1 and V in 2 to the total power is α 1 and α 2 , respectively, that is: From Equations (10), (13) and (14), the following can be obtained: . Furthermore, (9a) can be written now asx 3 4 , and i o =x 4 /R l are specified, all the stationary values can be calculated using: Stationary waveforms can be drawn from Model (6) and the expressions (15). They are shown in Figure 4.

The Design Procedure
General expressions for selecting inductors and capacitor values are obtained in this Section. First, it is shown how equilibrium point (stationary state) analysis can be used for selecting component values. Then, such n observation is applied to select inductors' and capacitors' values.

Analysis of Ripple Signals in the Stationary State
Suppose there is a ripple signal x (voltage, current, or other) with a constant stationary average value as is depicted in Figure 5. Note that in the stationary state, maximum and minimum signal values repeat from switching period to switching period. Within a switching period, x has two different derivatives. It can be written as: where v 1 and v 2 are two periodic signals defined by: That is, f 1 and f 2 are the derivatives of x during an interval of size T 1 and T 2 , respectively. Note that v 1 and v 2 are not necessarily actual switching signals. For example, in the case of having: where u is a switching signal and k is a constant, Equation (18) can be written as: where v 1 = u and v 2 = 1 − u. Ifẋ = f (x), then by fundamental theorem of calculus: Since the interest is the ripple magnitude, it can be written as: Hence the ripple magnitude of x can be calculated using f 1 (x) or f 2 (x) using any of the following expressions: The absolute value in the expressions (22) is due to f 1 (x) or f 2 (x) being able to be positive or negative without affecting the ripple magnitude. If f 1 (x) or f 2 (x) are constant, then at least one of the expressions (22) is easy to calculate. If both f 1 (x) and f 2 (x) are not constant, it can be assumed that T 1 and T 2 are small enough to make f 1 ≈ f 1 (x) and f 2 ≈ f 2 (x). Thus, Equation (22) is simplified to: Usually, f 1 and f 2 depend on a circuit component. Hence, for a given T 1 or T 2 and a required maximum ripple, a minimum value of the circuit component can be obtained. In some cases, it is easier to use (23a), and in other cases, the use of (23b) is preferable. The analysis presented in this subsection can be extended to cases when a ripple signal experiment has more than two different derivatives within a switching period. The key observation is that in the stationary state, signals increase and decrease by the same amount within a switching period.

General Expressions for Calculating Components' Values
From (6a), L 1 can be calculated as follows. During the time where u 1 = 0, current x 1 is described by (2a). This time is (1 −ū 1 )/F s , where F s is the switching frequency. Considering the observations made in the previous subsection, this results in: From Equation (8a), ∆x 1 can be expressed in terms of the duty cycle of switch S 1 : If a maximum value for ∆x 1 is specified as ∆x 1 max , then from (25), it results that the minimum value for L 1 to achieve this maximum ripple is given by: Following the same reasoning, from (6b) and (8b), it can be obtained that: Along the same lines used to derive L 1 min and L 2 min , from Equation (6d), an expression for the output capacitor C o can be obtained: To calculate C p , consider the time when u 1 = 1. From Equation (6c), during this time,ẋ 3 = −x 1 /c p . This time is given by (1 −d 1 )/F s ; hence, using the same reasoning as before:

Design Example
Let us suppose that it is desired to create a DC bus of 186.6 V to feed a 500 W load from two 24 V voltage sources. To this aim, a converter with parameters shown in Table 1 can be designed.
Using Expressions (10) and (13c) and the data of Table 1, it is obtained that: Then, from the expressions (15), the following results: and from Expressions (8) and (15), the following results:

Parameter Value
186.6 V Output power (P o ) 500 W Sources power contribution (α 1 , α 2 ) 0.5, 0.5 Switching Frequency (F s ) 100 kHz Max current ripple on L 1 (∆x 1 ) 3.5% ofx 1 Max current ripple on L 2 (∆x 2 ) 3.5% ofx 2 Max voltage ripple on C p (∆x 3 ) 5% ofx 3 Max voltage ripple on C o (∆x 4 ) 1% ofx 4 Having states' stationary values, from Table 1, it can be obtained that: Now, Expressions (26)-(29) can be applied to obtain: To select the switching devices, it is necessary to find the maximum stress on these devices, as well as the RMS current that will be flowing through them. From the operation modes depicted in Figure 5, the stress on the devices can be determined. The maximum anode-cathode (Vak)of the diodes are: For the transistor, the maximum reverse voltages are given by: Finally, the RMS current values can be determined using:

Experimental Validation of the Design Procedure
To evaluate the converter's practical characteristics and make sure that the model and design procedure are adequate, a converter prototype was constructed. The design parameters were those used in Section 4.3. Based on the design results given by (34), the prototype component values were selected as: L 1 = 500 µH, L 2 = 500 µH, C p = 10 µF, C o = 10 µF Using the expressions (15) and Equations (26)- (29), it is possible to calculate the stationary values of signals in the ideal case (whiteout losses of any type). The ideal signals expected stationary values result in being: and the ideal expected ripples are: Power transistor SiC-MOSFET (ROHM-SCH2080KE) was selected. Such a transistor has a maximum drain to source voltage of 1200 V and a drain current of 40 A. The diodes selected (GeneSiC: GC20MPS12) have a repetitive peak reverse voltage of 1200 V and a forward current of 32 A. The test bench is shown in Figure 6. To evaluate the converter open-loop robustness, several combinations of input sources, buffer capacitors, and load values were tested.     Due to both input sources having the same voltage, the stationary states of both inductor currents were very similar. Figure 9 shows the voltage through the inductor L 1 and the current ripple. This last signal was indirectly obtained through the oscilloscope built-in mathematical function Vdt. In this way, the amplitude of inductor current could be better appreciated. It resulted in ∆x 1 ≈ 0.34 A, which is very close to the expected value given in (40). Note that there were some sparks in the inductor current. These were due to the effect of the parasitic inductances and capacitances of the PCB traces or the parasitic elements in the switching device near the switching node. Such sparks could be reduced by optimizing the PCB or using a snubber [52,53]. The corresponding signals for inductor L 2 were very similar. In Figure 10, the ripple voltage in capacitor C p is shown for two cases. In Figure 10a, the capacitor value is C p = 0.56 µF, and a ripple of 47.8 V is obtained. Figure 10b shows that the ripple obtained using a C p = 10 µF is 2.67 V. Note that the two cases are in concordance with Expression (29). According to this equation, the ripple corresponding to C p = 0.56 µF should be ∆V C p = 50.42 V, and for C p = 10 µF, the ripple should be ∆V C p = 2.94 V.  Stress voltages in switching devices are shown in Figures 11 and 12. It can be observed from Figure 11 that the peak voltage efforts by switching transistors is around 75 V for S 2 and 115 V for S 1 . From Figure 12, it can be observed that the maximum reverse voltages are 115 V for diode D 1 and 186 V for diode D 2 . It is important to analyze power losses in the converter devices based on the temperature they reach during operation. This analysis provides information for determining the heat sinks and cooling elements to be used and possible locations for installation. Figure 13 shows the temperature reached by transistors under two different cases. In the first case, both input sources had the same voltages, and in the second case, one input source was bigger than the other. In both cases, the power dissipated was 500 W, and a fan was used to keep the temperature at acceptable levels. Figure 14 shows the temperature reached by the diodes when one input source is bigger than the other and the power dissipated by the load is 500 W. Note that when a source was bigger than the other, a transistor-diode pair reached a higher temperature. Figure 13. Thermal image of the MOSFETs at different input voltages under an assisted operation at 500 W. Using the methodology for analyzing power loss on switching devices presented in [54], power loss on converter components was calculated. To this end, the drain source resistor RDS on of the transistors and the forward voltage of the diodes were employed. These parameters were taken form the manufacturers' datasheets. For passive components, power loss were determined by their equivalent series resistance (ESR) and their direct current measurement. Results are summarized in the charts of Figure 15 and Table 2. Figure 15a shows the power loss for each component when V in 1 = V in 2 = 24 V, and Figure 15b shows the power loss when V in 1 = 48 V and V in 2 = 24 V. In Table 2, both cases are summarized.     Table 3 shows the results for three of these combinations. Note that as was expected, in the case for both input voltages being 24 V and the load being 68 Ω, a higher current was required from the source in comparison with other cases. Table 3. Experimental results for several combinations of input sources and load values.

Indirect Control in Multi-Input Converters
It is known that the boost converter is a non-minimum phase system [55,56]. That is, if the output voltage is controlled directly, the system can become unstable. It is possible to linearize the average model and design a controller for the linearized model. However, this process yields a slow control. There is a control idea for the boost converter and boost related topologies. This idea is to control the output voltage indirectly by controlling the inductor current. Different forms of developing this idea have been regarded as double control loop [57], backstepping control [58], indirect control [59], sliding-mode control [60,61], predictive control [62,63], etc. Here, it is presented by decomposing the boost converter control problem into two parts. By doing so, it is natural to extrapolate the approach to the multi-input converter.
Consider the simplified diagram of the boost converter shown in Figure 16. The output voltage control of the boost converter can be decomposed into two problems. One problem is to control the RCcircuit of Figure 17b with i RC as the input. The other problem is to stabilize the switching circuit of Figure 17a. To address the problem of stabilizing the circuit of Figure 17a, observe that it can be modeled by: which can be stabilized by: substituting (42) in (41), the closed loop equilibrium point is: when i re f is set to: then by (43):ī where the barabove the integral term means the stationary value of the integral term. The circuit shown in Figure 17b can be modeled by: It can be observed from Figure 17a,b that i RC = ui L . Ifī L is given by (45), then i RC of the circuit shown in Figure 17b satisfies:ī That is, if the control (42) is applied to the circuit of Figure 17a, then i RC in Figure 17b is given by (47). If i RC is given by (47), then by substituting in (46), it can be shown that in the stationary state This analysis can be extended to the multi-input converter of Figure 2. Instead of having one circuit of the kind shown in Figure 17a, there are two; however, there is only one RC output circuit. Hence, each input circuit contributes a portion of the i RC current needed in the output circuit. Using the notation for the multi-input converter, indirect control becomes: where i 1 re f and i 2 re f is the portion contributed to i RC of input V in 1 and V in 2 , respectively. If this portion contribution is denoted as β 1 and β 2 , that is: thus i 1 re f and i 2 re f are given by: Proportions β 1 and β 2 can be expressed in terms of α 1 and α 2 by noting from (46) that: Hence, from (15), (51), and (52) result β 1 = α 1 M 1 and β 2 = α 2 M 2 . Summing up, the control expressions for multi-input converter of Figure 2 modeled by (6) are given by:ũ with: Transistors duty cycles can be obtained from (8) and (53). However, to make sure that the duty cycles are within the interval [0, 0.5], it is necessary to introduce a saturation function. To this end, the following expressions can be used:

Control Simulation Results
The switched model (6) controlled by (53)- (55) was simulated in Simulink. Converter parameters shown in Table 1 and component values given in Expression (38) were employed. For the controller parameters, k p = 0.01 and k i = 200 were employed. A simulation that evaluated the startup performance, robustness under source and load variations, and changes of β 1,2 was conducted.
Results are shown in Figure 18. These results were obtained as follows. At t = 0, the simulations started with zero initial conditions to evaluate the start-up performance. To evaluate robustness under sources variations, at t = 5 mS, the source V in 1 was changed from 24 V to 18 V. At t = 10 mS, both input sources were changed. Source V in 1 changed from 18 V to its original value of 24 V, and V in 2 changed from 24 V to 20. To simulate what happens when more energy is required from one source, at t = 15 mS, the current contribution of V in 1 was increased to 70% by setting β 1 = 0.7 and β 2 = 0.3. Finally, to evaluate robustness under load variations, at t = 20 mS, the load was changed from 70 Ω to 90 Ω. All these change were introduced suddenly to simulate the worst scenario. With the exception of the load, in a practical situation, all these changes occurred smoothly.
It can be observed from Figure 18 that every time a parameter changes, the output voltage is perturbed. However, the controller recovered within 3-4 mS of its nominal value by changing inductors currents and buffer capacitor voltage. Simulation results showed that thanks to the controller, the multi-input converter was able to use different sources; it could also extract different currents from each source independently of its voltage. Furthermore, the controller exhibited a fast response and was robust under load and input voltage variations.

Conclusions
A low cost high gain step-up multi-input converter topology previously proposed was analyzed in this paper. The authors of the topology proposed the converter to be used in photovoltaic applications. However, it was shown in this work that with a proper control, as presented in this work, the converter could have a broader range of applications. In particular, the topology could be used as a low cost alternative to use several renewable sources, possibly backed up by a non-renewable source, allowing the prioritization of power sources at any given time.
A nonlinear model of the multi-input converter was developed. From this model, a detailed design procedure was derived. A 500 W multi-input converter prototype was constructed. With this prototype, the model validity was confirmed. In addition, converter efficiency and reliability were assessed using the prototype. Based on these results, it could be assured that the converter could have increased power, and with more inputs, higher efficiency could be achieved. Simulation results showed that the proposed controller had a fast performance and was robust under load and input voltage variations.
The ideas employed in this work to obtain the nonlinear model, the design procedure, and the controller could also be used for other multi-input converters as well. Acknowledgments: The authors would like to thank ROHM semiconductor for providing the SiC MOSFET SCH2080KEC samples used in this prototype.

Conflicts of Interest:
The authors declare no conflict of interest.