Optimal control of a high gain DC- DC converter

Received Nov 12, 2020 Revised Jan 24, 2022 Accepted Jan 31, 2022 Different types of power conditioning systems are mostly used with renewable energy systems. As the output of solar photovoltaic (PV) is low high voltage converters are required. These converters can be used for both standalone and grid tied systems. A high voltage gain DC-DC converter is proposed which can be used with photovoltaic modules to get high output voltage. This converter can provide a high voltage gain and a proportional integral derivative (PID) controller is used to maintain the high output voltage of the proposed converter coupled with solar photovoltaic module. The PID controller is first designed using classical Ziegler Nichol’s method (ZN) and optimal control of the converter is obtained by using different optimisation methods like particle swarm optimisation (PSO), bacteria foraging algorithm (BFA) and its comparison is presented in this paper.


INTRODUCTION
As there is fast depletion of fossil fuels owing to its higher cost, renewable energy systems have to be deployed to meet the power demand. But these renewable energy sources have to be integrated to the power system network. For integration of these renewable energy systems power conditioning systems like DC-DC converters or inverters are used. Renewable energy systems can be standalone type or grid tied type. The high voltage, high power converters are required as the efficiency and the output of the solar photovoltaic (PV) systems is low, hence high gain converters with higher power and efficiency are required. There are different topologies of high gain DC-DC converters, both single stage and multi-stage are found in literature. All these converters have both the advantages and limitations.
Mostly the high gain DC boost are used which are either non-isolated type or isolated type with transformer. The topologies with isolation transformers suffer from lower efficiency and higher cost, weight and volume in comparison to non-isolated type [1]. A switched capacitor topology of boost converter is used to get high output voltage gain and efficiency, component count is high and has higher value of input ripple current [2]. Other hybrid topologies found in literature with combination of switched capacitor and inductor have low voltage stress on diodes and switches, also has low ripple current. But the main drawback is that high output voltage cannot be gained [3], [4]. Other topologies found in literature have transformers for isolation and high voltage stress on switches. The major drawback of high frequency transformer is the loss due to parasitic capacitance. Also, the leakage inductance of the transformer leads to increase in voltage spikes ultimately increase in operating volage, due to the switching operation of the converter. Undesired volage regulation occurs due to drop across the leakage reactance with change in load. As for high voltage applications, magnetic materials size cannot be reduced as I will require high insulation [5]− [19]. So, a new non-isolated or transformer, high gain DC-DC converter is proposed as a solution.

CLOSED LOOP CONTROL OF THE PROPOSED CONVERTER
As solar PV system is an intermittent source, the output changes with the pv a controller is required to maintain a constant output. The proposed converter is coupled with PV module to give high output voltage. For closed loop control of a system proportional integral (PI) proportional derivative (PD), proportional integral derivative (PID), proportional derivative (PDD). Are used and to reduce the steady state error. PID controllers are mostly used to control non-linear systems. Three basic modes like proportional, integral and derivative mode are used in this controller. For higher order system the proportional controller cannot be used as basically they are used o stabilize first order systems. Forced oscillations cannot be controlled P and on-off controller as compared to PI controller. But the response of PI controller becomes sluggish when integral mode is included. PI controllers can be used for processes where faster response is not required. Prediction of error is not possible in his controller. But by including a derivative mode future prediction of error can be obtained and it also provides a faster response. Hence PID controller can be used in non-linear systems like solar PV and it gives a faster response to any change in the input (D). There is also reduction in error to almost zero (I). So PID controller has all the necessary characteristics required, hence used in his work for closed loop control of the converter. The output of the controller is given as: : Input error signal U(t) : Output signal

Design of PID controller using Ziegler Nichol's method
State averaging technique: In order to obtain the steady state and dynamic responses of the complete model of the converter. This technique is an important tool to design a power converter. By using his technique a relationship between the sources, outputs and control parameters can be obtained. Large scale changes in voltages and currents as the source, load or control inputs can be tracked by the averaged model and also complicated models can be easily obtained. In this method ̃ and ̃ are small perturbations in the output voltage and the switch duty ratio d; a small signal transfer function is given as ̃/̃ . At first for the switching states the sae variables are obtained. The inductor current and capacitor voltage are described as state variable vector x. The small ac perturbation added to the DC value given as = +̃. The state equations during switch ON and OFF time are given as: where A1 and A2 are state matrices, B1 and B2 are vectors, is the input voltage. Also during the switching states the output voltage can be described in terms of state variables is given as: where C1 and C2 are transposed vectors. The averaged model is given by the following equations over total switching period as: Finally the transfer function Tp(s) of the converter can be obtained in this form, By using the state space averaging (SSA) technique the averaged matrices are obtained as: The control transfer function is defined as the ratio of output voltage to duty ratio and it is obtained as (8).

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After state space averaging technique and state space analysis of the model and using MATLAB program/.m file, the root locus plot of the converter is obtained as shown in Figure 7 and from the root locus plot critical gain Kc and critical time period Tc was obtained [23]. To tune the parameters of the controller Ziegler Nichols method is used. The Ziegler Nichols tuning method is a heuristic method of tuning a PID controller [2]. To meet the performance requirements the controller parameters are tuned with the rules suggested by Ziegler and Nichols as shown in Table 1 [24].  The critical gain was found to be Kc=8.18x10 7 . Here in the proposed converter, we will use PID controller. So, the parameters are: In this work PID controller is used to keep the output voltage constant, PID controller is used in this work. The simulation diagram of the proposed converter with PID controller is shown in Figure 8 whose parameters are tuned using Ziegler Nichols method. The output voltage of the high gain converter is 300V for input voltage of 30V without controller is shown in Figure 9. Now with the use of PID controller; a constant voltage of 300V is to be maintained for different input voltages. The simulation result is shown in Figure 10. The Figure 10 shows output voltage of the converter is 299.2 V for input voltage of 30 V with PID controller. When the proposed converter is coupled with the PV module as shown in Figure 11, the output voltage of the converter changes as the PV output voltage change with the irradiance, hence PID controller is used to keep the output voltage constant at 300V. The simulation results are shown in Figures 12 and 13. Table 2 shows the output voltage of the converter with changing irradiance with PID controller designed by Ziegler Nichol's method.  In different areas of application artificial intelligence has emerged as an important tool to get a global optimum solution. Different heuristic algorithms have been developed and found in literature. Particle swarm optimization (PSO) is developed by Kennedy and Eberhart by considering the behavior of swarms of bees, birds. All the limitations of genetic algorithm (GA) are overcome by this algorithm and can be applied to applications here GA can be used [25].
The main characteristic of this algorithm is robustness in solving non-linear, non-differentiability and high dimensional problems. A very less time is required to get the optimal solution of fitness function. In this algorithm the movement of particles in search space is used o reach the optimal solution. By keeping the track of particle co-ordinates the best solution Pbest is obtained. The overall best solution gbest and its location obtained of the population. The velocity a each step changes to reach Pbest and gbest [26]- [29] as shown in Figure 14. The particle is represented by Xi = (xi1, xi2,……, xiD) in the d-dimension search space. The velocity of particle i is represented as Vi = (vi1, vi2, vi3, vi4,………………………………………….,viD). Acceleration is weighted by a random term, with separate random numbers being generated for acceleration toward Pbest and gbest. The position of the i th particle is then updated.

vid (t + 1) = ω × vid (t) + c1r1(Pid -xid (t)) + c2r2 (Pgd -xid(t))
xid (t + 1) = xid (t) + cvid(t + 1) Where, Pid and Pgd are Pbest and gbest. It is concluded that gbest version performs best in terms of median number of iterations to converge [25]. In Figure 14 the particles xj,g are the particles in search space with velocity vj,g moving with momentum updated to xj,g(t+1) and vj,g(t+1) respectively to reach the local best pbest & global best gbest solution. Figure 14. Description of velocity and position updates in particle swarm [25] In complex multimodal search spaces particle swarm optimization (PSO) technique used to find the optimal solution of non-differentiable cost functions. To optimise the parameters of the PID controller to get the constant output voltage from the proposed converter coupled with the intermittent source PV, PSO is used as ouput changes with irradiance. The objective function is expressed as (12).  ( 1) , In this work PSO program has been written in the MATLAB 7.18.0.m file. The simulation model of PV module coupled with the DC-DC converter is developed in SIMULINK environment. This simulation runs for every particle in the search space to compute the objective function. The simulation results are shown in Figures 15 and 16. Figures 15 and 16 shows the output voltage waveforms of the converter, where the PID controller is optimized using PSO so that the output voltage remains constant for different values of input voltage due to different irradiance values. The results are shown in Table 3 for output voltage of the converter with changing irradiance with PID controller tuned by PSO.  − Bacterial foraging algorithm (BFA) This is another optimization algorithm which uses he foraging technique of E. coli bacteria. In this algorithm four components of bacteria foraging is used, they are chemo taxis, swarming, reproduction and elimination-dispersal. Using this energy intake per unit time of E.coli bacteria is increased. − Chemo-taxis In this process two movements swimming and tumbling of E. coli is simulated, where i alternate between these movements. Suppose θi(j,k,l) represents ith bacterium at jth chemo tactic, kth reproductive and lth elimination-dispersal step. C(i) is the size of the step taken in the random direction specified by the tumble (run length unit). Hence in this process of chemo taxis the movement of the bacterium may be represented by (3).
Where Jcc(θ, P(j, k, l) is the objective function value to be added to the actual objective function (to be minimized) to present a time varying objective function, S is the total number of bacteria, p is the number of variables to be optimized, which are present in each bacterium and θ=[θ1,θ2,…,θp is a point in the p dimensional search domain.

− Reproduction
In this method a healthy E. coli bacterium asexually splits into two bacteria at the location where the unhealthy bacteria die to maintain the swarm size.

− Elimination and dispersal
In this process a group of bacteria can be killed or dispersed to a new location due to change in environment.

− Size of population 'S'
It takes more computation time if population size is increased but optimal solution can be obtained as he concentration of will be more a particular location. Length of chemo tactic step 'C(i)' has to be selected carefully, if a large value is selected then local minima can be missed out else if low value then it will take more time to converge. C(i) is like a "step size" for the algorithm. − Chemo tactic step 'Nc' If the size of Nc and number of reproduction steps 'Nre' is too small, the algorithm may converge earlier. Again, the elimination and dispersal number 'Ned' should not have a low value then random elimination-dispersal events will not be accounted for finding better nutrients. The formation of swarms depending on the availability of the nutrients affects the cell-to-cell attractant functions 'Jcc'. And the selection of proper value of the elimination dispersal probability ped leads to attainment of global optimum solution [30]− [31]. The BFA program has been written in the MATLAB 7.18.m file. The PV system model coupled with the DC-DC converter under study is developed in SIMULINK environment. The simulation is done for every particle in the search space for computing the objective function as shown in Figures 17 and  18. Figures 17 and 18 shows the output voltage waveforms of the converter, where the PID controller is tuned using BFA so that the output voltage remains constant for different values of input voltage due to different irradiance values. The results are shown in Table 4 for output voltage of the converter with changing irradiance with PID controller tuned by BFA.

CONCLUSION
In this work a DC-DC converter is coupled with the PV system to obtain the high output voltage. The designed high gain DC-DC converter is connected to the PV module, which gives high output voltage but it changes with the change in irradiance. In order to keep the output voltage constant at 300 V for different input voltages with changing irradiance a PID controller is used. At first Ziegler Nichols method is used to tune its parameters. Then the controller parameters are optimized PSO, the results shows the steady state is reached earlier than the parameters obtained by Ziegler Nichol's method. Also, the output voltage remains constant at 299.9 V for very low input voltage to the converter. The parameters of the PID controller are also optimized using BFA and the output voltage of the converter remains constant at 299.8 V for different input voltage from solar PV, also the peak overshoot is also reduced compared to PSO. A comparative analysis of these techniques, results and analysis are presented. From this work it can be concluded that this converter can be used with PV systems and a controller can be designed using the optimization techniques which will lead to an optimal control of the output as per the requirements.