Energy Conversion Optimization Method in Nano-Grids Using Variable Supply Voltage Adjustment Strategy Based on a Novel Inverse Maximum Power Point Tracking Technique (iMPPT)

Abstract

This paper introduces a novel power supply voltage adjustment strategy that can determine the optimum voltage value based on the amount of absorbed power. The novel automatic voltage adjustment technique was called inverse maximum power point tracking (iMPPT). The proposed control strategy consists of a modified maximum power point tracking (MPPT) algorithm (more precisely the P&O method). In this case, the modified MPPT technique establishes the minimum value of the input absorbed power of a consumer load served by a switched-mode power supply (SMPS). The iMPPT adjusts the input power by modifying the input voltage of the main power supply. The served loads are connected to the variable power supply via an interfacing power electronics converter that performs the automatic voltage regulation function (AVR). The optimal value of the input voltage level can be achieved when the input power of the automatic voltage regulation converter is at a minimum. In that case, the energy conversion efficiency ratio is at a maximum, and the overall losses related to the front-end power stage are at a minimum. The proposed technique can also be considered a Maximum Efficiency Tracking (MET) method. By performing the inverse operation of a maximum power point tracking algorithm on the input demanded power of a switched mode power supply (SMPS), the optimum input voltage level can be determined when the maximum energy conversion ratio (related to a given load level) is achieved. The novel proposed iMPPT method can improve the energy conversion ratio from 85% up to approximately 10% in the case of an output power level of 800 W served by a synchronous buck converter at the input voltage level of 350 V. The total amount of recovered power in this situation can be approximately 100 W.

1. Introduction

In the context of modern society, when green and renewable energy is preferred in residential applications, the discussion of powering electrical appliances and devices with direct current (DC) is more and more prevalent in technical literature [1,2,3,4]. Energy storage and conversion efficiency aspects are the main arguments that make direct current nano-grids feasible in residential applications [3]. The nano-grid can be considered the total amount of electrical wiring and equipment in a home or residential place. The micro-grid can connect multiple residential sites or homes in a hood.

The served loads (energy consumers) within the residential nano-grid are connected to the main high-voltage nano-grid bus via a DC-DC power electronic converter that acts as a switched-mode power supply. Automatic voltage level regulation is the main function of the interfacing DC-DC power electronic converter [3].

This paper proposes a new power supply method for residential nano-grids in which the main bus voltage level can be variable if the final loads are connected to the bus via a switched mode power supply (or power electronics converter) that performs the automatic voltage level regulation electronic function [5]. There are also different types of converters that serve as renewable energy production and storage means, but they also perform the automatic voltage level regulation function. The public AC grid can also be connected to the variable voltage DC bus of the nano-grid via a fully bi-directional PWM controlled inverter—rectifier converter (transistor-based controlled rectifier) [6]. A variable voltage adjustment decision unit is also part of the nano-grid. This unit can be a real-time embedded computing system that measures the power on the main bus and determines the optimum voltage level related to the demanded power level and the available power from the renewables and the public AC grid [7,8,9]. The specialized embedded computing device has digital I/Os and analog I/Os that can control the master power electronics converter that drives the main grid bus. The whole architecture of the variable voltage nano-grid can be observed in the following figure (Figure 1) [7,8].

The following elements are part of the depicted architecture (Figure 1 and Figure 2):

0

Variable voltage nano-grid architecture

1

Variable DC bus voltage

2

Photovoltaic panels (PV) array

3

Boost converter for the PV array

4

Micro wind turbine

5

Micro wind turbine boost converter

6

Battery bank

7

Battery bank bi-directional converter

8

AC public grid

9

AC public grid bi-directional power stage

10

AC-DC PWM controlled inverter/rectifier

11

DC-DC bi-directional converter

12

DC-Bus voltage reference

13

AC loads

14

AC loads buck/step-down converters

15

DC loads

16

DC loads buck/step-down converters

17

Logic control unit/specialized embedded computing system

17a

Power measurement module

17b

Power value logic decisional module

17c

Inverse maximum power point tracking (iMPPT) algorithm that minimizes “P_B”

17d

Inverse maximum power point tracking (iMPPT) algorithm that maximizes “P_B”

18

Centralized loads measurement module

19

Centralized unidirectional power sources measurement module

20

Centralized bidirectional power sources measurement module

21

PV array power measurement module

22

Wind turbine power measurement module

23

Battery bank power measurements module

24

AC public grid power measurement module

25

DC loads power measurement module

26

AC loads power measurement module

The proposed architecture of the variable voltage nano-grid [7,11] and the voltage adjustment strategy are parts of the pending patent called “Micro-grid of variable DC voltage and its control method” [7,8]. This is a Romanian patent pending application at OSIM (the National State Office of Trademarks and Inventions) applied on 19 December 2019 called “Micro-grid of variable DC voltage and its control method”. The patent application was also published in the “Official Bulletin of Industrial Property” in the section “Patents” [8,11].

The inverse maximum power point tracking (iMPPT) strategy [8,10] is a modified “Perturb and Observe” (P&O) algorithm that minimizes the input power of the interfacing power electronics converter by adjusting the main power supply or bus voltage level. An optimum bus voltage level will be determined by the iMPPT algorithm and is related to the amount of power demanded by the final served loads. The same algorithm can also act as a classic perturb and observe MPPT method when the nano-grid works in bi-directional mode, and the excessive power is injected into the public AC grid or battery [7,8,11].

The main advantage of using iMPPT consists of the fact that the overall losses related to the energy conversion process can be minimized by selecting the optimum voltage level related to the power level demanded by the attached loads [12,13].

2. Methodology and Experimental Setup for Proving the Main Concept

The main concept of this paper consists of the fact that the input power, absorbed by an electric load supplied via a switched-mode power supply, can be reduced if an optimum value of the input voltage is set. The maximum conversion efficiency possible, related to the output power, can be reached at the optimum input voltage level. Finally, an automatic voltage adjustment strategy can be implemented to improve the conversion efficiency of the considered power stage. The iMPPT method in this case can be considered a maximum efficiency tracking (MET) method [9,10,11]. The only requirement for the variable DC bus interfacing converter is that it can operate in a wide range of input voltage levels. A synchronous buck converter will be used (Figure 3) [14,15,16,17].

The synchronous (step-down) buck interfacing converter will act as a switched-mode power supply between the variable voltage DC bus and the final served load. The output voltage “V_out” of the converter will be kept constant by a “closed-loop” control strategy based on a proportional—integral (PI) regulator [18,19]. The PI controller performs the automatic voltage level regulation function on the output voltage level “V_out” [20,21,22]. The output voltage will remain constant regardless of the variation in the output load resistance “R_L” or input voltage “V_in” [18,19,20,21,22].

The PI controller will automatically adjust the duty cycle “d” so that the output voltage level measured by the feedback sensor “V_m” will match the reference voltage “V_ref” (Figure 4) [18,19,23,24].

According to the “open-loop” working mode equation of the considered power electronics converter (Equation (1)), the input voltage “V_in” and the output current “I_out” will act as perturbation parameters in the “closed-loop” system [8,18,19,24].

$${\mathrm{V}}_{\mathrm{out}}={\mathrm{V}}_{\mathrm{in}}·\mathrm{d} \to {\mathrm{I}}_{\mathrm{L}}·{\mathrm{R}}_{\mathrm{L}}={\mathrm{V}}_{\mathrm{in}}·\mathrm{d} \to { \mathrm{I}}_{\mathrm{L}}=\frac{{\mathrm{V}}_{\mathrm{in}}·\mathrm{d} }{{\mathrm{R}}_{\mathrm{s}}}=\frac{{\mathrm{V}}_{\mathrm{ies}}}{{\mathrm{R}}_{\mathrm{L}}}$$

(1)

By modifying the input voltage “V_in” or the final load resistance “R_L”, which directly affects the output current “I_out”, the PI controller will need to determine dynamically a new value for the duty cycle “d”, directly adjusting the output voltage “V_out” so that it will match the reference voltage “V_ref” even for new conditions imposed by perturbations (Equation (2)). The overall system error will be convergent to zero to match the closed-loop system stability conditions (Figure 5) [8,18,19,24].

$$\begin{array}{r}\mathrm{e}\left(\mathrm{t}\right)={ \mathrm{V}}_{\mathrm{ref}}-{ \mathrm{V}}_{\mathrm{m}}{ \mathrm{where} \mathrm{V}}_{\mathrm{m}}\approx { \mathrm{V}}_{\mathrm{out}}\\ \mathrm{e}\left(\mathrm{t}\right)={ \mathrm{V}}_{\mathrm{ref}}-{ \mathrm{V}}_{\mathrm{out}}\end{array}$$

(2)

Considering the duty cycle “d” as being time dependent “d(t)”, the overall system error “e(t)” can be determined from the following equation [8,18,19,24]:

$$\underset{\mathrm{t} \to \infty }{\lim }\mathrm{e}\left(\mathrm{t}\right)=0 \to \mathrm{e}\left(\mathrm{t}\right)={ \mathrm{V}}_{\mathrm{ref}}-[{\mathrm{V}}_{\mathrm{in}} · \mathrm{d}\left(\mathrm{t}\right)]=0$$

(3)

The system stability condition will be

$$\underset{\mathrm{t} \to \infty }{\lim }\mathrm{e}\left(\mathrm{t}\right)=0 \to { \mathrm{V}}_{\mathrm{ref}}=\left[{\mathrm{V}}_{\mathrm{in}} · \mathrm{d}\left(\mathrm{t}\right)\right] \to { \mathrm{V}}_{\mathrm{ref}}={ \mathrm{V}}_{\mathrm{out}}={\mathrm{V}}_{\mathrm{m}}$$

(4)

The PI controller adjusts the duty cycle by the following equation [8,18,19,24]:

$$\mathrm{d}\left(\mathrm{t}\right)={\mathrm{K}}_{\mathrm{p}}·\mathrm{e}\left(\mathrm{t}\right)+{\mathrm{K}}_{\mathrm{i}}·{{\displaystyle \int }}_0^{\mathrm{t}}\mathrm{e}\left(\mathrm{t}\right)·\partial \mathrm{t}$$

(5)

Considering the buck converter as the “plant” in the system, the expression becomes

$$\mathrm{d}\left(\mathrm{t}\right)={\mathrm{K}}_{\mathrm{p}}·\left({\mathrm{V}}_{\mathrm{ref}}-{\mathrm{V}}_{\mathrm{ies}}\right)+{\mathrm{K}}_{\mathrm{i}}·\underset{0}{\overset{\mathrm{t}}{{\displaystyle \int }}}\left({\mathrm{V}}_{\mathrm{ref}}-{\mathrm{V}}_{\mathrm{ies}}\right)·\partial \mathrm{t}$$

(6)

For the steady-state condition, when “e(t) = 0”, the expression becomes [8,18,19,24]

$$\mathrm{d}\left(\mathrm{t}\right)={\mathrm{K}}_{\mathrm{i}}·\underset{0}{\overset{\mathrm{t}}{{\displaystyle \int }}}0·\partial \mathrm{t}=\mathrm{const}.$$

(7)

As a short conclusion over the presented control theory, the PI controller automatically adjusts the duty cycle “d(t)” over time in such a way that the overall system error converges to zero. It means that the output voltage “V_out” matches the initial value of the reference voltage “V_ref”. The average effect that can be observed at the input of the converter is that it dynamically adjusts its “internal impedance” in such a way that the output voltage “V_out” remains constant regardless of the external system perturbations [18].

If the output load impedance remains constant and the output voltage remains constant, the system works in “constant power” (CP) mode [25,26]. The working principle of a power electronics converter in CP mode can be equivalent to a simple circuit in which the series resistance “r” controls the amount of current absorbed by the load and the final load resistance “R_L” controls the voltage drop at the output of the circuit. Both resistances are variable, and those are adjusted in such a way that the power consumption from the main power supply “E” remains constant even if the main power supply voltage changes (Figure 6). An ideal average model of the converter can be equivalent to the series resistance “r” which represents the converter working in automatic voltage regulation mode, and the final load “R_L” represents the real served load which absorbs the load current “I_L” [25,26].

Assuming the fact that the power supply voltage level (E₁ and E₂) and the final load resistance (R_L1 and R_L2) can be modified, while the series resistance “r” remains constant, the following equations will describe the constant power (CP) mode [8,25,26]:

$$\Delta {\mathrm{P}}_{\mathrm{L}}=({\mathrm{V}}_{\mathrm{L}2}·{\mathrm{I}}_{\mathrm{L}2})-({\mathrm{V}}_{\mathrm{L}1}·{\mathrm{I}}_{\mathrm{L}1})=0\to { \mathrm{P}}_{\mathrm{L}}=\mathrm{f}\left(\mathrm{V},\mathrm{I}\right)=\mathrm{const}.$$

(8)

$$\Delta {\mathrm{P}}_{\mathrm{L}}=\frac{{\mathrm{E}}_2{}^2}{{\mathrm{R}}_{\mathrm{L}2}}-\frac{{\mathrm{E}}_1{}^2}{{\mathrm{R}}_{\mathrm{L}1}}=0$$

(9)

$$\Delta {\mathrm{P}}_{\mathrm{L}}={\mathrm{P}}_{\mathrm{L}2}-{\mathrm{P}}_{\mathrm{L}1}=\frac{{\left[{\mathrm{E}}_2-\left({\mathrm{I}}_{\mathrm{L}2}·\mathrm{r}\right)\right]}^2}{\left({\mathrm{R}}_{\mathrm{L}2}+\mathrm{r}\right)}-\frac{{\left[{\mathrm{E}}_1-\left({\mathrm{I}}_{\mathrm{L}1}·\mathrm{r}\right)\right]}^2}{\left({\mathrm{R}}_{\mathrm{L}1}+\mathrm{r}\right)}=0$$

(10)

If the amount of absorbed power changes over time, the finite difference between the final and initial value of the load power “ΔP_L” shall be different than zero. If the power remains constant, the finite difference is zero. Based on this assumption, the equations (Equations (8)–(10)) were expressed. This means that the electrical power can be expressed in multiple ways, considering or not the series “r” resistance. Another important fact to mention is that electrical power can be expressed as a product of two variables: voltage and current. By representing the electrical power in an “I-V” axis system, a rectangular surface can be obtained. The side dimensions of the rectangular surface are determined by the voltage and current values. The area of the rectangular surface represents electrical power. The same surface area can be represented in two ways, for example, the long side that represents the voltage shall be greater than the side that represents the current and vice versa. Practically, electrical power can be represented as a high value of current and a low value of voltage or a high value of voltage and a low value of current; thus, in both cases, the same amount of electrical power can be obtained, and the same surface area can be plotted in the “I-V” plane in both cases (Figure 7) [17,20].

The input voltage level in this working mode (CP) can be modified without affecting the output power supplied by the converter to the final load (Figure 8).

The difference “ΔP_C” between the input power “P_in” and the output power “P_out” represents the total amount of power losses related to the conversion process (Equation (11)).

$$\Delta {\mathrm{P}}_{\mathrm{C}}={\mathrm{P}}_{\mathrm{in}}-{\mathrm{P}}_{\mathrm{out}}\to {\mathrm{P}}_{\mathrm{out}}={\mathrm{P}}_{\mathrm{in}}-\Delta {\mathrm{P}}_{\mathrm{C}}$$

(11)

The overall losses include reactive losses, switching and conduction losses and parasitic losses (ex. parasitic resistance, inductance, and capacity) (Figure 9) [8,27,28,29].

The ratio of the output power “P_out” and the input power “P_in” represents the conversion efficiency, and it can be described as a non-dimensional percentage (Equation (12)). The efficiency ratio represents a two-variable function, which can be plotted as a surface in a tridimensional axis system or a two-dimensional family of characteristic curves [8,27,28,29].

$$\begin{array}{c}\mathsf{\eta }=\frac{{\mathrm{P}}_{\mathrm{out}}}{{\mathrm{P}}_{\mathrm{in}}}\to \langle \mathsf{\eta }⟩=\left[\%\right]\\ \mathsf{\eta }=\frac{{\mathrm{P}}_{\mathrm{out}}}{{\mathrm{V}}_{\mathrm{in}}·{\mathrm{I}}_{\mathrm{in}}}\to \mathsf{\eta }=\mathrm{f}\left({\mathrm{V}}_{\mathrm{in}},{ \mathrm{P}}_{\mathrm{out}}\right)\end{array}$$

(12)

By changing the input voltage in a constant power (CP) working mode of a switched mode power supply, the total amount of power loss can be reduced significantly. The input power will decrease as the total power loss related to the conversion process decreases. A minimum point on the “losses vs input voltage” plot will be represented. That point corresponds to the maximum efficiency and minimum input power. The input voltage related to the minimum input power represents the optimum voltage corresponding to the actual output load power (Equation (13)) (Figure 10 and Figure 11) [8,27,28,29].

$$\begin{array}{c}\underset{\mathsf{\eta } \to 1}{\lim }{\mathrm{P}}_{\mathrm{in}}\left(\mathsf{\eta }\right)=\frac{{\mathrm{R}}_{\mathrm{out}}·{\mathrm{I}}_{\mathrm{out}}{}^2}{\mathsf{\eta }}=\frac{{\mathrm{V}}_{\mathrm{out}}·{\mathrm{I}}_{\mathrm{out}}}{\mathsf{\eta }}=\frac{{\mathrm{P}}_{\mathrm{in}}-\Delta {\mathrm{P}}_{\mathrm{C}}}{\mathsf{\eta }}\\ \to {\mathrm{P}}_{\mathrm{in}}={\mathrm{P}}_{\mathrm{out}}\to \Delta {\mathrm{P}}_{\mathrm{C}}\approx 0\end{array}$$

(13)

To prove the main concept described within this paper, an experimental setup has been implemented using specialized laboratory equipment (Figure 12) such as:

-

programmable DC voltage power supply Itech IT6534C (light blue mark)

-

AXIO MET AX-582B multimeter for measuring input current (blue mark)

-

AXIO MET AX-582B multimeter for measuring input voltage (pink mark)

-

power resistor with a value of 1 Ω for simulating the cable length (gray mark)

-

National Instruments MyRIO 1900—embedded development kit (orange mark)

-

synchronous buck step-down converter (dark yellow mark)

-

AXIO MET AX-582B multimeter for measuring output current (dark blue mark)

-

AXIO MET AX-582B multimeter for measuring output voltage (dark brown mark)

-

adjustable sources for powering annex modules (light green mark)

-

IT8616 programmable electronic load (light yellow mark)

The experimental setup block diagram is shown in Figure 13.

According to the experimental setup block diagram, the input of the synchronous buck converter was connected to a high-voltage DC programmable power supply (Itech IT6534C). Two AXIO MET AX-582B multimeters were used on the input side of the converter to measure the input voltage and the input current. On the input side, a one-ohm high-power resistor was also connected in series with the converter to simulate the cable’s length loss in a residential nano-grid. On the output side of the converter, a high-power programmable electronic load was connected. Also, on the output side of the converter, two AXIO MET AX-582B multimeters are used to measure the output current and voltage of the power electronics converter. The PWM digital gate signal, for the converter’s main switching transistors was provided by the National Instruments MyRIO 1900 development kit. MyRIO acts as a digital signal processor (DSP) because it also measures the output voltage of the converter via a voltage measuring module based on Avago ACPL 7900 isolated amplifier (Figure 14), and it runs the automatic voltage regulation control strategy, developed in National Instruments LabVIEW [8].

Based on the voltage feedback, MyRIO generated the desired duty cycle for the PWM gate signal of the synchronous buck converter [30] (Figure 15). A proportional—integral controller-based strategy was implemented in My Rio’s memory, using the NI LabVIEW graphical programming environment (Figure 16) [8].

After implementation of the experimental setup, the following test steps will be performed to collect numerical data-points for further study on the variable voltage supplying diverse DC consumer loads [8]:

-

At the input of the converter, the initial input voltage value will be set at 100 V and data-points will be collected in a summary table; after this step the input voltage value will be increased by another 25 V and data-points will be collected again in the summary table until the final value of 350 V is reached.

-

The output voltage will be kept constant at 72 V using the NI MyRIO embedded controller, with the control law implemented. The control loop runs at a 10 ms time step within LabVIEW and MyRIO.

-

The PID controller coefficients were set empirically based on the concept of minimizing the closed loop system’s error in the shortest possible time interval (a Ziegler-Nichols PID tuning method approach based on Equations (3), (4) and (6)). The determined controller coefficients are kP = 0.001 and kI = 0.008, kD = 0 and the saturation or limit range for the PWM signal and the duty cycle is [0, 0.95]. The controller is a classical PID model implemented in LabVIEW based on the classical mathematical approach of a simple (non-parallel) PID controller [18,19].

-

Using the programmable electronic load, constant power values will be set at the output of the power electronics converter.

-

At each new value of the output power, a complete input voltage sweep cycle will be performed, and the data-point will be recorded in a summary table.

-

Using the input and output values of voltage and current, the input and output power, the conversion efficiency, the overall converter losses and the input impedance values will be determined.

-

Based on all data-points and the determined parameters, functional characteristic curves of the power electronics converter will be plotted and analyzed in different situations.

After examining the real data-points collected from the physical experimental setup, a Hardware-In-the-Loop (HIL) (Figure 17) simulation of the automated variable supply voltage adjustment strategy setup will be implemented using the following equipment:

-

Two PCs running NI 2021 SP1 Software Bundle including VeriStand (same version).

-

MyRIO-1900 embedded controller.

-

NI PXIe-1071 Hardware In the Loop Real-Time simulation computer.

-

MathWorks Matlab-Simulink R2018b + NI VeriStand model framework.

The NI MyRIO-1900 embedded controller will run the iMPPT variable voltage adjustment strategy. The control law will be implemented within NI MyRIO using the Matlab-Simulink and NI VeriStand model framework (Figure 18). NI VeriStand will be installed on both computers, one for MyRIO and one for PXI [8].

The PXI will run a nano-grid model with three DC loads supplied via a synchronous step-down buck converter. In this simulation (HIL simulation) the “converter + load” group will be emulated as a variable resistor that changes its impedance according to the input voltage level based on a look-up table. The values within the look-up table were determined based on the real input current and voltage values (Figure 19 and Figure 20) [8].

The HIL simulation models were initially tested in offline simulations within the Simulink. The nano-grid model was implemented using Simulink SimScape. Three different power values were set for the DC consumer loads [8].

The final VeriStand results shall match the offline simulation model results.

The iMPPT-based automatic voltage adjustment technique sweeps the input voltage from the initial value of 100 V and stores the input voltage and input power values in one bi-dimensional array. A minimum input power value will be identified and its corresponding optimum input voltage value. The corresponding optimum input voltage value will be set by the special technique “incremental and settle” [31] to allow steady-state operation and data acquisition in the computing system. The workflow of the iMPPT-based automatic voltage adjustment technique can be observed in (Figure 21), [8].

In the next step the main results will be presented.

3. Results

To analyze the behavior of the electronic converter powered with variable voltage [31], the first step is to plot the functional characteristic curves of the studied converter.

Three nominal output power ratings were selected from the three voltage ranges (low, medium and high): P_{out_1} = 216 W, P_{out_2} = 288 W and P_{out_3} = 576 W. For these three cases, the main experimental results will be presented.

In the first case (

Table 1. Experimental results for case 1—Output power P_out = 216 W—constant.

Vin [V]	Iin [A]	Pin [W]	Vout [V]	Iout [A]	Pout [W]	η [%]	ΔPC [W]	Zin [Ω]
100	2.253	225.300	72.000	3.000	216.000	95.872	9.300	44.385
125	1.793	224.125	72.000	3.000	216.000	96.375	8.125	69.716
150	1.489	223.350	72.000	3.000	216.000	96.709	7.350	100.739
175	1.274	222.950	72.000	3.000	216.000	96.883	6.950	137.363
200	1.118	223.600	72.000	3.000	216.000	96.601	7.600	178.891
225	0.996	224.100	72.000	3.000	216.000	96.386	8.100	225.904
250	0.896	224.000	72.000	3.000	216.000	96.429	8.000	279.018
275	0.815	224.125	72.000	3.000	216.000	96.375	8.125	337.423
300	0.747	224.100	72.000	3.000	216.000	96.386	8.100	401.606
325	0.691	224.575	72.000	3.000	216.000	96.182	8.575	470.333
350	0.642	224.700	72.000	3.000	216.000	96.128	8.700	545.171

), it was found that the maximum conversion efficiency ratio η_MAX = 96.883% was obtained at the level of input voltage of V_in = 175 V, (Figure 22). At the same voltage level, both the minimum input power P_{in_min} = 222.95 W and the minimum overall loss at the converter level ΔP_{c_min} = 6.95 W were recorded (Figure 23 and Figure 24). For the input voltage level of V_in = 100 V, the minimum value of the efficiency was recorded η_min = 95.872% [31].

For the same voltage level, both the maximum input power P_{in_MAX} = 225.3 W and the maximum value of losses recorded, namely ΔP_{c_MAX} = 9.3 W, were obtained. Both the recovered value of conversion efficiency and overall converter loss levels can be determined using the following expressions:

$${\Delta \mathsf{\eta }}_{\mathrm{REC}}={\mathsf{\eta }}_{\mathrm{MAX}}-{\mathsf{\eta }}_{\min }=96.883-95.872=1.011\%$$

(14)

$${\Delta \mathrm{P}}_{\mathrm{c}\_\mathrm{REC}}={\Delta \mathrm{P}}_{\mathrm{c}\_\mathrm{MAX}}-{\Delta \mathrm{P}}_{\mathrm{c}\_\min }=9.3-6.95=2.35 \mathrm{W}$$

(15)

In the second case (

Table 2. Experimental results for case 2—Output power P_out = 288 W—constant.

Vin [V]	Iin [A]	Pin [W]	Vout [V]	Iout [A]	Pout [W]	η [%]	ΔPC [W]	Zin [Ω]
100	3.041	304.100	72.000	4.000	288.000	94.706	16.100	32.884
125	2.406	300.750	72.000	4.000	288.000	95.761	12.750	51.953
150	1.997	299.550	72.000	4.000	288.000	96.144	11.550	75.113
175	1.707	298.725	72.000	4.000	288.000	96.410	10.725	102.519
200	1.492	298.400	72.000	4.000	288.000	96.515	10.400	134.048
225	1.324	297.900	72.000	4.000	288.000	96.677	9.900	169.940
250	1.192	298.000	72.000	4.000	288.000	96.644	10.000	209.732
275	1.085	298.375	72.000	4.000	288.000	96.523	10.375	253.456
300	0.996	298.800	72.000	4.000	288.000	96.386	10.800	301.205
325	0.919	298.675	72.000	4.000	288.000	96.426	10.675	353.645
350	0.855	299.250	72.000	4.000	288.000	96.241	11.250	409.357

), it was found that the maximum efficiency η_MAX = 96.677% was obtained at the input voltage level of V_in = 225 V (Figure 25). At the same voltage level, both the minimum input power P_{in_min} = 297.9 W and the minimum overall converter loss, named ΔP_{c_min} = 9.9 W, were recorded (Figure 26 and Figure 27). At the input voltage level of U_in = 100 V, the minimum value of the conversion efficiency was recorded, named η_min = 94.706%. At the same voltage level, both the maximum value of the input power P_{in_MAX} = 304.1 W and the maximum value of converter loss, namely ΔP_{c_MAX} = 16.1 W, were obtained. Thus, both the recovered amount of conversion efficiency and overall converter losses can be determined based on the following equations:

$${\Delta \mathsf{\eta }}_{\mathrm{REC}}={\mathsf{\eta }}_{\mathrm{MAX}}-{\mathsf{\eta }}_{\min }=96.677-94.706=1.971\%$$

(16)

$${\Delta \mathrm{P}}_{\mathrm{c}\_\mathrm{REC}}={\Delta \mathrm{P}}_{\mathrm{c}\_\mathrm{MAX}}-{\Delta \mathrm{P}}_{\mathrm{c}\_\min }=16.1-9.9=6.2 \mathrm{W}$$

(17)

For the third case (

Table 3. Experimental results for case 3—Output power P_out = 576 W—constant.

Vin [V]	Iin [A]	Pin [W]	Vout [V]	Iout [A]	Pout [W]	η [%]	ΔPC [W]	Zin [Ω]
100	6.438	643.800	72.000	8.000	576.000	89.468	67.800	15.533
125	5.004	625.500	72.000	8.000	576.000	92.086	49.500	24.980
150	4.112	616.800	72.000	8.000	576.000	93.385	40.800	36.479
175	3.490	610.750	72.000	8.000	576.000	94.310	34.750	50.143
200	3.040	608.000	72.000	8.000	576.000	94.736	32.000	65.789
225	2.693	605.925	72.000	8.000	576.000	95.061	29.925	83.550
250	2.417	604.250	72.000	8.000	576.000	95.324	28.250	103.434
275	2.195	603.625	72.000	8.000	576.000	95.423	27.625	125.285
300	2.003	600.900	72.000	8.000	576.000	95.856	24.900	149.775
325	1.847	600.275	72.000	8.000	576.000	95.956	24.275	175.961
350	1.719	601.650	72.000	8.000	576.000	95.736	25.650	203.607

), the maximum conversion efficiency η_MAX = 95.956% was obtained at the input voltage level of V_in = 325 V, (Figure 28). At the same voltage level, both the minimum input power P_{in_min} = 600.75 W and the minimum overall converter loss ΔP_{c_min} = 24,275 W were recorded (Figure 29 and Figure 30). At the input voltage level named U_in = 100 [V], the minimum value of the conversion efficiency was recorded, η_min = 89.468%. At the same voltage level, both the maximum value of the input power P_{in_MAX} = 643.8 W and the maximum value of the converter loss, namely ΔP_{c_MAX} = 67.8 W, were obtained. Therefore, the recovered value of the conversion efficiency, as well as of the overall converter losses, can be determined using the following equations:

$${\Delta \mathsf{\eta }}_{\mathrm{REC}}={\mathsf{\eta }}_{\mathrm{MAX}}-{\mathsf{\eta }}_{\min }=95.956-89.468=6.488\%$$

(18)

$${\Delta \mathrm{P}}_{\mathrm{c}\_\mathrm{REC}}={\Delta \mathrm{P}}_{\mathrm{c}\_\mathrm{MAX}}-{\Delta \mathrm{P}}_{\mathrm{c}\_\min }=67.8-24.275=43.525 \mathrm{W}$$

(19)

As a preliminary conclusion over this step, the main concept was proved because, according to the results represented above, the conversion efficiency and overall losses can be reduced by selecting the optimum input voltage level, related to the minimum value of the input power.

Considering this assumption (that the optimum input voltage corresponds to the minimum value of the input power), an automatic voltage adjustment technique was developed first in offline simulations. The MathWorks Matlab-Simulink environment was used to develop the offline simulation model and the Real-Time HIL online simulation hardware-ready model. Two models were built, one for the nano-grid model and one for the automatic voltage adjustment technique.

The virtual DC loads were simulated based on real data recorded in the previous step in the summary tables. More precisely, the impedance of a variable load was adjusted according to a look-up table that contained the real voltage values (Figure 31).

By using the “impedance related to voltage” functional characteristic curves of the three cases studied earlier, three variable loads can be simulated by using look-up tables. Those loads emulate the real behavior of the “converter + load” group in the variable input voltage powering conditions and constant power at the output [32,33]. Three variable loads were connected sequentially to the nano-grid within the DC grid Simulink model (Figure 32 and Figure 33).

The iMPPT algorithm automatically adjusts the main bus input voltage level so that the absorbed power by the variable loads is always at a minimum for that corresponding demanded load power level (Figure 34).

The optimum voltage level corresponds to the minimum input power and to the minimum overall loss level. The automatic voltage adjustment technique tracks the minimum input power and sets the voltage level accordingly. The first step to identify the minimum power value is a voltage sweeping operation performed by the iMPPT algorithm. The input power and voltage values are stored in a bi-dimensional array, and then, the minimum value of the input power is determined from the array. In the same row, the optimum input voltage level is stored in the bi-dimensional array (Figure 35). The index of the optimum input voltage value will be determined, and the corresponding value will be extracted from the array (Figure 35) [8].

The iMPPT algorithm also decides if the voltage level shall be increased or decreased to reach the target level set by the bi-dimensional array fetching operation. If the main absorbed grid power value changes, the iMPPT re-sets the voltage adjustment operation and scans for a new optimum voltage level (Figure 36) [8].

The input absorbed current level from the nano-grid will be also set at an optimum value, not an absolute minimum value (Figure 37).

The math operations are always performed in a steady state. The voltage adjustment technique increases or decreases the voltage discontinuously. The adjustment technique we have novelty called “incremental and settle” (Figure 38 and Figure 39). The data acquisition is also performed in a steady state. A square synchronization signal was used.

In the first experimental step, the voltage level was set manually from the high-power DC power supply front panel knob. In the second experimental step, an offline Simulink simulation model was developed to obtain the automatic voltage adjustment technique. The offline simulation was performed using real tabular data recorded under real lab conditions using a physical prototype [8].

In the third experimental step, a real-time online Hardware-In-the-Loop (HIL) simulation will be performed (Figure 40).

Both Simulink Models (the nano-grid model and the voltage adjustment model) were imported onto two separate computers running the NI VeriStand and NI Software bundle. For the voltage adjustment model, NI MyRIO was used, and the following dashboard was created (Figure 41):

For the NI PXIe-1071, the nano-grid Simulink model was also imported in VeriStand, and a dashboard has been created (Figure 42).

The results of the online HIL simulation correspond to the results obtained in the offline simulation within Matlab-Simulink. It means that the automatic voltage adjustment control strategy runs on the NI MyRIO 1900 embedded controller (Figure 43).

The automatic voltage adjustment technique determines the new optimum voltage value each time the absorbed power level changes.

The implementation step of the automated voltage adjustment technique on the NI MyRIO 1900 embedded device was tested and confirmed using the Hardware-In-the-Loop (HIL) real-time simulation procedure (Figure 44 and Figure 45).

4. Discussion

The same test scenarios were performed for 11 different power levels (from 72 W to 792 W). It was found that the variable voltage powering mode applies to a wide range of power levels but predominantly its loss recovery effect can be observed only at higher power levels. The highest amount of power was recovered at the output power level P_out = 792 W, with a total amount of overall loss of ΔP_{C_REC} = 96 W, equivalent to approximately Δη_REC~9.82% percent of nominal efficiency of the converter (Figure 46 and Figure 47).

Also based on empirical data, both the family of characteristics “efficiency depending on input voltage” and “the map of efficiency values related to input voltage and output power” will be plotted (Figure 48 and Figure 49) [8].

For the implementation step of the automatic input voltage adjust technique of the grid or power supply, the families of characteristic curves alternative to the representation “efficiency related to input voltage” will be used, namely, two more sets of characteristic curves will be plotted, namely “current related to input voltage” and “input impedance related to input voltage” (Figure 50 and Figure 51) [8].

The recovered values for overall losses and conversion efficiency percentages were determined according to equations (14) and (15), for example. The results were recorded in a summary table (

Table 4. Recovered values of overall power losses and conversion efficiency percentage.

Pout [W]	72	144	216	288	360	432	504	576	648	720	792
ΔηREC [%]	3.294	1.252	1.011	1.971	3.236	4.318	5.423	6.488	7.867	8.804	9.822
ΔPC_REC [W]	2.55	1.925	2.35	6.2	12.925	21	31.3	43.525	61	77	96

).

The input current of the power electronics converter is adjusted parabolically, like the constant power (CP) working mode I-V plot when input voltage changes (Figure 6). It means that the synchronous buck converter operates in constant power mode (CP) [8].

The proportional integral controller automatically adjusts the duty cycle in such a way that the output voltage always remains constant. By adjusting the duty cycle in the power electronics converter, its input impedance will be modified (Figure 50). The optimal voltage corresponds to the optimal impedance and the minimum input power [8].

5. Conclusions

Finally, the main conclusions can be summarized as follows:

-

Based on representative functional characteristics of consumers (e.g., characteristic curve I = f(V)), it is possible to identify the operating mode [34].

-

In the case of a regulated voltage source, the equivalent circuit can be reduced to a constant power consumer load from the point of view of the input side [34].

-

The converter adjusts the impedance to serve the final consumer load at the optimal parameters [34].

-

The input voltage level can be adjusted in such a way that the conversion efficiency of a regulated switched mode power supply can be improved.

-

The novel proposed iMPPT method can improve the energy conversion ratio from 85% up to approximately 10% in case of an output power level of 800 W served by a synchronous buck converter at the input voltage level of 350 V. The total amount of recovered power in this situation can be approximately 100 W.

6. Patents

Based on the principle of optimization of conversion efficiency presented in the previous chapter, a strategy for the automatic voltage regulation for the power supply side level was proposed in a patent application filed with OSIM on 19 December 2019 entitled “Micro-grid of variable DC voltage and its control method” (Figure 1 and Figure 2).

The application was published in the “Official Bulletin of Industrial Property” in the section “Patents” [7,11]. Both the operating logic diagram of the automatic voltage level adjustment strategy and the residential microgrid structure can be observed in the following figures (Figure 1 and Figure 2).