Battery-Supercapacitor Hybrid Energy Storage Systems for Stand-Alone Photovoltaic

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Figure 1. Block diagram of PV systems with energy storage

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Figure 2. Diagram of the simulation of the PV system with hybrid storage in MATLAB-Simulink

2.1 GPV modelling

Figure 4(a) represents the electrical model of a PV cell consisting of a photocurrent and a diode describing the properties of the semiconductor [8]. A series resistances $\mathrm{R}_s$ represents an internal resistance, and a parallel resistance $\mathrm{R}_p$  showing a leakage current [6].

The mathematical equation expressing the charging current can be given as [9]:

$\mathrm{I}=\mathrm{I}_{\mathrm{ph}}-\mathrm{I}_{\mathrm{s}}\left[\mathrm{e}^{\left(\frac{\mathrm{qv}+\mathrm{IR}_{\mathrm{s\quad}}}{\mathrm{kT}_{\mathrm{c}} \mathrm{A}}\right)}-1\right]-\left(\frac{\mathrm{v}+\mathrm{IR}_{\mathrm{s}}}{\mathrm{R}_{\mathrm{p}}}\right)$     (1)

Based on Eq. (1) we see that the physical behavior of the PV cell is also related to temperature and solar radiation [6], so for a given temperature of 25℃, and variable radiation. The I-V and P-V curves are generated as shown in Figure 3.

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Figure 3. Characteristics of the PV system with variable solar radiation

2.2 Battery modelling

The model is shown in Figure 4(b), it consists of a voltage source corresponding to the open circuit voltage source $\mathrm{E}_0$ in series with an equivalent internal resistance $\mathrm{R}_S$  [10].

The terminal voltage of the battery is given by the Eq. (2) [11].

$\mathrm{V}_{\mathrm{b}}=\mathrm{E}_0-\mathrm{I}_{\mathrm{b}} \cdot \mathrm{R}_{\mathrm{s}}$     (2)

The state of charge (SOC) of the battery is also defined by:

SOC $=1-\frac{Q_d}{C_b}$      (3)

where $C_b$ is the nominal capacity (Ah) of the battery, and $\mathrm{Q}_{\mathrm{d}}$ is the amount of charge missing from $C_b$ [2].

2.3 Supercapacitor modelling

This is a simple electrical circuit model consisting of a resistor $\mathrm{R}_{\mathrm{sc}}$ and a capacitor $\mathrm{C}_{\mathrm{sc}}$ in series (Figure 4(c)). This model is commonly used for the study of energy systems.

The expression for the SC voltage $\mathrm{V}_{\mathrm{SC}}$ is given by Eq. (4) [11].

$\mathrm{V}_{\mathrm{sc}}=\mathrm{V}_1-\mathrm{R}_{\mathrm{sc}} \cdot \mathrm{I}_{\mathrm{sc}}=\frac{\mathrm{Q}_{\mathrm{sc}}}{\mathrm{C}_{\mathrm{sc}}}-\mathrm{R}_{\mathrm{sc}} \cdot \mathrm{I}_{\mathrm{sc}}$    (4)

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Figure 4. Electrical model of a) PV cell, b) Battery, c) Supercapacitor

2.4 DC/DC converter modeling

Choppers are static DC-DC converters whose function is to provide a variable DC voltage from a fixed DC voltage. This energy conversion is carried out thanks to a high-frequency “chopping” characterized by high efficiency.

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Figure 5. DC/DC converter diagram

If the output voltage is lower than the input voltage, the chopper is called a buck (BUCK). Otherwise, it is said to be a booster (BOOST). The study of the simplest static DC-DC converters is divided into three families of converters:

-Boost converter,

-Buck converter,

-Buck-Boost converter,

Figure 5 illustrates a photovoltaic system adapted by a Boost-type converter feeding a resistive load R.

The capacitor is charged by the current coming from the generator and stored in the inductor during the first phase of operation. It is used to smooth the output voltage.

$V_{C h}=E\left(\frac{1}{1-\alpha}\right)$     (5)

$I_{C h}=(1-\alpha) I$    (6)

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Figure 6. Electric circuit of the Boost converter

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Figure 7. DC/DC Buck converter

In this type, the average output voltage is higher than the input voltage. It is also called a parallel converter. This structure requires a controlled switch and is in parallel with the source (Figure 6). The duty cycle, chopping period (T), has two states.

A state of energy accumulation: when the switch k is closed (on the state), this leads to an increase in the current in the inductance and therefore the storage of a quantity of energy in the form of magnetic energy. The diode is then blocked and the load is disconnected from the power supply. This phase lasts from (0) to ($\alpha \mathrm{T}$);

When the switch k is open, the inductance is then in series with the Generator and its e.m.f is added to that of the generator (booster effect). The current flowing through the inductor then crosses the diode, the capacitor and the load. This results in a transfer of the energy accumulated in the inductor to the capacitor. This phase lasts from ($\alpha \mathrm{T}$) to (T). The equations for voltage and current in the load are:

This type of converter is used to produce a higher voltage than that supplied by the photovoltaic generator (Figure 7).

The proposed system (Figure 1) contains two transistor switches, the method of controlling which determines the mode of bidirectional converter operation (buck or boost):

A. Discharge mode:

For the first phase T: closed, D: open

$\left\{\begin{array}{l}\frac{d I_L}{d t}=\frac{1}{L}\left(V_e-V_s\right) \quad(0<t<\alpha T s) \\ \frac{d V e}{d t}=\frac{1}{C}\left(I_L\right)-\frac{V s}{R C}\end{array}\right.$   (7)

For the second phase T: open, D: closed

$\left\{\begin{array}{l}\frac{d I_L}{d t}=-\frac{1}{L}(V s) \\ \frac{d V e}{d t}=\frac{1}{C}\left(I_L\right)-\frac{V s}{R C}\end{array} \quad(\alpha T s<t<T s)\right.$  (8)

So, the output voltage is

$V_s=\alpha \ V e$   (9)

B. Charge mode:

For the first phase T: closed, D: open

$\left\{\begin{array}{l}\frac{d I_L}{d t}=\frac{1}{L}\left(V_e\right) \\ \frac{d V e}{d t}=-\frac{V s}{R C}\end{array}\left(0<t<\alpha T_s\right)\right.$   (10)

For the second phase T: open, D: closed

$\left\{\begin{array}{l}\frac{d I_L}{d t}=\frac{1}{L}\left(V_e\right)+\frac{1}{L}\left(V_s\right)(\alpha T s<t<T s) \\ \frac{d V e}{d t}=\frac{1}{C}\left(I_L\right)-\frac{V e}{R C}\end{array}\right.$    (11)

Output voltage is:

$V s=V e \cdot \frac{1}{1-\alpha}$   (12)

2.5. MPPT for GPV

Since the considered DC microgrid is a PV-based system, it is essential to operate the PV source at its maximum power point [12].

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(a)

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(b)

Figure 8. (a) Principle of the P&O method algorithm, (b) P&O Algorithm

By adjusting the duty cycle of the DC/DC converter [13]. The Perturber & Observer method is based on perturbation by increasing or decreasing the module's operating voltage $\mathrm{V}_{\mathrm{PV}}$ of a small amplitude around its initial value and analyzing the behavior of the resulting power variation $\mathrm{P}_{\mathrm{PV}}$ [14]. If ΔP is positive, it means that the operating point is to the left of the MPP. If, on the contrary, the power decreases, it implies that the system has already exceeded the MPP (Figure 8-a) [2]. The parameter used in this method is the PV generator voltage ($\mathrm{V}_{\mathrm{K}}$) which must be calculated at time k and the current of the PV generator ( $\mathrm{I}_{\mathrm{K}}$ ) and then we have the $\mathrm{P}_{\mathrm{K}}$  which is compared to the previous disturbance time cycles ( $\mathrm{P}_{\mathrm{K-1}}$ ). If the comparison value shows zero, the maximum power point has been reached (Figure 8-b). As soon as the MPP is reached, V oscillates around the ideal operating voltage [15-17].

In this scenario, variable solar irradiation is used, as shown in Figure 10, and the load is assumed to be constant throughout the experiment and equal to 500 watts. Figure 11 shows the performance of the proposed PMS under variable solar irradiation such that when the panels produce enough energy to meet the load requirements; the surplus is used to recharge the supercapacitor and batteries.

The Parameters of PV generator, Supercapacitor, and battery are shown in Tables 1, 2, and 3 respectively.

Table 1. PV generator properties

Table 2. Supercapacitor properties

Table 3. Battery properties

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Figure 10. Solar radiation and temperature

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Figure 11. Responses of $P_{p V}, P_L, P_{b a t}$ and $\mathrm{P}_{\mathrm{SC}}$ with variable solar radiation

Figure 12 and Figure 13 show the responses of and with varying solar irradiance. These results illustrate the efficiency of the supercapacitor when storing or releasing energy to reduce stress on the battery. As early t=2s, the power delivered by the panels increases from 400 to 600 watts and thus becomes higher than the consumption. During this short period, the power delivered by the battery tends towards a negative value (Figure 12) which is logical since the demand is met by the energy from the panels. And the battery switches from discharge to charge mode.

There is an excess of power during this period which is stored in the SC. The power of the panels continues to increase as the irradiation increases in steps and it is observed that at each sudden change in power, the Supercapacitor reacts to the change in a very short time.

In this time interval from 0 to 2s and from 9 to 12s, the power of the panels is lower than the power demanded by the load. The battery supplies the power deficit to the load (positive current and battery power respectively in Figure 13. At t=9s. There is a sharp drop in panel power from 700 to 470 watts in a very short time. On this same delay, the battery switches from charge to discharge mode with respect SOC (state of charge), and the Supercapacitor reacts to the deficit first and feeds the load and the battery afterwards, and at t = 2s we see that the surplus is used to recharge the supercapacitor first, then the batteries. With each sudden decrease or increase in the panel power, the supercapacitor continues to respond by supplying or absorbing power in a short period of time.

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Figure 12. Vbat, Ibat, SOCbat, and P_Batr

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Figure13. Vsc, Isc, SOCsc, and $P_{s c}$