Photovoltaic and Electrochemical Equivalent Circuit Model

As a society we are desperately in need of clean energy sources, but further­more, we are desperately in need of portable energy sources which can be used in automotive applications. The clean and renewable aspects of using photovoltaic devices to absorb sunlight, when coupled with the use of elec­trochemical reactions to produce hydrogen fuel for mobile applications, is an attractive solution to our long term energy problems. The combination of pho­tovoltaic and electrochemical cells has recently attracted a lot of attention in what is commonly referred to as an “artificial leaf”.

Leaves, and other photosynthetic organisms, directly convert the energy of sunlight into hydrogen fuel by splitting water to produce oxygen and hy­drogen. Subsequently, this chemical fuel is fixed through their combination with carbon dioxide to produce carbohydrates. This last step, however, does not change the energy of the system, and is just a convenient way for na­ture to store energy and the main extraction of energy from sunlight is in the creation of hydrogen fuel. In other words, the energy storage in photo­synthesis is primarily due to water splitting, and carbohydrate production is simply a method of storing the hydrogen. The solar water-splitting cells, or artificial leaves, consist of a silicon photovoltaic interfaced to hydrogen – and oxygen-evolving catalysts. This offers a direct method of converting solar en­ergy directly into fuel, in a manner similar to that of real leaves. In order to drive the electrochemical process of water-splitting we must provide enough energy to overcome both the energetic barrier of water oxidation (1.23 V) and any overpotentials needed to drive the catalysis. Typically, solar cells will not have a large enough voltage to drive water splitting, and a number of solar cells in series will be required. Therefore, multi-junction or tandem solar cell designs are required for these kinds of devices.

To model these photovoltaic-electrochemical systems we can use an equiv­alent circuit model, capable of predicting the performance of the system and, crucially, the efficiency of the system. Note that the equivalent circuit ap­proach to solar cells is well established, as we looked at the use of equivalent circuit models of solar cells in Chapter 4. To recap, the light absorption by the solar cell produces a current source of excited electrons and holes. Some of this current is lost due to internal recombination, and this recombination is represented by a diode in the equivalent circuit model, ID. Mechanical defects and material dislocations also cause losses within the solar cell, and these are captured through the insertion of the shunt resistance, Rsh. Internal electrical resistances, particularly at the interface of the semiconductor and the metal contacts, can also result in losses and this is captured by the insertion of a series resistor, Rs . These are depicted on the left-hand side of the equivalent circuit in Figure 7.3. On the right-hand side of the circuit is the electrochem­ical cell which shares both the same voltage and current as the photovoltaic cell. The electrochemical reaction can only occur if the voltage produced is greater than the thermodynamic potential (pth = 1.23 V), the overpotential for each reaction, and any losses due to the resistance of the solution between the electrodes. The overpotential for the two reactions, both the oxygen and hydrogen reactions, are captured using diodes in the equivalent circuit (which interestingly enough is consistent with Tafel’s law, relating the rate of an elec­trochemical reaction to the overpotential), while the resistance of the solution is a simple resistor. We can now solve this equivalent circuit model and find the operating current in the photovoltaic-electrochemical system.

The equivalent circuit of a solar cell has already been covered in Chapter 4, but here it is a little different. Not only do we have the electrochemical part, which we’ll come to in a minute, but we also have to combine more than one solar cell in series. A single solar cell would be unable to generate the required voltage to drive water splitting, and cannot be used for direct solar to fuel conversion systems. However, tandem or multi-junction cells are a single pho­tovoltaic device made up of multiple p-n junctions, of different semiconductor materials, where the voltage across the multi-junction device is the addition of the voltage across each individual p-n junction. Furthermore, the p-n junc­tions, and the different semiconductor materials used for each p-n junction, absorb different regions of the spectrum of sunlight which further increases the efficiency. It should be noted that currently the cost of such multi-junction solar cells is prohibitively high for all but the most specialized applications; it is unclear if or when this technology might become more economically viable. The current density from a multi-junction solar cell is the same as for a single junction solar cell, except we introduce the number of cells (here, junctions) in series, N:

where JL is the photogenerated current density, Jo is the reverse saturation current density, q is the elementary charge, V is the voltage across the cell, N is the number of cells (or junctions) in series, Rs is is the series specific resistance, n is the diode ideality factor, k is the Boltzmann constant, T is the cell temperature, and Rsh is the shunt specific resistance. On the right-side of the circuit is the electrochemical cell, where the voltage required to drive the electrochemical reaction must exceed the thermodynamic potential, pth, of
the water splitting reaction by an amount sufficient for the reaction kinetics to occur. To drive the reaction kinetics the overpotential for each reaction (given by Tafel’s law) must be met

Vo = to ln( J (7.7)

for the oxygen reaction, and

vh = th ln ^ jh) (7.8)

for the hydrogen reaction, where to is the Tafel slope for the oxygen reaction, J is the current density from the solar cell, J° is the exchange current density for the oxygen reaction, th is the Tafel slope for the hydrogen reaction, and JH is the exchange current density for the hydrogen reaction. We add to this the resistance of the solution, and obtain the required voltage to drive water splitting as

We can now couple the two equations, Equation 7.6 for the photovoltaic cell and Equation 7.9 for the electrochemical cell, to obtain a single solution for the current density in the coupled photovotaic-electrochemical system

J = Jl-

(7.10)

In particular, the above equation simply matches the current and voltage output from the photovoltaic device to the current and voltage input to the electrochemical cell. In this steady-state configuration it can be shown that the efficiency of the solar power to fuel conversion is of the form

pthJ Vsf = P—

P sun

where pthJ represents the intensity (power per unit area, Wcm-2) devoted to the water splitting process, and Psun represents the solar intensity.

In order to solve the above transcendental equation we must use a numer­ical method. In other words, above we have one equation and one unknown

FIGURE 7.4 Schematic of how the Newton-Raphson method works.

(the current density), but the solution is not obvious. However, we solved a similar equation in chapter 4 when looking at equivalent circuits. In that case we used the bisection method which is a very reliable and accurate method of solving such problems. Here we’ll use a much faster, although arguably less reliable, method called the Newton-Raphson method. The Newton-Raphson method is depicted graphically in Figure 7.4, where the function we are trying to solve is the dark black curve and we are trying to find where the function crosses the x-axis, and is equal to zero. We start with an initial guess xo which is not the correct solution, and find the value of the function, f (xo), and its derivative at this point. Note the derivative is represented by the straight line. The next approximation to the solution, xi is obtained by finding where the slope crosses the x-axis. This is repeated, with each step yielding successively better approximations, until an acceptable solution is found. Mathematically, this is often written as the solution of the function

and the equation for finding successively better approximations is of the form

f (ХП) /„ , QN

Xn+1 = Xn – ^—гг (7.13)

J (xn)’

In terms of our system, therefore, we need not only the function, f (xn), but also the derivative of the function, f(xn)r. The function we are solving is simply Equation 7.10, which upon putting the current density on the right – hand side to make the left-hand side zero gives us the function

The derivative of this function is

and we can now enter this into a spreadsheet and iteratively solve for the current density in our photovoltaic-electrochemical system.

The spreadsheet implementation of the above model is depicted in Figure 7.5. The constants required for the mode, including the photovoltaic properties and the electrochemical properties, are included at the top of the spreadsheet. Note that these constants are taken from the literature, but different systems could have quite different parameters. Below the constants is the area of the spreadsheet devoted to solving the Newton-Raphson method. This consists of three columns. In the first column is the approximate solution of the current density for a given iteration (as we move down the rows in the spreadsheet we perform subsequent iterations). The second and third columns contain the function and the derivative of the function, respectively.

Photovoltaic and electrochemical

constants	r V		-el system
	A	В		C	D	E		F	G	H
і	z 3 I 1 0-	ilectroct	mlc<	зі equivalent c	rcuit
2
3	Constants	V
4		V
5	J L	Э.00Е-002		J o*H	1.00E-01			N	3
6	J 0	4.00E-013		R sol	3.00E+00			к	1.38E-019
7	mu th	1.23		R s	1.50E+00			T	300
3	tau О	0.06		n	1.
9	J 0*0	1.00E-013		R sh	1.00E+00
10	tau H	0.03		P sun	1.36E-00
11						7
12
13	J	Ml____	d f(jy d J		Efficiency v
14	1.00E-016	0.027972327	-3 667E+012		0.129939404
15	7.7288E-015	0.026376228	-4.744E+Q1Q
16	5.6Э74Е-01Э	0.024805358	-650412648
17	3 8702E-011	0.023254712	-9474194.746
18	2.4932E-009	0.021727376	-147065.8343
19	1 502ЭЕ-007	0 02022439	-2441.778915
20	8.4329E-006	0.018738328	-44.59560702
21	0 000426616	0.016829402	-1.970466294
22	0.00696943a	0.006191346	-1.155879559
23	0.014325832	4.7279E-005	-1.140594791
24	0 014367262	1.5Э19Е-009	-1.140520948
25	0 014367264	"5	r-q0520946
26	0.014367264	0	Xf<^!Q946

For the Newton-Raphson method both the function that we are solving, f(J), and its derivative are required.

In cell A14 is an initial guess at a solution. Note that the Newton-Raphson method can be quite sensitive to this initial guess, and if the method becomes unstable (giving very large and silly numbers, for example, that continue to grow towards infinite) then you might want to try changing this initial guess. In cell B14 we calculate the function using the code

=$B$5 – $B$6*(EXP((1.6E-019*($B$7+$B$8*LN(A14/$B$9)

+ $B$10*LN(A14/$E$5) +

A14*($E$6+ $H$5*$E$7)))/($E$8*$H$5*$H$6*$H$7)) – 1) –

($B$7+$B$8*LN(A14/$B$9) + $B$10*LN(A14/$E$5) +

A14*($E$6+ $H$5*$E$7))/($H$5*$E$9) – A14

In cell C14 we calculate the derivative of the function, using the present iter­ation’s estimate for the current density, using the code

=-($B$6*1.6E-019*($B$10/A14 + $B$8/A14 +

$E$6+$H$5*$E$7)*EXP(1.6E-019*($B$10*LN(A14/$E$5) +

$B$8*LN(A14/$B$9) +

A14*($E$6+$H$5*$E$7) + $B$7)/($E$8*$H$5*$H$6*$H$7)))/
($E$8*$H$5*$H$6*$H$7) –

($B$10/A14+$B$8/A14+$E$6+$H$5*$E$7)/($H$5*$E$9) – 1

The previous estimate is updated, to obtain successively better estimates using both the function and its derivative, and this is reflected in cell A15 which takes the initial guess in cell A14 and updates it using the code

=A14-B14/C14

After only a few iterations the Newton-Raphson method typically gives a very good solution to the transcendental equation (Equation 7.10). Once we have the current density we can obtain the predicted efficiency of the photovoltaic – electrochemical system using

Vsf = (7.16)

Psun

where pthJ represents the intensity (power per unit area, Wcm-2) devoted to the water splitting process, and Psun represents the solar intensity. This is calculated in cell E14. In other words, this model can take fundamental vari­ables from both a photovoltaic system and a separate electrochemical system, and ultimately predict the efficiency of a coupled photovoltaic electrochemical system.