1. Introduction

There is a need in the modern world for sustainable means of producing clean energy economically, on a very large scale. The planet's human population is inexorably increasing toward the 10 billion mark [1–3]. The rapid growth in population presents both opportunities for companies

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

seeking to broaden markets for their products, and challenges for governments, as the growing populations demand their share of prosperity. It is well known that prosperity generating economic growth requires energy [4–6]. To meet the demands for prosperity, carbon based fossil fuel consumption has increased accordingly, resulting in unacceptable levels of air pollution in major conurbations in both advanced and developing countries [7–10]. Much of the pollution comes from burning fossil fuels inside internal combustion engines (ICEs) of motor vehicles and ships. Coal burning thermal power plants used for electricity generation also contribute substantially to the rise in air pollution [11, 12].

Scientific research has hitherto yielded various solutions to the clean energy challenge using innovative approaches ranging from development of hybrid gasoline-electric motor vehicles (HEVs), plug-in hybrid gasoline-electric motor vehicles (PHEVs), pure battery electric vehicles (BEVs), fuel cell electric vehicles (FCEVs), advanced catalysts for reducing exhaust emissions as well as carbon capture technologies applicable to coal burning thermal power plants [13–17]. These solutions however, work only to mitigate the problem of carbon based fossil fuel emissions and do not address the fundamental problem, namely, how to circumvent carbon based fossil fuels in energy generation and ground transport applications. Although HEV, PHEV, BEV and FCEV technologies offer promise in reducing pollution at least locally, they represent at best, an incomplete remedy to a major problem. HEVs and PHEVs still require combustion of gasoline inside an ICE while BEVs require copious quantities of electrical energy for battery charging, generated by power plants connected to the electric grid. The power plants supplying the electric grid can be hydroelectric or nuclear, but far more often, are fossil fuel burning thermal power plants that are only incrementally more efficient than internal combustion engines in motor vehicles [18]. Since hydroelectric generating capacity in the United States of America (U.S.A.) has already been reached and construction of new nuclear power stations is fraught due to well substantiated fears of radiological leaks, it becomes apparent that the only way to meet the increased demand for electricity from electric vehicle proliferation is by constructing more fossil fuel burning thermal power plants [19, 20].

The FCEVs exist at present in small numbers as vehicle prototypes that function primarily as technology demonstrators [21, 22]. FCEVs are unique however, because they represent the only motor vehicle technology that uses hydrogen (H2) fuel to generate electric energy to power a motor driving the wheels of the vehicle. Although in existence in various forms since the 1960s, FCEVs have not proliferated for manifold reasons, the principal ones being the absence of means for safely storing hydrogen fuel on board, coupled with a lack of means to economically generate sufficiently pure hydrogen (H2(g)) fuel to prevent poisoning sensitive catalysts that might be present in the fuel cells [23, 24]. The existing methods of storing hydrogen on board motor vehicles utilize cryogenic storage of liquid hydrogen (H2(l)), storage of hydrogen (H2(g)) gas at pressures as high as 70 MPa (10,153 psi) in cylinders made from composite material, and storage as a metal hydride (MHX) in tanks filled with porous metal sponge or powder comprised of light group 1 and 2 metals and/or transition metal elements, namely, Titanium (Ti) or Nickel (Ni) [25–30]. Such direct hydrogen storage methods however, are impractical due to the high cost of suitable transition metals Ti and Ni, and moreover, because an infrastructure is needed to supply hydrogen directly in large volume to fill liquid or gas tanks or to saturate or replenish the metal sponge within the storage reservoir inside a motor vehicle, a procedure fraught with all of the well known safety risks associated with handling large volumes of elemental hydrogen [31, 32]. Furthermore, the existing industrial method of generating hydrogen (H2(g)) gas using steam reforming of natural gas, the latter containing mostly methane (CH4), produces significant quantities of carbon monoxide (CO) even after application of the shift reaction, the latter meant to transform the CO into carbon dioxide (CO2) [33, 34]. The presence of even minute quantities of CO on the parts per million (ppm) order of magnitude in H2(g) fuel, results in rapid poisoning of sensitive platinum (Pt) catalysts present in the latest generation of low operating temperature, proton exchange membrane (PEM) fuel cells [35]. Catalysts based on a mixture of platinum and ruthenium (Pt-Ru) developed to overcome the sensitivity of pure Pt to carbon monoxide poisoning are not cost effective for large scale application in motor vehicle transport applications due to the dearth of ruthenium [36]. Since hydrogen production by conventional steam reforming methods generates significant quantities of CO and CO2, it becomes difficult to justify using the approach to generate hydrogen (H2) fuel for FCEVs given that the purpose of advancing such technology is to eliminate carbon based fossil fuel emissions.

seeking to broaden markets for their products, and challenges for governments, as the growing populations demand their share of prosperity. It is well known that prosperity generating economic growth requires energy [4–6]. To meet the demands for prosperity, carbon based fossil fuel consumption has increased accordingly, resulting in unacceptable levels of air pollution in major conurbations in both advanced and developing countries [7–10]. Much of the pollution comes from burning fossil fuels inside internal combustion engines (ICEs) of motor vehicles and ships. Coal burning thermal power plants used for electricity generation also contribute substan-

Scientific research has hitherto yielded various solutions to the clean energy challenge using innovative approaches ranging from development of hybrid gasoline-electric motor vehicles (HEVs), plug-in hybrid gasoline-electric motor vehicles (PHEVs), pure battery electric vehicles (BEVs), fuel cell electric vehicles (FCEVs), advanced catalysts for reducing exhaust emissions as well as carbon capture technologies applicable to coal burning thermal power plants [13–17]. These solutions however, work only to mitigate the problem of carbon based fossil fuel emissions and do not address the fundamental problem, namely, how to circumvent carbon based fossil fuels in energy generation and ground transport applications. Although HEV, PHEV, BEV and FCEV technologies offer promise in reducing pollution at least locally, they represent at best, an incomplete remedy to a major problem. HEVs and PHEVs still require combustion of gasoline inside an ICE while BEVs require copious quantities of electrical energy for battery charging, generated by power plants connected to the electric grid. The power plants supplying the electric grid can be hydroelectric or nuclear, but far more often, are fossil fuel burning thermal power plants that are only incrementally more efficient than internal combustion engines in motor vehicles [18]. Since hydroelectric generating capacity in the United States of America (U.S.A.) has already been reached and construction of new nuclear power stations is fraught due to well substantiated fears of radiological leaks, it becomes apparent that the only way to meet the increased demand for electricity from electric vehicle proliferation is by constructing more fossil

The FCEVs exist at present in small numbers as vehicle prototypes that function primarily as technology demonstrators [21, 22]. FCEVs are unique however, because they represent the only motor vehicle technology that uses hydrogen (H2) fuel to generate electric energy to power a motor driving the wheels of the vehicle. Although in existence in various forms since the 1960s, FCEVs have not proliferated for manifold reasons, the principal ones being the absence of means for safely storing hydrogen fuel on board, coupled with a lack of means to economically generate sufficiently pure hydrogen (H2(g)) fuel to prevent poisoning sensitive catalysts that might be present in the fuel cells [23, 24]. The existing methods of storing hydrogen on board motor vehicles utilize cryogenic storage of liquid hydrogen (H2(l)), storage of hydrogen (H2(g)) gas at pressures as high as 70 MPa (10,153 psi) in cylinders made from composite material, and storage as a metal hydride (MHX) in tanks filled with porous metal sponge or powder comprised of light group 1 and 2 metals and/or transition metal elements, namely, Titanium (Ti) or Nickel (Ni) [25–30]. Such direct hydrogen storage methods however, are impractical due to the high cost of suitable transition metals Ti and Ni, and moreover, because an infrastructure is needed to supply hydrogen directly in large volume to fill liquid or gas tanks or to saturate or replenish the metal sponge within the storage reservoir inside a motor vehicle, a procedure fraught with all of the

tially to the rise in air pollution [11, 12].

146 Recent Improvements of Power Plants Management and Technology

fuel burning thermal power plants [19, 20].

Despite challenges, hydrogen (H2) which is stored in near limitless quantity in seawater is the only alternative fuel that is more abundant and environmentally cleaner with the potential of having a lower cost than nonrenewable carbon based fossil fuels. We have shown in previous published work that a novel apparatus and method for safely generating hydrogen fuel at the time and point of use from ordinary salinated (sea) or desalinated (fresh) water (H2O) will enable a vehicle range exceeding 300 miles per fueling using direct combustion of the H2 fuel in appropriately configured internal combustion engines of the Otto or Diesel types, which is comparable to the vehicle ranges presently achieved with gasoline or Diesel fuels, while providing a sustainable, closed clean energy cycle [37]. The novel hydrogen generation apparatus enables hydrogen (H2(g)) fuel to be produced on demand in the motor vehicle using a controlled chemical reaction where liquid water (H2O(l)) is made to react with solid sodium (Na(s)) metal reactant to produce hydrogen (H2(g)) gas and sodium hydroxide (NaOH(s)) byproduct according to Eq. (1).

$$2\text{Na}\_{(s)} + 2\text{H}\_2\text{O}\_{(l)} \to \text{H}\_{2(g)} + 2\text{NaOH}\_{(s)}\tag{1}$$

The high purity hydrogen (H2(g)) fuel produced on demand by the novel hydrogen generation apparatus can be used to safely power FCEVs without contaminating the sensitive Pt catalysts present in PEM fuel cells or any other types of catalysts in fuel cells, because the hydrogen is not derived from carbon based fossil fuels, and therefore does not contain even trace amounts of carbon monoxide or sulfur compounds. The seawater reactant can be concentrated to as much as 252.18 grams of sea salt solute per kilogram of seawater solution to provide a fusion temperature TEu ¼ –21.2 �C (251.95 K), that is equivalent to the eutectic temperature of a 23.18% by weight NaCl in NaCl-H2O solution [38, 39]. The concentrated sea salt in seawater solution allows the hydrogen generator to operate reliably over a wide ambient temperature range from –21.2 �C (251.95 K) to 56.7 �C (329.85 K) prevailing in the 48 conterminous states of the U.S.A. [37]. The sodium hydroxide (NaOH) byproduct of the hydrogen generating chemical reaction is stored temporarily within the hydrogen generation apparatus and is recovered during motor vehicle refueling. The NaOH is subsequently reprocessed by electrolysis to recover the sodium (Na) metal for reuse in generating hydrogen fuel.

In this chapter, we describe in detail our company's design approach for constructing a novel, scalable, self-contained electrolytic sodium (Na) metal production plant that uses electric power sourced from the sun. The solar powered electrolytic production plant is meant to form an integral part of a hydrogen fuel, sustainable, closed clean energy cycle in conjunction with the novel, hydrogen generation apparatus, enabling Na metal to be produced cost effectively without negative impact to the environment [37].

### 2. Sodium metal production plant characteristics

For its successful implementation, the hydrogen fuel, sustainable, closed clean energy cycle requires a means of producing quantities of sodium (Na) metal cost effectively on a large scale by electrolysis of sodium hydroxide (NaOH), the latter created as a byproduct of hydrogen (H2(g)) fuel generation inside motor vehicles according to Eq. (1). The electrolysis is performed either on pure sodium hydroxide (NaOH) or on a mixture of NaOH and sea salt, the latter consisting primarily of sodium chloride (NaCl), according to Eqs. (2) and (3) [40–42].

$$4\text{Na}^+ + 4\text{OH}^- \rightarrow 4\text{Na}\_{(l)} + 2\text{H}\_2\text{O}\_{(g)} + \text{O}\_{2(g)}\tag{2}$$

$$\text{2Na}^+ + \text{2Cl}^- \rightarrow \text{2Na}\_{(l)} + \text{Cl}\_{2(g)} \tag{3}$$

The electrical cost of electrolysis can be estimated from the standard reduction potentials of the oxidation and reduction half reactions that occur at the anode and cathode, respectively of the electrolysis cell when implementing Eqs. (2) and (3) [43].

$$\begin{aligned} \text{Cathode (reduction): } \text{ Na}^+\_{\text{(aq)}} + \text{e}^- &\to \text{Na}\_{\text{(s)}} \end{aligned} \qquad \qquad \qquad \begin{aligned} \text{E}^\circ\_\text{r} &= -2.71 \text{ V} \end{aligned} \tag{4}$$

$$\text{Anode (oxidation): }\,\,\text{2Cl}^{-}\_{\text{(aq)}} \rightarrow \text{Cl}\_{2(g)} + \,\text{2e}^{-} \qquad\qquad\qquad\qquad E\_{\text{o}}^{\circ} = -1.36\,\text{V} \tag{5}$$

$$4\text{OH}^-\_{\text{(aq)}} \rightarrow \text{O}\_{2(g)} + 2\text{H}\_2\text{O} + 4\text{e}^- \qquad\qquad E\_\text{o}^\circ = -0.40\text{ V} \tag{6}$$

From Eqs. (4)–(6), the minimum potentials of Eov � ¼ –4.07 V and Eov � ¼ –3.11 V are needed to electrolyze NaCl and NaOH, respectively. These voltages are significantly higher than the potential of Eov � ¼ –1.23 V needed to electrolyze ordinary H2O(l) to produce H2(g) at the cathode and O2(g) at the anode, however, the benefit from not having to store volatile H2(g) in very large industrial quantities and to transport it between the production plants and refueling stations, outweighs the added electrical cost of producing the solid Na(s) metal.

In the United States of America, the only clean renewable source of energy available in sufficient abundance to implement Eqs. (2) and (3) on a large scale is the radiant energy from the sun that illuminates vast tracts of flat, arid, desert land in West Texas, New Mexico, Arizona and Southern California. The weather in the southwestern U.S.A. is mostly warm and arid with high solar irradiance all year and therefore, constitutes the ideal location for constructing scalable, self-contained solar powered electrolytic sodium (Na) metal production plant units by the thousands [44–48]. Each sodium (Na) metal production plant has to be capable of operating autonomously as a self-contained factory, requiring minimal maintenance and resources. The diagram showing all of the material and energy inputs and outputs of the self-contained sodium (Na) metal production plant is presented in Figure 1.

Figure 1. Self-contained sodium (Na) metal production plant operating resources diagram.

In this chapter, we describe in detail our company's design approach for constructing a novel, scalable, self-contained electrolytic sodium (Na) metal production plant that uses electric power sourced from the sun. The solar powered electrolytic production plant is meant to form an integral part of a hydrogen fuel, sustainable, closed clean energy cycle in conjunction with the novel, hydrogen generation apparatus, enabling Na metal to be produced cost effectively

For its successful implementation, the hydrogen fuel, sustainable, closed clean energy cycle requires a means of producing quantities of sodium (Na) metal cost effectively on a large scale by electrolysis of sodium hydroxide (NaOH), the latter created as a byproduct of hydrogen (H2(g)) fuel generation inside motor vehicles according to Eq. (1). The electrolysis is performed either on pure sodium hydroxide (NaOH) or on a mixture of NaOH and sea salt, the latter

The electrical cost of electrolysis can be estimated from the standard reduction potentials of the oxidation and reduction half reactions that occur at the anode and cathode, respectively of the

electrolyze NaCl and NaOH, respectively. These voltages are significantly higher than the

and O2(g) at the anode, however, the benefit from not having to store volatile H2(g) in very large industrial quantities and to transport it between the production plants and refueling stations,

In the United States of America, the only clean renewable source of energy available in sufficient abundance to implement Eqs. (2) and (3) on a large scale is the radiant energy from the sun that illuminates vast tracts of flat, arid, desert land in West Texas, New Mexico, Arizona and Southern California. The weather in the southwestern U.S.A. is mostly warm and arid with high solar irradiance all year and therefore, constitutes the ideal location for constructing scalable, self-contained solar powered electrolytic sodium (Na) metal production plant units by the thousands [44–48]. Each sodium (Na) metal production plant has to be

<sup>ð</sup>aq<sup>Þ</sup> <sup>þ</sup> <sup>e</sup>� ! Naðs<sup>Þ</sup> <sup>E</sup>�

<sup>ð</sup>aq<sup>Þ</sup> ! Cl2ðg<sup>Þ</sup> <sup>þ</sup> 2e� <sup>E</sup>�

<sup>ð</sup>aq<sup>Þ</sup> ! O2ðg<sup>Þ</sup> <sup>þ</sup> 2H2O <sup>þ</sup> 4e� <sup>E</sup>�

� ¼ –1.23 V needed to electrolyze ordinary H2O(l) to produce H2(g) at the cathode

� ¼ –4.07 V and Eov

4Na<sup>þ</sup> þ 4OH� ! 4Naðl<sup>Þ</sup> þ 2H2Oðg<sup>Þ</sup> þ O2ðg<sup>Þ</sup> (2)

2Na<sup>þ</sup> þ 2Cl� ! 2Naðl<sup>Þ</sup> þ Cl2ðg<sup>Þ</sup> (3)

<sup>r</sup> ¼ �2:71 V (4)

<sup>o</sup> ¼ �1:36 V (5)

<sup>o</sup> ¼ �0:40 V (6)

� ¼ –3.11 V are needed to

consisting primarily of sodium chloride (NaCl), according to Eqs. (2) and (3) [40–42].

without negative impact to the environment [37].

148 Recent Improvements of Power Plants Management and Technology

2. Sodium metal production plant characteristics

electrolysis cell when implementing Eqs. (2) and (3) [43].

4OH�

outweighs the added electrical cost of producing the solid Na(s) metal.

Cathode ðreductionÞ: Na<sup>þ</sup>

potential of Eov

Anode ðoxidationÞ: 2Cl�

From Eqs. (4)–(6), the minimum potentials of Eov

In Figure 1, electric power for the Na metal production plant is produced using photovoltaic (PV) device panels spatially arrayed and electrically interconnected on a vertical tower structure that maximizes the use of scarce real estate or land area. Up to N<sup>P</sup> ¼ 30,000 PV panels, each having an active area <sup>A</sup><sup>P</sup> <sup>¼</sup> 1 m2 are mechanically assembled onto the tower, yielding a total PV device panel array active area given as <sup>A</sup>PA <sup>¼</sup> <sup>N</sup><sup>P</sup> · <sup>A</sup><sup>P</sup> <sup>¼</sup> (30,000 PV panels) · (1 m2 ) ¼ 30,000 m2 . Sodium hydroxide (NaOH) or a mixture of NaOH and NaCl recovered from motor vehicle hydrogen generators during refueling, must be supplied to the electrolytic cells to replenish the consumed reactants. When operating the hydrogen generation apparatus in warm tropical climates, NaOH exclusively can be recovered from motor vehicle hydrogen generators during refueling because desalinated (fresh) water (H2O(l)) can safely be used as a reactant without risk for it to freeze. Sodium (Na) metal is produced at the electrolytic cell cathode, and steam (H2O(g)), oxygen (O2(g)) gas and some chlorine (Cl2(g)) gas are produced at the cell anode, the latter resulting from electrolysis of the NaCl in sea salt. The H2O(g) and O2(g) can be released directly to the atmosphere while the Cl2(g) must be collected, condensed to a liquid and stored in bottles, for subsequent sale to customers that consume chlorine including the paper and polymer (plastic) manufacturing industries [49]. It is also possible to collect and condense the steam generated at the anode and use the liquid water (H2O(l)) for crop irrigation in arid, desert environments where water resources are limited. The layout of the self-contained sodium (Na) metal production plant is shown in Figure 2.

The self-contained sodium (Na) metal production plant shown in Figure 2, consists of a solar tower that comprises a photovoltaic (PV) device panel array active area given as APA ¼ 30,000 m2 . It also consists of a prefabricated Quonset or Q-type metal building having a semicircular cross section, assembled onto a concrete pad foundation that houses electrical switch gear, voltage step down DC-DC converters, the sodium hydroxide (NaOH) electrolytic cells, sodium (Na) metal packaging unit and chlorine (Cl2) gas separation and bottling unit. A control room permits monitoring the operation of the plant. Two above ground storage tanks are shown located outside of the Q-type metal building with one on either side, for storing aqueous sodium hydroxide (NaOH(aq)) solution. Each above ground storage reservoir has a liquid volume capacity of VNaOH(aq) ¼ 37,850 L (10,000 Gal), and is used to replenish the NaOH in the electrolytic cells after the reactant in the cells has been consumed by electrolysis.

Figure 2. Layout of the self-contained sodium (Na) metal production plant (NOT TO SCALE).

Many factors influence the production yield of sodium (Na) metal during a normal day of plant operation. The two most important factors include the power conversion efficiency of the PV device panels and the magnitude and duration of solar irradiance incident on the PV panels. Other factors affecting the Na metal yield include the power conversion efficiency of the voltage step down DC-DC converter and the efficiency of the electrolytic cell in recovering the Na metal from fused NaOH(l) or from a mixture of fused NaOH(l) and NaCl(l). The efficiency of the voltage step down DC-DC converter depends mainly on how much power is dissipated or lost in the solid state transistors as a result of high frequency on-off switching. The efficiency of the electrolytic cell in this work is viewed in terms of the number of electrons from the electrolytic cell current ICELL, flowing through an electrolytic cell actually needed to produce an atom of Na metal. An alternate definition of the electrolytic cell efficiency might consider the theoretical electric power required to be supplied to the electrolytic cell to produce one mole of Na metal as calculated from the known Gibbs free energy of NaOH(l), divided by the actual power required to be supplied to the electrolytic cell to produce one mole of the Na metal, a definition that we do not consider here. Ideally, each electron flowing through the electric circuit comprising the electrolytic cell should produce one atom of Na metal. In the analysis that follows, it will be assumed that the voltage step down DC-DC converters and the electrolytic cells have efficiencies ηDC-DC ¼ 100% and ηCELL ¼ 100%, respectively.

#### 2.1. Scalable photovoltaic tower concept

The solar tower comprising the photovoltaic (PV) device panel array supplies electric power to the sodium (Na) metal producing electrolytic cells. The amount of electric energy supplied by the solar tower is a key determinant of the quantity of Na metal that can be electrochemically separated from the NaOH reactant. The energy conversion efficiency of the photovoltaic (PV) device panels is therefore a critical parameter for determining the amount of Na metal that can be produced by the self-contained sodium (Na) metal production plant. Ideally, the photovoltaic (PV) device panels of the solar tower should have maximum optical to electric energy conversion efficiency approaching the thermodynamic limit ηPVmax ¼ 93% [50, 51]. According to the Shockley-Queisser theory, such a high conversion efficiency requires PV devices comprising manifold semiconductor junctions [52, 50]. Contemporary, commercially available single junction, monocrystalline silicon photovoltaic (PV) devices attain energy conversion efficiencies ranging between ηPV ¼ 15–18% for front-illuminated silicon devices and up to ηPV ¼ 21.5% for backilluminated silicon devices as summarized in Table 1.


a PV panel performance data sourced from respective product datasheets. ASTM AM 1.5G solar irradiance of 1000 W/m2 .

Table 1. Performance of commercial PV device panels in 2016<sup>a</sup> .

Many factors influence the production yield of sodium (Na) metal during a normal day of plant operation. The two most important factors include the power conversion efficiency of the PV device panels and the magnitude and duration of solar irradiance incident on the PV panels. Other factors affecting the Na metal yield include the power conversion efficiency of the voltage step down DC-DC converter and the efficiency of the electrolytic cell in recovering the Na metal from fused NaOH(l) or from a mixture of fused NaOH(l) and NaCl(l). The efficiency of the voltage step down DC-DC converter depends mainly on how much power is dissipated or lost in the solid state transistors as a result of high frequency on-off switching. The efficiency of the electrolytic cell in this work is viewed in terms of the number of electrons from the electrolytic cell current ICELL, flowing through an electrolytic cell actually needed to produce an atom of Na metal. An alternate definition of the electrolytic cell efficiency might consider the theoretical electric power required to be supplied to the electrolytic cell to produce one mole of Na metal as calculated from the known Gibbs free energy of NaOH(l), divided by the actual power required to be supplied to the electrolytic cell to produce one mole of the Na metal, a definition that we do not consider here. Ideally, each electron flowing through the electric circuit comprising the electrolytic cell should produce one atom of Na metal. In the analysis that follows, it will be assumed that the voltage step down DC-DC converters and the electrolytic cells have efficiencies ηDC-DC ¼ 100% and ηCELL ¼ 100%, respectively.

Figure 2. Layout of the self-contained sodium (Na) metal production plant (NOT TO SCALE).

150 Recent Improvements of Power Plants Management and Technology

The solar tower comprising the photovoltaic (PV) device panel array supplies electric power to the sodium (Na) metal producing electrolytic cells. The amount of electric energy supplied by the solar tower is a key determinant of the quantity of Na metal that can be electrochemically

2.1. Scalable photovoltaic tower concept

In Table 1, the PV device panels from the Sunpower manufacturing company stand out as having the highest efficiency, due to the use of back-illumination of the silicon device layer. Multijunction devices that offer higher efficiencies remain at a research and development stage and have not reached a level of maturity or cost effectiveness to be ready for release as commercial PV panel products [53]. Our company, AG STERN, LLC is researching the development of advanced, high efficiency PV devices based on novel, very high transmittance, back-illuminated, silicon-on-sapphire semiconductor substrates expected capable of transmitting 93.7% of the total solar irradiance into the semiconductor device layer and therefore capable of achieving an energy conversion efficiency ηPV ¼ 90% with proper engineering of the semiconductor photon absorbing layers and PV device structure [54–57].

The solar tower providing power to the electrolytic cells must be scalable in electric power output, and thus capable of allowing PV device panels to be replaced as higher efficiency ones become available, without affecting the overall operation of the self-contained Na metal production plant in any way, other than increasing the Na metal yield. Since the highest performing PV device panels currently offered commercially are listed in Table 1, it is possible to calculate the expected power output of the solar tower comprising a PV device panel array active area of <sup>A</sup>PA <sup>¼</sup> 30,000 m2 , under ASTM direct normal air mass (AM) 1.5D standard terrestrial solar spectral irradiance with a total irradiance IrrAM1.5D <sup>¼</sup> 887 W/m<sup>2</sup> , as a function of the PV device efficiency, as shown in Figure 3.

Figure 3. Electric power output for a solar tower PV device panel array with area <sup>A</sup>PA <sup>¼</sup> 30,000 m2 under ASTM direct normal AM 1.5D standard terrestrial solar spectral irradiance as a function of the PV device efficiency.

It is clear in Figure 3, that single junction, monocrystalline silicon PV devices having an efficiency ηPV ¼ 21.5% can be arrayed to generate electric power given as PST ¼ 5.72 MW for electrolysis. Once ηPV ¼ 90% efficient PV device panels will become available, the electric power output can be increased to a substantial PST ¼ 23.9 MW, entailing that just 50 of the self-contained sodium (Na) metal production plants can generate peak power given as PST-50 ¼ 50 · 23.9 MW ¼ 1195 MW, matching the power generating capacity of a large commercial nuclear power station.

#### 2.2. Solar irradiance conditions

The electric power output of the solar tower comprising photovoltaic (PV) device panels depends on the magnitude and duration of solar irradiance incident on the PV panels, in addition to the PV device energy conversion efficiency described in Section 2.1. It can be assumed that the PV device energy conversion efficiency is constant and might only decrease in value slowly over time [58]. In contrast, the solar irradiance incident on the PV device panels can vary on a daily basis and is determined by two principal factors, namely, the solar geometry and prevailing atmospheric or meteorological conditions.

The sun is effectively a large hydrogen (H2) fusion reactor, spherical in shape with a radius <sup>R</sup>sun <sup>¼</sup> 6.96 · 108 m located at a mean distance from the earth given as <sup>r</sup>sun-m <sup>¼</sup> 1.496 · 1011 <sup>m</sup> [59, 60]. The total power output of the sun is given as <sup>P</sup>sun <sup>¼</sup> 3.8 · 1026 W that radiates isotropically in all directions, resulting in a surface temperature of the solar black body Tsun ¼ 5800 K. Of the total solar power output, earth receives only <sup>P</sup>earth <sup>¼</sup> 1.7 · <sup>10</sup><sup>17</sup> W which if converted to electric power, vastly exceeds the earth's energy needs [61]. The sun therefore constitutes an excellent potential source of clean radiant energy to be harnessed on earth.

The solar geometry has a key role in determining how much radiant energy from the sun will be incident on the PV devices located on earth. To understand how the solar geometry influences the solar irradiance at the earth's surface, it will be assumed that earth follows a stable elliptical orbit around the sun described by Eq. (7) [59].

$$r\_{\text{sun}}(\theta) = \frac{a(1 - e^2)}{1 + e \cdot \cos(\theta)}\tag{7}$$

$$\text{Periodion: } r\_{\text{sun}}(0^\circ) = a(1 - e)$$

$$\text{Aphelion: } r\_{\text{sun}}(180^\circ) = a(1 + e)$$

In Eq. (7), the distance rsun, represents the distance from the center of the sun to the center of the earth and the angle θ, represents the angle between the present position of the earth in orbit around the sun, and the perihelion position when it is closest to the sun. The eccentricity e ¼ 0.01673, describes the shape of the elliptical orbit of the earth around the sun and the value a, represents the length of the semi-major axis of the orbit defined as the mean distance from the center of the sun to the center of the earth given as a ¼ {rsun(0�) þ rsun(180�)} / 2 ¼ <sup>r</sup>sun-m <sup>¼</sup> 1.496 · <sup>10</sup><sup>11</sup> m. The period of earth's elliptical rotation around the sun is approximately Tes ¼ 365.24 days and the period of rotation around its own axis is approximately Tea ¼ 86,400 seconds or 24 hours, the latter known as a mean solar day [62–66]. The earth's angular velocity about its own axis is given as <sup>ω</sup>ea <sup>¼</sup> 7.292115 · <sup>10</sup>�<sup>5</sup> rad/sec, although the rotation is slowing over time [67]. The obliquity or tilt of the earth's axis of rotation with respect to a line perpendicular to the plane of its elliptical orbit around the sun is given as ε ¼ 23.44�, although slight precession and nutation of the axis exists [60]. The plane of the sun passes through the center of the sun and remains parallel to the earth's equator during the elliptical orbit. The summer and winter solstices occur approximately on June 21 and December 21, respectively. The spring and fall equinoxes occur approximately on March 21 and September 21, respectively, and are characterized by the length of day being equal to the length of night and the earth's equator coinciding with the plane of the sun.

It is clear in Figure 3, that single junction, monocrystalline silicon PV devices having an efficiency ηPV ¼ 21.5% can be arrayed to generate electric power given as PST ¼ 5.72 MW for electrolysis. Once ηPV ¼ 90% efficient PV device panels will become available, the electric power output can be increased to a substantial PST ¼ 23.9 MW, entailing that just 50 of the self-contained sodium (Na) metal production plants can generate peak power given as PST-50 ¼ 50 · 23.9 MW ¼ 1195 MW, matching the power generating capacity of a large commercial

Figure 3. Electric power output for a solar tower PV device panel array with area <sup>A</sup>PA <sup>¼</sup> 30,000 m2 under ASTM direct

normal AM 1.5D standard terrestrial solar spectral irradiance as a function of the PV device efficiency.

152 Recent Improvements of Power Plants Management and Technology

The electric power output of the solar tower comprising photovoltaic (PV) device panels depends on the magnitude and duration of solar irradiance incident on the PV panels, in addition to the PV device energy conversion efficiency described in Section 2.1. It can be assumed that the PV device energy conversion efficiency is constant and might only decrease in value slowly over time [58]. In contrast, the solar irradiance incident on the PV device panels can vary on a daily basis and is determined by two principal factors, namely, the solar

The sun is effectively a large hydrogen (H2) fusion reactor, spherical in shape with a radius <sup>R</sup>sun <sup>¼</sup> 6.96 · 108 m located at a mean distance from the earth given as <sup>r</sup>sun-m <sup>¼</sup> 1.496 · 1011 <sup>m</sup> [59, 60]. The total power output of the sun is given as <sup>P</sup>sun <sup>¼</sup> 3.8 · 1026 W that radiates isotropically in all directions, resulting in a surface temperature of the solar black body Tsun ¼ 5800 K. Of the total solar power output, earth receives only <sup>P</sup>earth <sup>¼</sup> 1.7 · <sup>10</sup><sup>17</sup> W which if converted to electric power, vastly exceeds the earth's energy needs [61]. The sun therefore constitutes an excellent potential source of clean radiant energy to be harnessed on earth.

The solar geometry has a key role in determining how much radiant energy from the sun will be incident on the PV devices located on earth. To understand how the solar geometry

geometry and prevailing atmospheric or meteorological conditions.

nuclear power station.

2.2. Solar irradiance conditions

The solar declination angle δ, is the angle made by a ray of the sun (passing through the center of the earth and the center of the sun), and the equatorial plane of the earth. The solar declination angle has a range given as -23.44� ≤ δ ≤ þ23.44�. On the summer solstice day when δ ¼ þ23.44�, the sun shines most directly on the earth's latitude ϕT-CAN ¼ þ23.44�, known as the Tropic of Cancer. On the winter solstice day when δ ¼ –23.44�, the sun shines most directly on the earth's latitude ϕT-CAP ¼ –23.44�, known as the Tropic of Capricorn. The longitude at the Greenwich prime meridian λPM ¼ 0� [68]. The solar declination angle can be calculated using a geocentric reference frame with the sun located on a celestial sphere and earth located at the center of the celestial sphere, according to Eq. (8) [60].

$$
\delta = \arcsin(\sin \lambda\_{\text{e}} \cdot \sin \varepsilon) \tag{8}
$$

In Eq. (8), λ<sup>e</sup> represents the ecliptic longitude of the sun, that is the sun's angular position along its apparent orbit in the plane of the ecliptic. The plane of the ecliptic is tilted by the obliquity of the ecliptic angle ε ¼ 23.44� representing the angle between the plane of the ecliptic and the plane of the celestial equator, the latter being earth's equator projected onto the celestial sphere. The ecliptic intersects the equatorial plane at the points corresponding to the spring and fall equinoxes occurring approximately on March 21 and September 21, respectively with λ<sup>e</sup> ¼ 0� at the March 21 equinox. The Eq. (8) is usually formulated in terms of the day number in a year to account approximately for the elliptic nature of earth's orbit around the sun, as given by Eq. (9) [69].

$$\begin{aligned} \delta &= [0.006918 - 0.399912 \cos(\Gamma) + 0.070257 \sin(\Gamma) - 0.006758 \cos(2\Gamma) + 0.000907 \sin(2\Gamma) \\ &- 0.002697 \cos(3\Gamma) + 0.00148 \sin(3\Gamma)](180/\pi) \end{aligned} \tag{9}$$

In Eq. (9), the solar declination angle δ, is given as a function of the day angle Γ ¼ 2π(Nday � 1) / 365, with day number <sup>N</sup>day, in a year where <sup>N</sup>day <sup>¼</sup> 1 corresponds to January 1st. The Eq. (9) provides sufficient accuracy due to the relatively small eccentricity value e in Eq. (7). The Eq. (9) shows that the solar illumination of the earth's surface throughout the year occurs most directly between tropical latitudes, namely, the Tropic of Cancer (δ ≈ þ23.44�) and the Tropic of Capricorn (δ ≈ �23.44�). At the time of the vernal and autumnal equinoxes, the earth's equator is located in the plane of the sun and every location on earth receives 12 hours of sunlight.

The PV devices can only generate significant electric power during daylight hours therefore, it is essential to verify that a specific geographic location on earth as defined by the latitude ϕ, and longitude λ, will receive sufficient hours of high incident solar irradiance during the year, before constructing the self-contained sodium (Na) metal production plant. To accurately describe the sun's position in the sky from sunrise to sunset as well as the resulting number of daylight hours, the apparent solar time (AST) is used to express the time of the day. The sun crosses the meridian of the observer at the local solar noon when AST ¼ 12, however, the latter time does not coincide exactly with the 12:00 noon time of the locality of the observer. A conversion between the local standard time (LST) and the AST can be made by using equation of time (ET) and longitude corrections. The equation of time accounts for the length of the day variation as a result of the slight eccentricity of the earth's orbit around the sun as well as the tilt of the earth's axis of rotation with respect to a line perpendicular to the plane of its elliptical orbit around the sun. The ET is given by Eq. (10) and has the dimension of minutes [69, 70].

$$\text{ET} = [0.000075 + 0.001868 \cos(\Gamma) - 0.032077 \sin(\Gamma) - 0.014615 \cos(2\Gamma) - 0.04089 \sin(2\Gamma)](229.18) \tag{10}$$

The relation between the AST and LST can be made using Eq. (11) that includes the ET and longitude correction which accounts for the fact that the sun traverses 1� of longitude in 4 minutes.

$$\text{AST} = \text{LST} + \text{ET} + (4 \,\text{min}/\text{deg})(\lambda\_\text{S} - \lambda\_\text{L}) - \text{DST} \tag{11}$$

In Eq. (11), the local longitude λL, represents the exact longitude value for the location relative to the Greenwich prime meridian of λPM ¼ 0�. The standard longitude λS, is calculated from λ<sup>L</sup> as λ<sup>S</sup> ¼ 15� · (λ<sup>L</sup> / 15�) where the quantity in the parentheses is rounded to an integer value. In Eq. (11), if λ<sup>L</sup> is east of λ<sup>S</sup> then the longitude correction is positive and if λ<sup>L</sup> is west of λ<sup>S</sup> then the longitude correction is negative, while DST ¼ 0 minutes or 60 minutes depending on whether daylight savings time applies. If in effect, DST occurs from March until November.

The hour angle h, represents the angular distance in degrees between the longitude λ, of the observer and the longitude whose plane contains the sun, and is calculated according to Eq. (12) [60].

and fall equinoxes occurring approximately on March 21 and September 21, respectively with λ<sup>e</sup> ¼ 0� at the March 21 equinox. The Eq. (8) is usually formulated in terms of the day number in a year to account approximately for the elliptic nature of earth's orbit around the sun, as

δ ¼½0:006918 � 0:399912cosðΓÞ þ 0:070257sinðΓÞ � 0:006758cosð2ΓÞ þ 0:000907sinð2ΓÞ

In Eq. (9), the solar declination angle δ, is given as a function of the day angle Γ ¼ 2π(Nday � 1) / 365, with day number <sup>N</sup>day, in a year where <sup>N</sup>day <sup>¼</sup> 1 corresponds to January 1st. The Eq. (9) provides sufficient accuracy due to the relatively small eccentricity value e in Eq. (7). The Eq. (9) shows that the solar illumination of the earth's surface throughout the year occurs most directly between tropical latitudes, namely, the Tropic of Cancer (δ ≈ þ23.44�) and the Tropic of Capricorn (δ ≈ �23.44�). At the time of the vernal and autumnal equinoxes, the earth's equator is located in the plane of the sun and every location on earth receives 12 hours of sunlight.

The PV devices can only generate significant electric power during daylight hours therefore, it is essential to verify that a specific geographic location on earth as defined by the latitude ϕ, and longitude λ, will receive sufficient hours of high incident solar irradiance during the year, before constructing the self-contained sodium (Na) metal production plant. To accurately describe the sun's position in the sky from sunrise to sunset as well as the resulting number of daylight hours, the apparent solar time (AST) is used to express the time of the day. The sun crosses the meridian of the observer at the local solar noon when AST ¼ 12, however, the latter time does not coincide exactly with the 12:00 noon time of the locality of the observer. A conversion between the local standard time (LST) and the AST can be made by using equation of time (ET) and longitude corrections. The equation of time accounts for the length of the day variation as a result of the slight eccentricity of the earth's orbit around the sun as well as the tilt of the earth's axis of rotation with respect to a line perpendicular to the plane of its elliptical orbit around the sun. The ET is given by Eq. (10) and has the dimension of minutes [69, 70].

ET ¼ ½0:000075 þ 0:001868cosðΓÞ � 0:032077sinðΓÞ � 0:014615cosð2ΓÞ � 0:04089sinð2ΓÞ�ð229:18Þ

The relation between the AST and LST can be made using Eq. (11) that includes the ET and longitude correction which accounts for the fact that the sun traverses 1� of longitude in 4

In Eq. (11), the local longitude λL, represents the exact longitude value for the location relative to the Greenwich prime meridian of λPM ¼ 0�. The standard longitude λS, is calculated from λ<sup>L</sup> as λ<sup>S</sup> ¼ 15� · (λ<sup>L</sup> / 15�) where the quantity in the parentheses is rounded to an integer value. In Eq. (11), if λ<sup>L</sup> is east of λ<sup>S</sup> then the longitude correction is positive and if λ<sup>L</sup> is west of λ<sup>S</sup> then the longitude correction is negative, while DST ¼ 0 minutes or 60 minutes depending on whether daylight savings time applies. If in effect, DST occurs from March until November.

AST ¼ LST þ ET þ ð4 min=degÞðλ<sup>S</sup> � λLÞ � DST (11)

(10)

� <sup>0</sup>:002697cosð3ΓÞ þ <sup>0</sup>:00148sinð3ΓÞ�ð180=π<sup>Þ</sup> (9)

given by Eq. (9) [69].

154 Recent Improvements of Power Plants Management and Technology

minutes.

$$h = 15 \cdot (\text{AST} - 12) \tag{12}$$

In Eq. (12), the multiplier of 15� arises because the earth rotates around its own axis 15� in 1 hour, and the AST has a value 0 < AST < 24 hours. When the sun reaches its maximum angle of elevation at the longitude of the observer, it corresponds to the local solar noon time where h ¼ 0�. The solar elevation angle αs, above the observer's horizon and its complement, the solar zenith angle θsz, can be calculated according to Eq. (13) [60].

$$\sin(\alpha\_s) = \cos(\theta\_{sz}) = \sin(\varphi)\sin(\delta) + \cos(\varphi)\cos(\delta)\cos(h) \tag{13}$$

The solar azimuth angle γs, in the horizontal plane of the observer is given by Eq. (14) [60].

$$\cos(\nu\_s) = \frac{\sin(\delta)\cos(\rho) - \cos(\delta)\sin(\rho)\cos(h)}{\cos(a\_s)}\tag{14}$$

The Eq. (14) uses the convention of the solar azimuth angle defined as positive clockwise from north meaning that east corresponds to 90�, south corresponds to 180� and west corresponds to 270�. The solar azimuth angle provided by Eq. (14) should be interpreted as 0� ≤ γ<sup>s</sup> ≤ 180� when h < 0�, meaning the sun is located east of the observer, and interpreted as 180� ≤ γ<sup>s</sup> ≤ 360� when h > 0�, meaning the sun is located west of the observer. The sunrise equation allows to calculate the number of hours of sunlight per day that depends on the solar declination angle δ, and on latitude ϕ, as shown in Eq. (15), which is derived from Eq. (13) by setting α<sup>s</sup> ¼ 0� and solving for h.

$$H\_{\rm ss} = -H\_{\rm sr} = \frac{1}{15} \arccos[-\tan(\varphi)\tan(\delta)]\tag{15}$$

In Eq. (15), the hour angle h, has a value �180� < h < 0� at sunrise and 0� < h < þ180� at sunset, and when divided by 15�/hour, yields the number of hours before and after the local solar noon time (AST ¼ 12) when h ¼ 0�, corresponding to sunrise Hsr, and sunset Hss, respectively. Using Eqs. (9) and (15), it is possible to calculate for any geographic location on earth the number of daylight hours on any specific day of the year. The maximum angle of solar elevation αs-max, above the observer's horizon that occurs at the local solar noon time is calculated according to Eqs. (16) and (17) that are derived from Eq. (13) by setting h ¼ 0�.

$$
\alpha\_{s-\text{max}} = 90^{\circ} - (\varphi - \delta) \tag{16}
$$

$$
\alpha\_{s-\text{max}} = 90^{\circ} + (\varphi - \delta) \tag{17}
$$

The Eq. (16) is applicable for the northern hemisphere while Eq. (17) is applicable for the southern hemisphere. Formulations for the solar declination, elevation, azimuth and zenith angles which account with greater accuracy for the elliptic nature of earth's orbit around the sun exist in the scientific literature often as part of solar position algorithms however, they can be rather complex [71–76]. The air mass can be calculated as a function of the true solar zenith angle θsz, given by Eq. (13), considering the effect of atmospheric refraction, according to Eq. (18) [77].

$$\text{AM} = \frac{1.002432 \text{cos}^2(\theta\_{\text{sx}}) + 0.148386 \cos(\theta\_{\text{sx}}) + 0.0096467}{\cos^3(\theta\_{\text{sx}}) + 0.149864 \cos^2(\theta\_{\text{sx}}) + 0.0102963 \cos(\theta\_{\text{sx}}) + 0.000303978} \tag{18}$$

The direct normal total (spectrally integrated) solar irradiance as a function of the air mass (AM) and atmospheric conditions that include the effects of elevation, can be calculated using the Parameterization Model C developed by Iqbal, given in Eq. (19) [78–80].

$$Irr\_n = 0.9751 \cdot E\_0 \cdot Irr\_0 \cdot \pi\_r \pi\_o \pi\_g \pi\_w \pi\_a \tag{19}$$

In Eq. (19), Irr<sup>0</sup> <sup>¼</sup> 1367 W/m<sup>2</sup> , represents the solar constant or total irradiance of the AM 0 standard solar spectral irradiance in space prior to attenuation by earth's atmosphere and the factor 0.9751 is included because the spectral interval of 0.3–3.0 μm is used by the detailed SOLTRAN model from which the Parameterization Model C is derived. The factor E0, represents the effect of the eccentricity of earth's orbit around the sun on the solar constant Irr0, due to the periodically varying distance between the earth and sun, as given by Eq. (20) [69].

$$\begin{aligned} E\_0 &= \left(r\_{\text{sun}-m}/r\_{\text{sun}}\right)^2 = 1.000110 + 0.034221 \cos(\varGamma) + 0.001280 \sin(\varGamma) \\\\ &+ 0.000719 \cos(2\varGamma) + 0.000077 \sin(2\varGamma) \end{aligned} \tag{20}$$

In Eq. (19), the symbol τ<sup>r</sup> represents transmittance by Rayleigh scattering, τ<sup>o</sup> represents transmittance by ozone, τ<sup>g</sup> represents transmittance by uniformly mixed gases, τ<sup>w</sup> represents transmittance by water vapor and τ<sup>a</sup> represents transmittance by aerosols. The expressions for τr, τo, τg, τw, τ<sup>a</sup> are given in Eqs. (21–25).

$$\tau\_{\mathbf{r}} = \exp[-0.0903 \cdot \text{AM}\_{\mathbf{a}}^{0.84} (1.0 + \text{AM}\_{\mathbf{a}} - \text{AM}\_{\mathbf{a}}^{1.01})] \tag{21}$$

$$\tau\_o = 1 - \left[ 0.1611 \cdot \text{U}\_3 (1.0 + 139.48 \cdot \text{U}\_3)^{-0.305} - 0.002715 \cdot \text{U}\_3 (1.0 + 0.044 \text{U}\_3 + 0.0003 \text{U}\_3^2)^{-1} \right] \tag{22}$$

$$
\pi\_{\rm g} = \exp[-0.0127 \cdot \text{AM}\_{\rm a}^{0.26}] \tag{23}
$$

$$
\pi\_{\rm w} = 1 - 2.4959 \cdot \mathcal{U}\_1 [(1.0 + 79.034 \mathcal{U}\_1)^{0.6828} + 6.385 \mathcal{U}\_1]^{-1} \tag{24}
$$

$$\tau\_{\mathbf{a}} = \exp[-k\_{\mathbf{a}}^{0.873}(1.0 + k\_{\mathbf{a}} - k\_{\mathbf{a}}^{0.7088})\mathbf{A}\mathbf{M}\_{\mathbf{a}}^{0.9108}] \tag{25}$$

In Eqs. (21), (23) and (25), AMa represents the air mass at the actual atmospheric pressure P and is given as AMa ¼ AM · (P / P0) where AM, calculated using Eq. (18), corresponds to standard atmospheric pressure P<sup>0</sup> ¼ 101325 Pa. The actual atmospheric pressure P, can be calculated using the isothermal atmosphere formula in Eq. (26).

$$P = P\_0 \cdot \exp\left[\frac{-\mathcal{g}\_0 \cdot M\_{\text{air}} \cdot h\_{\text{PV}}}{R\_{\text{g}}T}\right] \tag{26}$$

The Eq. (26) can be used to calculate the atmospheric pressure P in pascals (Pa) at an elevation hPV, in meters (m) above mean sea level, assuming a constant ambient atmospheric temperature T in Kelvin (K), over the difference in elevations. The molar mass of air Mair ¼ 0.028964 kg/mol, earth's gravitational acceleration near the surface <sup>g</sup><sup>0</sup> <sup>¼</sup> 9.80665 m/sec2 , and the universal gas constant R<sup>g</sup> ¼ 8.3144621 J/K∙mol [43, 81, 82]. In Eq. (22), U<sup>3</sup> ¼ l<sup>o</sup> · AM, where l<sup>o</sup> represents the vertical ozone layer thickness in centimeters (cm) at normal temperature and surface pressure (NTP), and AM is given by Eq. (18). The vertical ozone layer thickness is assumed in the present work to have a mean annual value l<sup>o</sup> ¼ 0.35 cm (NTP). In Eq. (24), U<sup>1</sup> ¼ w<sup>0</sup> ðP=101325Þ <sup>0</sup>:<sup>75</sup> <sup>ð</sup>273=T<sup>Þ</sup> <sup>0</sup>:<sup>5</sup> · AM, where w' represents the precipitable water vapor thickness in centimeters (cm) under the actual atmospheric conditions with pressure P in pascals (Pa), and temperature T in Kelvin (K). In Eq. (25), k<sup>a</sup> represents the aerosol optical thickness and has a range 0 < k<sup>a</sup> < 1, depending on how clear or aerosol saturated the atmosphere is. The value of k<sup>a</sup> is generally measured at the optical wavelengths of 380 nm and 500 nm due to low ozone absorption and calculated using the formula, k<sup>a</sup> ¼ 0:2758 � k<sup>a</sup>jλ¼380 nm þ 0:35 � k<sup>a</sup>jλ¼500 nm. The aerosol optical thickness can vary daily, however, it is assumed in the present work to have a mean annual value k<sup>a</sup> ¼ 0.1 reflecting clear days.

[71–76]. The air mass can be calculated as a function of the true solar zenith angle θsz, given by

The direct normal total (spectrally integrated) solar irradiance as a function of the air mass (AM) and atmospheric conditions that include the effects of elevation, can be calculated using

0 standard solar spectral irradiance in space prior to attenuation by earth's atmosphere and the factor 0.9751 is included because the spectral interval of 0.3–3.0 μm is used by the detailed SOLTRAN model from which the Parameterization Model C is derived. The factor E0, represents the effect of the eccentricity of earth's orbit around the sun on the solar constant Irr0, due to the periodically varying distance between the earth and sun, as given by Eq. (20) [69].

In Eq. (19), the symbol τ<sup>r</sup> represents transmittance by Rayleigh scattering, τ<sup>o</sup> represents transmittance by ozone, τ<sup>g</sup> represents transmittance by uniformly mixed gases, τ<sup>w</sup> represents transmittance by water vapor and τ<sup>a</sup> represents transmittance by aerosols. The expressions

<sup>τ</sup><sup>g</sup> <sup>¼</sup> exp½�0:<sup>0127</sup> � AM0:<sup>26</sup>

<sup>a</sup> <sup>ð</sup>1:<sup>0</sup> <sup>þ</sup> <sup>k</sup><sup>a</sup> � <sup>k</sup><sup>0</sup>:<sup>7088</sup>

In Eqs. (21), (23) and (25), AMa represents the air mass at the actual atmospheric pressure P and is given as AMa ¼ AM · (P / P0) where AM, calculated using Eq. (18), corresponds to standard atmospheric pressure P<sup>0</sup> ¼ 101325 Pa. The actual atmospheric pressure P, can be

<sup>P</sup> <sup>¼</sup> <sup>P</sup><sup>0</sup> � exp �g<sup>0</sup> � <sup>M</sup>air � <sup>h</sup>PV

RgT 

cos3ðθszÞ þ <sup>0</sup>:149864cos2ðθszÞ þ <sup>0</sup>:0102963cosðθszÞ þ <sup>0</sup>:<sup>000303978</sup> (18)

<sup>2</sup> <sup>¼</sup> <sup>1</sup>:<sup>000110</sup> <sup>þ</sup> <sup>0</sup>:034221cosðΓÞ þ <sup>0</sup>:001280sinðΓ<sup>Þ</sup>

<sup>a</sup> <sup>ð</sup>1:<sup>0</sup> <sup>þ</sup> AMa � AM1:<sup>01</sup>

�0:<sup>3035</sup> � <sup>0</sup>:<sup>002715</sup> � <sup>U</sup>3ð1:<sup>0</sup> <sup>þ</sup> <sup>0</sup>:044U<sup>3</sup> <sup>þ</sup> <sup>0</sup>:0003U<sup>2</sup>

<sup>0</sup>:<sup>6828</sup> <sup>þ</sup> <sup>6</sup>:385U1�

<sup>a</sup> <sup>Þ</sup>AM0:<sup>9108</sup>

Irrn ¼ 0:9751 � E<sup>0</sup> � Irr<sup>0</sup> � τrτoτgτwτ<sup>a</sup> (19)

, represents the solar constant or total irradiance of the AM

(20)

<sup>a</sup> Þ� (21)

�<sup>1</sup> (24)

<sup>a</sup> � (23)

<sup>a</sup> � (25)

3Þ �1 � (22)

(26)

Eq. (13), considering the effect of atmospheric refraction, according to Eq. (18) [77].

the Parameterization Model C developed by Iqbal, given in Eq. (19) [78–80].

þ 0:000719cosð2ΓÞ þ 0:000077sinð2ΓÞ

<sup>τ</sup><sup>r</sup> <sup>¼</sup> exp½�0:<sup>0903</sup> � AM<sup>0</sup>:<sup>84</sup>

τ<sup>w</sup> ¼ 1 � 2:4959 � U1½ð1:0 þ 79:034U1Þ

<sup>τ</sup><sup>a</sup> <sup>¼</sup> exp½�k<sup>0</sup>:<sup>873</sup>

calculated using the isothermal atmosphere formula in Eq. (26).

In Eq. (19), Irr<sup>0</sup> <sup>¼</sup> 1367 W/m<sup>2</sup>

E<sup>0</sup> ¼ ðrsun�<sup>m</sup>=rsunÞ

156 Recent Improvements of Power Plants Management and Technology

for τr, τo, τg, τw, τ<sup>a</sup> are given in Eqs. (21–25).

τ<sup>o</sup> ¼ 1 � ½0:1611 � U3ð1:0 þ 139:48 � U3Þ

AM <sup>¼</sup> <sup>1</sup>:002432cos2ðθszÞ þ <sup>0</sup>:148386cosðθszÞ þ <sup>0</sup>:<sup>0096467</sup>

In Table 2, geographic and climate characteristics are specified for four prospective locations of the scalable, self-contained solar powered electrolytic sodium (Na) metal production plant including El Paso, Texas; Alice Springs, Australia; Bangkok, Thailand and Mbandaka, Democratic Republic of Congo (DRC).


a Accurate geographic coordinates and elevations above mean sea level for largest commercial airports of respective cities. b [83]. c [84–87].

Table 2. Geographic and climate characteristics for four prospective plant locations.

InTable 2, the geographic coordinates use the sign convention ofþ/– latitudeϕ, for locations north and south of the equator, respectively, and þ/– longitude λ, for locations east and west of the prime meridian (λPM ¼ 0�). According to the Köppen-Geiger climate classification system, El Paso has an arid, desert, cold (BWk) climate, Alice Springs has an arid, desert, hot (BWh) climate, Bangkok has a tropical, savanna (Aw) climate and Mbandaka has a tropical, rainforest (Af) climate [88]. The mean annual local solar noon air mass values in Table 2 are calculated using Eqs. (9) and (16)–(18), and the length of day results provided by Eq. (15), show that proximity to the equator for the selfcontained sodium (Na) metal production plant maximizes the solar irradiance and provides uniform hours of daylight per day throughout the year. Tropical regions however, experience higher mean monthly temperatures and consequently higher mean annual precipitable water vapor levels with a long rainy season and sun obscuring rain clouds. Other factors that can significantly affect incident solar irradiance on the PV device panels include the geographic elevation above mean sea level and aerosols comprising solid and liquid sunlight obscuring particulates in the air that can occur naturally due to dust storms and volcanic activity or from human activity such as slash and burn agriculture [89]. According to the accumulated world meteorological data, the southwestern region of the U.S.A. and the central region of Australia, both constitute nearly ideal locations for the self-contained sodium (Na) metal production plants due to the existence of an arid, desert climate, vast tracts of flat open land, sparse human population and many clear days with high solar irradiance throughout the year [47, 48, 90, 91].

## 3. Sodium metal production plant architecture

The architecture of the self-contained sodium (Na) metal production plant has to provide an optimal balance between high performance, reliability and cost effective operation. The highest performance can be achieved by constructing the solar tower of the plant shown in Figure 2, in a manner that allows the optical k-vectors from the sun to be normally incident onto the photovoltaic (PV) device panels throughout the entire period of daylight from sunrise to sunset. The Figure 4 shows a solar tower architecture with fixed PV device panels, wherein the panels can rotate and tilt with the solar tower as a single unit to follow the sun's overhead trajectory.

In Figure 4, the solar tower is fabricated using modular sections comprised of high strength, lightweight aluminum alloy that can be fitted end to end and bolted together until the final slant height of the structure S<sup>h</sup> ¼ 300 m is achieved. Up to NB-L/R ¼ 100 branches that constitute levels, comprised of modular sections each having a final length B<sup>l</sup> ¼ 50 m, extend out on each side from the central column of the solar tower resulting in a width for the structure of approximately S<sup>w</sup> ¼ 100 m. The PV device panels are mounted along the top and bottom of the branches projecting from the central column of the solar tower. The projecting branches of the solar tower also enable ground glass, light diffusing panels to be placed between the rows of solar panels that allow sunlight to penetrate and effectively illuminate the land area beneath the solar tower to support crop cultivation including rice paddy fields. The central column of the solar tower is fixed at the base to a static pylon and approximately midway up the height of the column, to a boom for elevating the solar tower to a tower elevation angle π/2 - αs, that is the complement of the solar elevation angle αs, the latter given in Eq. (13). The opposite end of the boom is fixed to the center of a railroad flatcar mounted on a linear train track that is capable of being displaced along the track to elevate and lower the solar tower. An electric motor module mounted midway up the height of the column enables the PV device panels to collectively rotate left and right about the axis of the central column to track the solar azimuth angle γ<sup>s</sup> , the latter given in Eq. (14), thereby allowing the optical k-vectors from the sun to impinge at normal incidence onto the PV device panels throughout most of the day of operation. When the sun has set at the end of the day or when inclement weather of sufficient severity is expected, the solar tower can be lowered to be parallel and nearly flush with the ground as shown in Figure 4, with metal louvers drawn over the PV device panels to safeguard against damage to the panels from storms, strong winds and flying debris that can occur, albeit rarely, in the southwestern U.S.A. [92, 93].

Figure 4. Solar tower architecture (NOT TO SCALE).

vapor levels with a long rainy season and sun obscuring rain clouds. Other factors that can significantly affect incident solar irradiance on the PV device panels include the geographic elevation above mean sea level and aerosols comprising solid and liquid sunlight obscuring particulates in the air that can occur naturally due to dust storms and volcanic activity or from human activity such as slash and burn agriculture [89]. According to the accumulated world meteorological data, the southwestern region of the U.S.A. and the central region of Australia, both constitute nearly ideal locations for the self-contained sodium (Na) metal production plants due to the existence of an arid, desert climate, vast tracts of flat open land, sparse human popula-

The architecture of the self-contained sodium (Na) metal production plant has to provide an optimal balance between high performance, reliability and cost effective operation. The highest performance can be achieved by constructing the solar tower of the plant shown in Figure 2, in a manner that allows the optical k-vectors from the sun to be normally incident onto the photovoltaic (PV) device panels throughout the entire period of daylight from sunrise to sunset. The Figure 4 shows a solar tower architecture with fixed PV device panels, wherein the panels can rotate and tilt with the solar tower as a single unit to follow the sun's overhead trajectory.

In Figure 4, the solar tower is fabricated using modular sections comprised of high strength, lightweight aluminum alloy that can be fitted end to end and bolted together until the final slant height of the structure S<sup>h</sup> ¼ 300 m is achieved. Up to NB-L/R ¼ 100 branches that constitute levels, comprised of modular sections each having a final length B<sup>l</sup> ¼ 50 m, extend out on each side from the central column of the solar tower resulting in a width for the structure of approximately S<sup>w</sup> ¼ 100 m. The PV device panels are mounted along the top and bottom of the branches projecting from the central column of the solar tower. The projecting branches of the solar tower also enable ground glass, light diffusing panels to be placed between the rows of solar panels that allow sunlight to penetrate and effectively illuminate the land area beneath the solar tower to support crop cultivation including rice paddy fields. The central column of the solar tower is fixed at the base to a static pylon and approximately midway up the height of the column, to a boom for elevating the solar tower to a tower elevation angle π/2 - αs, that is the complement of the solar elevation angle αs, the latter given in Eq. (13). The opposite end of the boom is fixed to the center of a railroad flatcar mounted on a linear train track that is capable of being displaced along the track to elevate and lower the solar tower. An electric motor module mounted midway up the height of the column enables the PV device panels to collectively rotate left and right about the axis of the central column to track the solar azimuth angle γ<sup>s</sup> , the latter given in Eq. (14), thereby allowing the optical k-vectors from the sun to impinge at normal incidence onto the PV device panels throughout most of the day of operation. When the sun has set at the end of the day or when inclement weather of sufficient severity is expected, the solar tower can be lowered to be parallel and nearly flush with the ground as shown in Figure 4, with metal louvers drawn over the PV device panels to safeguard against damage to the panels from storms, strong

winds and flying debris that can occur, albeit rarely, in the southwestern U.S.A. [92, 93].

tion and many clear days with high solar irradiance throughout the year [47, 48, 90, 91].

3. Sodium metal production plant architecture

158 Recent Improvements of Power Plants Management and Technology

## 3.1. Electrical design of the solar tower

The electrical design of the solar tower comprising PV device panels has to accommodate scalability in the power output level, where it is possible to supplant the existing PV device panels with newer and more efficient ones when they become available, without having to modify other components in the solar tower. It is therefore necessary to optimally dimension the electrical conductors embedded within the branches and central column that transmit the electric power generated by the photovoltaic (PV) device panels to the electrolytic cells, according to the magnitude of the current expected to be transmitted once ηPV ¼ 90% efficient PV device panels become available to be installed on the branches of the tower as shown in Figure 4. To develop an accurate design for the aluminum current carrying conductors of the solar tower, it becomes necessary to define the electrical interconnection topology of the photovoltaic (PV) device panels installed on the solar tower. In Figure 5, the equivalent circuit model of the photovoltaic (PV) device panel array installed on the solar tower is shown, with the corresponding straight line approximation of the current versus voltage curve for the PV device panel array.

In the circuit models shown in Figure 5, IPV represents the PV device current, VOC represents the PV device open circuit voltage, R<sup>P</sup> represents the parallel resistance of the PV device that ideally

Figure 5. Circuit model of the photovoltaic (PV) device panel array installed on the solar tower.

should be infinite, R<sup>S</sup> represents the series resistance of the PV device that ideally should be zero in value, IST represents the output current of the solar tower and VST represents the output voltage of the solar tower. The circuit models shown in Figure 5 are applicable for a single PV cell as well as for an array of PV device panels comprised of PV cells connected in series and/or in parallel [94]. The expressions for the Thevenin equivalent circuit voltage VTH and resistance RTH, are given both for the linear equivalent current source and voltage source circuits. The electrical interconnection topology of the PV device panels on the solar tower must achieve an optimal balance in operating parameters including the solar tower supply system maximum voltage VST-MAX and maximum current IST-MAX. It will be assumed in the further analysis and calculations that VST-MAX corresponds to the maximum power point (MPP) operating voltage of the PV device panel array (VMPP-PA) rather than to the open circuit voltage VOC, of the array and similarly, that IST-MAX corresponds to the MPP operating current of the PV device panel array (IMPP-PA) rather than to the short circuit current ISC, of the array. The assumptions can be considered valid since it is possible to show using Eqs. (27)–(29) that there exists only a small difference in the value between VMPP-PA and VOC and similarly between IMPP-PA and ISC if R<sup>P</sup> is large and R<sup>S</sup> is small. The Eqs. (27)–(29) describe the solar tower PV array model shown in Figure 5.

$$I\_{\rm PV} = I\_{\rm SC} \frac{(R\_{\rm S} + R\_{\rm P})}{R\_{\rm P}} \tag{27}$$

$$\frac{V\_{\rm MPP-PA}}{I\_{\rm SC} - I\_{\rm MPP-PA}} = R\_{\rm S} + R\_{\rm P} \tag{28}$$

$$\frac{V\_{\rm OC} - V\_{\rm MPP-PA}}{I\_{\rm MPP-PA}} = R\_{\rm S} \tag{29}$$

The Eq. (27), can be derived from the circuit model in Figure 5 by short circuiting the output terminals of the solar tower where IST ¼ ISC. The Eq. (28) is calculated from the solar tower PV array operating as a current source at the maximum power point (MPP), and Eq. (29) is calculated from the solar tower PV array operating as a voltage source at the MPP.

The solar tower must be capable of transmitting PST ¼ 23.9 MW of electric power as indicated in Figure 3, from the photovoltaic (PV) device panels to the electrolytic cells of the scalable, sodium (Na) metal production plant while maintaining reasonable design values for VST-MAX and IST-MAX that will not escalate the cost of the system. To achieve an optimal balance between the VST-MAX and IST-MAX parameters of the solar tower supply system, it is necessary to identify the most reliable and cost effective approach for maintaining the PV devices at or near their maximum power point (MPP) during operation. Many methods exist for providing a variable or dynamic load to PV devices and performing maximum power point tracking (MPPT), however, for the present application it is necessary to consider that since there are up to N<sup>P</sup> ¼ 30,000 PV device panels installed on the solar tower, it becomes uneconomical to use any approach that requires dedicated MPPT hardware for so many panels individually or even for small groups of panels and therefore, a collective solution is required for the entire PV device panel array having an area <sup>A</sup>PA <sup>¼</sup> 30,000 m2 . A reliable method of operating PV devices very near, if not exactly at the MPP can be implemented by controlling the output voltage of the PV devices [95]. The maximum power point output voltage of a single PV cell VMPP-C, is always very near in value to a temperature dependent operating voltage of the PV cell and this phenomenon can be used to implement a type of MPPT by controlling the output voltage VST, of the entire PV device panel array with area <sup>A</sup>PA <sup>¼</sup> 30,000 m2 . The output voltage VST of the entire PV device panel array can be controlled for example, by using a voltage step down pulse width modulated (PWM) DC-DC converter with closed loop feedback control that is operated at the proper duty cycle required to maintain the input voltage VIN, to the converter (which is equivalent to the output voltage VST, of the PV device panel array), at the value near to the MPP voltage. In practice, the MPP output voltage of a single junction, monocrystalline silicon PV cell is given as VMPP-C ≈ 0.4 V which is too low for direct input to a PWM DC-DC converter [95]. Commercial single junction, front-illuminated, monocrystalline silicon PV panels of the type shown in Table 1, typically contain 60 PV cell modules connected in series resulting in a maximum power point output voltage for the panel given as VMPP-P ¼ 30.1 V for the NU-U240F2 Sharp panel, VMPP-P ¼ 31.9 V for the LG280S1C-B3 LG panel and VMPP-P ¼ 31.7 V for the STP290S-20 Suntech panel. The maximum power point output voltage of the 96 module, back-illuminated SPR-X21-345 Sunpower panel is given as VMPP-P ¼ 57.3 V. The series connected PV cell modules use bypass diodes to prevent power loss from an entire chain of series connected cells if one cell becomes shaded, and receives less illumination than other cells causing its resistance to increase, by allowing energy to be collected from PV cells that are not shaded and are outside of the bypassed section of the chain containing the shaded cell(s) [96, 97]. If 15 of the standard 60 cell module single junction, front-illuminated, monocrystalline silicon PV panel types are connected together in series, then the solar tower supply system maximum voltage can be limited to approximately VST-MAX ≈ 450 VDC, which corresponds to a low voltage supply according to the International Electrotechnical Commission (IEC) that defines low voltage DC equipment as having a nominal voltage below 750 VDC [98]. A low

should be infinite, R<sup>S</sup> represents the series resistance of the PV device that ideally should be zero in value, IST represents the output current of the solar tower and VST represents the output voltage of the solar tower. The circuit models shown in Figure 5 are applicable for a single PV cell as well as for an array of PV device panels comprised of PV cells connected in series and/or in parallel [94]. The expressions for the Thevenin equivalent circuit voltage VTH and resistance RTH, are given both for the linear equivalent current source and voltage source circuits. The electrical interconnection topology of the PV device panels on the solar tower must achieve an optimal balance in operating parameters including the solar tower supply system maximum voltage VST-MAX and maximum current IST-MAX. It will be assumed in the further analysis and calculations that VST-MAX corresponds to the maximum power point (MPP) operating voltage of the PV device panel array (VMPP-PA) rather than to the open circuit voltage VOC, of the array and similarly, that IST-MAX corresponds to the MPP operating current of the PV device panel array (IMPP-PA) rather than to the short circuit current ISC, of the array. The assumptions can be considered valid since it is possible to show using Eqs. (27)–(29) that there exists only a small difference in the value between VMPP-PA and VOC and similarly between IMPP-PA and ISC if R<sup>P</sup> is large and R<sup>S</sup> is small.

Figure 5. Circuit model of the photovoltaic (PV) device panel array installed on the solar tower.

160 Recent Improvements of Power Plants Management and Technology

The Eqs. (27)–(29) describe the solar tower PV array model shown in Figure 5.

IPV ¼ ISC

VMPP�PA ISC � IMPP�PA

> VOC � VMPP�PA IMPP�PA

ðR<sup>S</sup> þ RPÞ R<sup>P</sup>

¼ R<sup>S</sup> þ R<sup>P</sup> (28)

¼ R<sup>S</sup> (29)

(27)

voltage DC supply system entails a minor risk of electric arcing through the air and an exemption from specialized protection equipment needed for high voltage. If 15 of the 96 cell module, back-illuminated SPR-X21-345 Sunpower PV panels are connected together in series, then the solar tower supply system maximum voltage VST-MAX ≈ 900 VDC, which exceeds by 150 V the nominal 750 VDC IEC standard of a low voltage supply. Once ηPV ¼ 90% efficient PV device panels will become available, it is expected that they will each have an area <sup>A</sup><sup>P</sup> <sup>¼</sup> 1 m<sup>2</sup> and the number of PV cell modules connected together in series will yield a maximum power point voltage for the panel given as VMPP-P ≈ 30 V that is similar in value to most of the commercial single junction, front-illuminated, monocrystalline silicon PV panels listed in Table 1, that contain 60 PV cell modules connected in series. Under an ASTM AM 1.5G standard terrestrial solar spectral irradiance with a total irradiance IrrAM1.5G <sup>¼</sup> 1000 W/m<sup>2</sup> , the corresponding maximum power point current of the ηPV ¼ 90% efficient PV panel can be calculated as IMPP-P ¼ (ηPV · IrrAM1.5G) / VMPP-P ¼ 30.0 A, which is a substantially larger current than the IMPP-P current values listed in Table 1, as might be expected for the more efficient PV panel unit. Therefore, it is necessary to design the current conductors within the scalable solar tower to be capable of transmitting the substantially larger current of more efficient PV panels once they will become available for installation onto the solar tower. The solar tower supply system maximum voltage can be set to VST-MAX ¼ 450 V, achieved by connecting in series 15 PV device panels with an efficiency ηPV ¼ 90% and VMPP-P ≈ 30 V, mounted on a single branch section of the solar tower apparatus as shown in Figure 6.

In Figure 6, each branch section supports the installation of 2 groups of 15 series connected PV panels having an efficiency ηPV ¼ 90% for a total NP-Bsec ¼ 30 PV panels per branch section. A total of 5 branch sections should be fastened together end to end, to yield NP-B ¼ 5 · NP-Bsec ¼ 150 PV panels per branch with 5 groups of 15 series connected PV panels installed along the top of the branch and another 5 groups of 15 series connected PV panels installed along the bottom of the branch. Each branch section has a length Bsec<sup>l</sup> ¼ 10 m, and contains embedded within the aluminum current conductors, yielding a branch length B<sup>l</sup> ¼ 50 m as shown in Figure 4. The groups of 15 series connected PV panels are electrically connected to a pair of aluminum conductors inside the branch, resulting in parallel electrical interconnection for the groups of 15 series connected PV panels that increases the electric current delivered by the solar tower. Each PV panel has a width P<sup>w</sup> ¼ 0.66 m and a height P<sup>l</sup> ¼ 1.5 m for a total PV panel area given as A<sup>P</sup> ¼ <sup>P</sup><sup>w</sup> · <sup>P</sup><sup>l</sup> <sup>¼</sup> 1 m2 . Since the height of each PV panel is given as P<sup>l</sup> ¼ 1.5 m, then each branch has an overall height given as B<sup>h</sup> ¼ 2 · P<sup>l</sup> ¼ 3 m. The solar tower that has a height S<sup>h</sup> ¼ 300 m as shown in Figure 4, can therefore accommodate up to 100 branches extending out from the left and right sides of the central column, yielding a total PV panel array area on the solar tower given as <sup>A</sup>PA <sup>¼</sup> <sup>2</sup> · (S<sup>h</sup> / <sup>B</sup>h) · <sup>N</sup>P-B · <sup>A</sup><sup>P</sup> <sup>¼</sup> 30,000 m2 .

The electric current from each branch section flows into the electrical conductors installed inside the central column of the solar tower as shown in Figure 6. Therefore, electric current that flows from the outermost branch section toward the central column increases as each branch section contributes additional current generated by the 30 PV panels mounted on it. If an ASTM direct normal AM 1.5D standard terrestrial solar spectral irradiance with a total irradiance IrrAM1.5D ¼ 887 W/m2 , is incident on the PV panels having an efficiency ηPV ¼ 90% and VMPP-P ¼ 30 V, then the maximum power point output current can be calculated as IMPP-P ¼ (ηPV · IrrAM1.5D) /

voltage DC supply system entails a minor risk of electric arcing through the air and an exemption from specialized protection equipment needed for high voltage. If 15 of the 96 cell module, back-illuminated SPR-X21-345 Sunpower PV panels are connected together in series, then the solar tower supply system maximum voltage VST-MAX ≈ 900 VDC, which exceeds by 150 V the nominal 750 VDC IEC standard of a low voltage supply. Once ηPV ¼ 90% efficient PV device panels will become available, it is expected that they will each have an area <sup>A</sup><sup>P</sup> <sup>¼</sup> 1 m<sup>2</sup> and the number of PV cell modules connected together in series will yield a maximum power point voltage for the panel given as VMPP-P ≈ 30 V that is similar in value to most of the commercial single junction, front-illuminated, monocrystalline silicon PV panels listed in Table 1, that contain 60 PV cell modules connected in series. Under an ASTM AM 1.5G standard terrestrial

maximum power point current of the ηPV ¼ 90% efficient PV panel can be calculated as IMPP-P ¼ (ηPV · IrrAM1.5G) / VMPP-P ¼ 30.0 A, which is a substantially larger current than the IMPP-P current values listed in Table 1, as might be expected for the more efficient PV panel unit. Therefore, it is necessary to design the current conductors within the scalable solar tower to be capable of transmitting the substantially larger current of more efficient PV panels once they will become available for installation onto the solar tower. The solar tower supply system maximum voltage can be set to VST-MAX ¼ 450 V, achieved by connecting in series 15 PV device panels with an efficiency ηPV ¼ 90% and VMPP-P ≈ 30 V, mounted on a single branch section of

In Figure 6, each branch section supports the installation of 2 groups of 15 series connected PV panels having an efficiency ηPV ¼ 90% for a total NP-Bsec ¼ 30 PV panels per branch section. A total of 5 branch sections should be fastened together end to end, to yield NP-B ¼ 5 · NP-Bsec ¼ 150 PV panels per branch with 5 groups of 15 series connected PV panels installed along the top of the branch and another 5 groups of 15 series connected PV panels installed along the bottom of the branch. Each branch section has a length Bsec<sup>l</sup> ¼ 10 m, and contains embedded within the aluminum current conductors, yielding a branch length B<sup>l</sup> ¼ 50 m as shown in Figure 4. The groups of 15 series connected PV panels are electrically connected to a pair of aluminum conductors inside the branch, resulting in parallel electrical interconnection for the groups of 15 series connected PV panels that increases the electric current delivered by the solar tower. Each PV panel has a width P<sup>w</sup> ¼ 0.66 m and a height P<sup>l</sup> ¼ 1.5 m for a total PV panel area given as A<sup>P</sup> ¼

overall height given as B<sup>h</sup> ¼ 2 · P<sup>l</sup> ¼ 3 m. The solar tower that has a height S<sup>h</sup> ¼ 300 m as shown in Figure 4, can therefore accommodate up to 100 branches extending out from the left and right sides of the central column, yielding a total PV panel array area on the solar tower given as

The electric current from each branch section flows into the electrical conductors installed inside the central column of the solar tower as shown in Figure 6. Therefore, electric current that flows from the outermost branch section toward the central column increases as each branch section contributes additional current generated by the 30 PV panels mounted on it. If an ASTM direct normal AM 1.5D standard terrestrial solar spectral irradiance with a total irradiance IrrAM1.5D ¼

the maximum power point output current can be calculated as IMPP-P ¼ (ηPV · IrrAM1.5D) /

, is incident on the PV panels having an efficiency ηPV ¼ 90% and VMPP-P ¼ 30 V, then

.

. Since the height of each PV panel is given as P<sup>l</sup> ¼ 1.5 m, then each branch has an

, the corresponding

solar spectral irradiance with a total irradiance IrrAM1.5G <sup>¼</sup> 1000 W/m<sup>2</sup>

the solar tower apparatus as shown in Figure 6.

162 Recent Improvements of Power Plants Management and Technology

<sup>A</sup>PA <sup>¼</sup> <sup>2</sup> · (S<sup>h</sup> / <sup>B</sup>h) · <sup>N</sup>P-B · <sup>A</sup><sup>P</sup> <sup>¼</sup> 30,000 m2

<sup>P</sup><sup>w</sup> · <sup>P</sup><sup>l</sup> <sup>¼</sup> 1 m2

887 W/m2

Figure 6. Solar tower branch sections showing groups of 15 series interconnected PV device panels installed.

(VMPP-P) ¼ 26.6 A. The result for IMPP-P ¼ 26.6 A, entails that each branch section will contribute IBsec ¼ 2 · IMPP-P ¼ 53.2 A. Therefore, the current in each branch increases by IBsec ¼ 53.2 A as it flows in from the outermost branch section toward the current conductors of the central column of the solar tower. The total current contributed by each branch of the solar tower is then calculated as I<sup>B</sup> ¼ 5 · IBsec ¼ 266 A, since a single branch comprises 5 branch sections. Each side of the solar tower has NB-L/R ¼ 100 branches for a total number of branches on the solar tower given as NB-ST ¼ 2 · NB-L/R ¼ 200 branches. It is convenient however, to install four electrical conductors labeled 1, 2, 3 and 4, in the central column of the solar tower as shown in Figure 6. The first pair of current conductors 1 and 2, transmits current from the branches mounted on the left side of the solar tower and the second pair of current conductors 3 and 4, transmits current from the branches mounted on the right side of the solar tower, so that each conductor pair only needs to transmit a maximum current IST-MAX ¼ 26,600 A.

The four electrical conductors installed within the modular sections of the central column of the solar tower that are located at or near the top of the solar tower, do not have to carry the maximum current IST-MAX ¼ 26,600 A, rather only the four electrical conductors installed in the bottom most modular section of the central column have to be dimensioned to transmit the maximum current. The diameters of aluminum electrical conductors within the central column sections scale linearly from a diameter DS1 ¼ 2 cm in stage 1 at the top to DS100 ¼ 41.6 cm for stage 100 at the bottom of the solar tower, the latter that transmits the maximum current IST-MAX ¼ 26,600 A. Figure 6, shows the calculated voltage drop across the aluminum electrical conductors in all 100 modular sections of the central column of the solar tower to be given as VST-DROP ¼ 3.86 V, corresponding to a low loss considering that the solar tower supply system maximum voltage VST-MAX ¼ 450 V.

The electrical design of the solar tower described provides manifold advantages including a mostly parallel electrical interconnection architecture for the PV device panels that allows the MPP of the PV device panels to be controlled collectively by controlling the output voltage VST of the PV device panel array using a voltage step down PWM DC-DC converter designed to have a constant output voltage VOUT, and controllable input voltage VIN, the latter supplied from the PV device panel array and equal to VST, shown in Figure 5. Other advantages include a low voltage DC solar tower supply system with optimal balance between the maximum voltage VST-MAX ≈ 450 V and maximum current IST-MAX ¼ 26,600 A, wherein two independent and electrically isolated power supply feeds are provided from the left side branches and right side branches of the solar tower apparatus, respectively. Yet another important advantage of the design includes a low center of gravity for the solar tower apparatus due to the modular sections comprising the central column being heavier near the base and lighter near the top as a result of larger diameter aluminum electrical conductors placed near the base of the tower.

### 3.2. Electrical design of the sodium hydroxide electrolysis plant

The solar tower apparatus described in Section 3.1 implements a mostly parallel electrical interconnection architecture for the PV device panels that yields two identical, independent and electrically isolated low voltage DC power supplies, each having a maximum voltage VST-MAX ≈ 450 V and maximum current IST-MAX ¼ 26,600 A, wherein the two power supply feeds emanate from the left side branches and right side branches of the solar tower apparatus, respectively as shown in Figure 6. The sodium hydroxide (NaOH) electrolysis cell however, requires a substantially lower voltage for operation. The quantity of sodium (Na) metal produced by the NaOH electrolytic cell depends fundamentally on the magnitude of the DC current flowing between the anode and cathode terminals of the electrolysis cell. The Eqs. (4)–(6) provide the standard reduction potentials of the oxidation and reduction half reactions that occur at the anode and cathode, respectively of the electrolytic cell. In practice, the electrolytic cell operating voltage has to be set at approximately VCELL ¼ 4 V for electrolysis of pure NaOH or VCELL ¼ 5 V for electrolysis of a mixture of NaOH and NaCl, the latter derived from sea salt. The higher voltage accounts for the overvoltage effects [99]. The maximum current that can be supplied to an electrolytic cell operating at approximately VCELL ≈ 5 V will be ICELL ≈ 100,000 A and is limited in large measure by the material cost as well as by the mass and physical dimensions of the electrical conductors required to transmit such magnitude of the current.

In a fused or molten state, NaOH(l) is highly corrosive and therefore the only conventional materials capable of withstanding prolonged exposure to its caustic effects at an elevated temperature include graphite, iron and nickel. Graphite however, cannot be used as an anode electrode because it will react with the oxygen (O2) generated to produce carbon dioxide (CO2) and become consumed in the process. Iron can withstand corrosion from fused NaOH(l), and consequently could be used to fabricate the electrolytic vessel however, as an anode electrode, the reaction with steam (H2O(g)) and O2 will quickly oxidize and erode iron. The only material suitable for fabricating both the anode and cathode electrodes remains nickel (Ni) which is significantly more expensive than both graphite and iron. The cost of the Ni electrodes therefore becomes an important factor in limiting the maximum current in the electrolytic cell. The other factor limiting the current in the electrolytic cell becomes the PWM DC-DC converter that must supply the large currents at the correct output voltage to the electrolytic cell, safely and reliably.

bottom most modular section of the central column have to be dimensioned to transmit the maximum current. The diameters of aluminum electrical conductors within the central column sections scale linearly from a diameter DS1 ¼ 2 cm in stage 1 at the top to DS100 ¼ 41.6 cm for stage 100 at the bottom of the solar tower, the latter that transmits the maximum current IST-MAX ¼ 26,600 A. Figure 6, shows the calculated voltage drop across the aluminum electrical conductors in all 100 modular sections of the central column of the solar tower to be given as VST-DROP ¼ 3.86 V, corresponding to a low loss considering that the solar tower supply system

The electrical design of the solar tower described provides manifold advantages including a mostly parallel electrical interconnection architecture for the PV device panels that allows the MPP of the PV device panels to be controlled collectively by controlling the output voltage VST of the PV device panel array using a voltage step down PWM DC-DC converter designed to have a constant output voltage VOUT, and controllable input voltage VIN, the latter supplied from the PV device panel array and equal to VST, shown in Figure 5. Other advantages include a low voltage DC solar tower supply system with optimal balance between the maximum voltage VST-MAX ≈ 450 V and maximum current IST-MAX ¼ 26,600 A, wherein two independent and electrically isolated power supply feeds are provided from the left side branches and right side branches of the solar tower apparatus, respectively. Yet another important advantage of the design includes a low center of gravity for the solar tower apparatus due to the modular sections comprising the central column being heavier near the base and lighter near the top as a result of larger diameter aluminum electrical conductors placed near the base of the tower.

The solar tower apparatus described in Section 3.1 implements a mostly parallel electrical interconnection architecture for the PV device panels that yields two identical, independent and electrically isolated low voltage DC power supplies, each having a maximum voltage VST-MAX ≈ 450 V and maximum current IST-MAX ¼ 26,600 A, wherein the two power supply feeds emanate from the left side branches and right side branches of the solar tower apparatus, respectively as shown in Figure 6. The sodium hydroxide (NaOH) electrolysis cell however, requires a substantially lower voltage for operation. The quantity of sodium (Na) metal produced by the NaOH electrolytic cell depends fundamentally on the magnitude of the DC current flowing between the anode and cathode terminals of the electrolysis cell. The Eqs. (4)–(6) provide the standard reduction potentials of the oxidation and reduction half reactions that occur at the anode and cathode, respectively of the electrolytic cell. In practice, the electrolytic cell operating voltage has to be set at approximately VCELL ¼ 4 V for electrolysis of pure NaOH or VCELL ¼ 5 V for electrolysis of a mixture of NaOH and NaCl, the latter derived from sea salt. The higher voltage accounts for the overvoltage effects [99]. The maximum current that can be supplied to an electrolytic cell operating at approximately VCELL ≈ 5 V will be ICELL ≈ 100,000 A and is limited in large measure by the material cost as well as by the mass and physical dimensions of the electrical conductors required to transmit such magnitude of the current.

In a fused or molten state, NaOH(l) is highly corrosive and therefore the only conventional materials capable of withstanding prolonged exposure to its caustic effects at an elevated

3.2. Electrical design of the sodium hydroxide electrolysis plant

maximum voltage VST-MAX ¼ 450 V.

164 Recent Improvements of Power Plants Management and Technology

It is necessary to provide two identical voltage step down PWM DC-DC converters to convert the power PST-L/R ¼ 11.95 MW delivered by the two independent low voltage DC power supplies of the solar tower, each having a maximum voltage VST-MAX ≈ 450 V and maximum current IST-MAX ¼ 26,600 A. The PWM DC-DC converter must be designed to safely convert the electric power supplied from the solar tower to magnitudes of DC voltage and DC current that are appropriate for supplying the NaOH electrolytic cells. The PWM DC-DC converters and electrolytic cells are housed together inside the prefabricated Q-type metal building indicated in Figure 4. If each PWM DC-DC converter is designed to supply a single NaOH electrolytic cell with a power conversion efficiency ηDC-DC ¼ 100%, then the maximum current in the single NaOH electrolytic cell would be calculated as IOUT ¼ (VST-MAX · IST-MAX) / VCELL ¼ (450 V) · (26,600 A) / 5 V ¼ 2,394,000 A, a value that is clearly beyond the practical realm. Consequently, multiple NaOH electrolytic cells have to be constructed and electrically connected in series to be supplied by a cell current ICELL ¼ 96,500 A that corresponds to approximately one mole of electrons supplied per second [43]. The output voltage of the PWM DC-DC converter is then calculated as VOUT ¼ (VST-MAX · IST-MAX) / ICELL ¼ (450 V) · (26,600 A) / 96,500 A ¼ 124 V. The number of NaOH electrolytic cells that must be connected in series to be supplied by a single PWM DC-DC converter is calculated as VOUT / VCELL ¼ 124 V/5V ≈ 25 cells. Therefore, the self-contained sodium (Na) metal production plant has a total of 50 NaOH electrolytic cells with 2 sets of 25 cells electrically connected in series and supplied by two identical PWM DC-DC converters that function as voltage step down converters to transform the output voltage of the solar tower VST, which represents a DC input voltage to the converter unit VIN ¼ VST-MAX ≈ 450 V, to a constant DC output voltage VOUT ¼ 124 V with high efficiency, while controlling the PWM DC-DC converter input voltage VIN, and thereby output voltage VST of the solar tower PV device panels to maintain their operation near the MPP.

The design of voltage step down PWM DC-DC converters that have a fixed output voltage VOUT and control the input voltage VIN that is variable, has been an active topic of research in the context of photovoltaic power systems applications [94, 95]. The requirements of the present application however, entail a specialized type of large scale direct photovoltaic power conversion for chemical electrolysis not hitherto contemplated or addressed in the scientific/ industrial literature. The design for the voltage step down PWM DC-DC converter with a fixed output voltage VOUT and variable input voltage VIN meant to supply the NaOH electrolytic cells is shown in Figure 7 to consist of a multiphase converter topology, with synchronous voltage step down converter circuits connected in parallel.

Figure 7. Multiphase voltage step down PWM DC-DC converter power supply for NaOH electrolytic cells.

The synchronous parallel multiphase voltage step down PWM DC-DC converter power supply shown in Figure 7 offers the inherent advantage of allowing the large load current IOUT ¼ ICELL ¼ 96,500 A to be split among the phases of the converter to match the current carrying capacity of the individual electronic switches, that in practice would consist of power insulated gate bipolar transistors (IGBTs) such as the Model 1MBI3600U4D-120, manufactured by the Fuji Electric Company with a maximum rated collector-emitter voltage and collector-emitter current given as VCE ¼ 1,200 V (at T<sup>C</sup> ¼ 25 �C) and ICE ¼ 3,600 A (at T<sup>C</sup> ¼ 80 �C), respectively. It is possible to use 64 such IGBT devices in up to N<sup>φ</sup> ¼ 32 phases of the multiphase PWM DC-DC converter with 2 IGBT devices per phase as shown in Figure 7, to deliver the required current IOUT ¼ ICELL ¼ 96,500 A to the 25 series connected NaOH electrolytic cells indicated as having resistances RC1, RC2, …, RC25, without exceeding the electrical ratings of the solid state switches. Using a multiphase converter topology with synchronous voltage step down converter circuits connected in parallel, also allows the current ICELL ¼ 96,500 A to be split among the 32 inductors L1–L32, present in each of the N<sup>φ</sup> ¼ 32 phases, thereby not requiring one single large inductor to transmit the full current IOUT, supplied by the PWM DC-DC converter to the load comprising the electrolytic cells. Current sharing occurs between the synchronous voltage step down converter circuits connected in parallel according to Eq. (30).

cells is shown in Figure 7 to consist of a multiphase converter topology, with synchronous

The synchronous parallel multiphase voltage step down PWM DC-DC converter power supply shown in Figure 7 offers the inherent advantage of allowing the large load current IOUT ¼ ICELL ¼ 96,500 A to be split among the phases of the converter to match the current carrying capacity of the individual electronic switches, that in practice would consist of power insulated gate bipolar transistors (IGBTs) such as the Model 1MBI3600U4D-120, manufactured by the Fuji Electric Company with a maximum rated collector-emitter voltage and collector-emitter current given as VCE ¼ 1,200 V (at T<sup>C</sup> ¼ 25 �C) and ICE ¼ 3,600 A (at T<sup>C</sup> ¼ 80 �C), respectively. It is possible to use 64 such IGBT devices in up to N<sup>φ</sup> ¼ 32 phases of the multiphase PWM DC-DC converter with 2 IGBT devices per phase as shown in Figure 7, to deliver the required current IOUT ¼ ICELL ¼ 96,500 A to the 25 series connected NaOH electrolytic cells indicated as having resistances RC1, RC2, …, RC25, without exceeding the electrical ratings of the solid state switches. Using a multiphase converter topology with synchronous voltage step down converter circuits connected in parallel, also allows the current ICELL ¼ 96,500 A to be split among the 32 inductors L1–L32, present in each of the N<sup>φ</sup> ¼ 32 phases, thereby not requiring one single large inductor to transmit the full current IOUT, supplied by the PWM DC-DC converter to the

Figure 7. Multiphase voltage step down PWM DC-DC converter power supply for NaOH electrolytic cells.

voltage step down converter circuits connected in parallel.

166 Recent Improvements of Power Plants Management and Technology

$$I\_{\rm O1} + I\_{\rm O2} + \dots + I\_{\rm ON} = I\_{\rm OUT} \tag{30}$$

Assuming the ideal case that current sharing between synchronous parallel voltage step down converter circuits is equal, then IO1 ¼ IO2 ¼ … ¼ ION ¼ IOUT / N<sup>φ</sup> where ION ¼ IOUT / N<sup>φ</sup> ¼ (96,500 A / 32) ¼ 3,016 A transmitted per phase. Thus, each of the 32 inductors L1–L32, can be designed for a current load ION ¼ 3,016 A, a value well within the capabilities of inductor manufacturers. The output voltage of the multiphase voltage step down PWM DC-DC converter can be set to a fixed value using a utility scale battery, and thus VBAT ¼ VOUT ¼ 124 V. The input voltage VIN, of the multiphase voltage step down PWM DC-DC converter power supply for electrolytic cells shown in Figure 7, that also corresponds to the output voltage VST, of the solar tower PV device array has to be controlled reliably to ensure that the PV device panel array operates at or near its maximum power point (MPP).

The synchronous parallel multiphase voltage step down PWM DC-DC converter power supply in Figure 7 with N<sup>φ</sup> ¼ 32 phases, constitutes a complicated nonlinear dynamical system that can be challenging to model, control and analyze accurately to ensure stable operation under wide ranging conditions [100]. The closed loop control system using voltage control and a single feedback loop that can be applied to the present converter is shown in Figure 8.

Figure 8. Closed loop control system for the multiphase voltage step down PWM DC-DC converter.

The control system for the multiphase voltage step down PWM DC-DC converter shown in Figure 8 consists of a sensor that senses the process parameter to be controlled, namely, the input voltage VIN to the PWM DC-DC converter that is also the output voltage VST, of the solar tower PV device panel array. The error amplifier compares the scaled process parameter VIN to the set point value VSET and computes the difference as an error signal VERROR. A proportional, integrator, differentiator (PID) circuit processes the error signal VERROR, and accordingly generates a control signal VCONTROL, that is supplied to a pulse width modulation (PWM) signal generation circuit to produce the control signals that have the correct frequency, duty cycle and phase shift. The PWM signal to the IGBT switches has to be replicated N<sup>φ</sup> ¼ 32 times and phase shifted by φShift ¼ 360� / N<sup>φ</sup> ¼ 11.25� using a driver circuit to control all the synchronous voltage step down converters connected in parallel in Figure 7. In Figure 9, the electronic circuit schematics for some of the different control blocks in Figure 8 are shown.

Figure 9. Electronic circuit schematics of the control system for the multiphase voltage step down PWM DC-DC converter.

The electronic circuits shown in Figure 9 are representative of the functional blocks of the PWM DC-DC converter control system. The sensor circuit can consist of a resistive voltage divider with a unity gain op-amp buffer that senses the voltage VIN, at the input of the PWM DC-DC converter according to Eq. (31).

$$V\_{\rm SEN} = V\_{\rm IN} \times (R2 \, / R1 + R2) \tag{31}$$

The error amplifier subtracts the input voltage VSEN, scaled by the resistive voltage divider of the sensor circuit, from the set point reference voltage VSET, to calculate an error voltage VERROR, according to Eq. (32).

$$V\_{\rm ERROR} = (R\Im/R1) \times (V\_{\rm SET} - V\_{\rm SEN}) \text{ where } R1 = R2 \text{ and } R3 = R4 \tag{32}$$

The PID unit receives the error voltage VERROR, as an input and produces a control voltage VCONTROL, given as Eq. (33).

Scalable, Self‐Contained Sodium Metal Production Plant for a Hydrogen Fuel Clean Energy Cycle http://dx.doi.org/10.5772/67597 169

$$V\_{\rm CONTROL}(t) = G\_{\rm P} \left( V\_{\rm ERROR}(t) \right) + G\_{\rm I} \int\_{0}^{t} V\_{\rm ERROR}(t)dt + G\_{\rm D} \frac{dV\_{\rm ERROR}(t)}{dt} \tag{33}$$

shifted by φShift ¼ 360� / N<sup>φ</sup> ¼ 11.25� using a driver circuit to control all the synchronous voltage step down converters connected in parallel in Figure 7. In Figure 9, the electronic

The electronic circuits shown in Figure 9 are representative of the functional blocks of the PWM DC-DC converter control system. The sensor circuit can consist of a resistive voltage divider with a unity gain op-amp buffer that senses the voltage VIN, at the input of the PWM

Figure 9. Electronic circuit schematics of the control system for the multiphase voltage step down PWM DC-DC converter.

The error amplifier subtracts the input voltage VSEN, scaled by the resistive voltage divider of the sensor circuit, from the set point reference voltage VSET, to calculate an error voltage

The PID unit receives the error voltage VERROR, as an input and produces a control voltage

VERROR ¼ ðR3 = R1Þ · ðVSET � VSENÞ where R1 ¼ R2 and R3 ¼ R4 (32)

VSEN ¼ VIN · ðR2 =R1 þ R2Þ (31)

DC-DC converter according to Eq. (31).

VERROR, according to Eq. (32).

VCONTROL, given as Eq. (33).

circuit schematics for some of the different control blocks in Figure 8 are shown.

168 Recent Improvements of Power Plants Management and Technology

In Eq. (33), the terms GP, G<sup>I</sup> and G<sup>D</sup> represent the gains of the proportional, integrator and differentiator circuits, respectively that can be tuned as needed to achieve optimal control characteristics for the PWM DC-DC converter. There exist myriad other ways of controlling the PWM DC-DC converter using only proportional (P) and integrator (I) control functions without the differentiator (D) circuit for example, however, the control system shown in Figures 8 and 9, represents a robust and reliable type of control of the PWM DC-DC converter. The PWM unit is shown to consist of a fixed frequency square wave voltage oscillator with an op-amp integrator circuit that converts the square wave into a triangular wave ramp signal which is then compared to the control voltage VCONTROL, from the PID circuit using a comparator, to generate the PWM signal for the IGBT electronic switches of the PWM DC-DC converter. The driver circuit that generates the N<sup>φ</sup> ¼ 32 copies of the PWM signal from Figure 8, with each signal and its complement phase shifted by φShift ¼ 11.25� for all 64 IGBT switches in the converter is not shown in Figure 9. It could be implemented however, using all digital logic. The control system allows the PWM DC-DC converter to be effectively controlled by setting just one parameter, namely, the reference voltage VSET, to a value that corresponds with the maximum power point (MPP) output of the solar tower PV device panel array to transmit the maximum power available from the PV devices to the NaOH electrolytic cells.

It is possible to gain insight into the operation of the synchronous parallel multiphase voltage step down PWM DC-DC converter with attached solar tower PV device panel array, from the most common modeling approach using small signal analysis based on state space averaging [101]. The open loop transfer function GV(s), of the voltage step down PWM DC-DC converter that yields the small signal response of the input voltage process parameter VIN ¼ VST to the duty cycle control variable d, can be calculated straightforwardly by making the simplifying assumption, namely, that the power supply has a single phase (N<sup>φ</sup> ¼ 1) rather than N<sup>φ</sup> ¼ 32 phases in parallel as depicted in Figure 7. The voltage step down PWM DC-DC converter from Figure 7 with only a single phase φ1, can be characterized by two pairs of differential equations that describe the current IO1, flowing through the inductor L1, and the voltage VIN, across the input capacitor C1, during the two distinct states of the converter when the switch SH1 is closed and open. The pair of differential equations corresponding to the switch SH1 being closed are given by Eqs. (34) and (35).

$$L\_1 \frac{dI\_{\rm O1}}{dt} = V\_{\rm IN} - V\_{\rm OUT} \tag{34}$$

$$C\_1 \frac{dV\_{\rm IN}}{dt} = \frac{V\_{\rm TH} - V\_{\rm IN}}{R\_{\rm TH}} - I\_{\rm O1} \tag{35}$$

The pair of differential equations corresponding to the switch SH1 being open are given by Eqs. (36) and (37).

$$L\_1 \frac{dI\_{\rm O1}}{dt} = -V\_{\rm OUT} \tag{36}$$

$$C\_1 \frac{dV\_{\rm IN}}{dt} = \frac{V\_{\rm TH} - V\_{\rm IN}}{R\_{\rm TH}} \tag{37}$$

The state space form of the above four differential equations describing the PWM DC-DC converter having just a single phase φ1, is given as Eqs. (38) and (39) for SH1 being closed and open, respectively.

$$
\frac{d}{dt} \begin{bmatrix} I\_{\text{O1}} \\ V\_{\text{IN}} \end{bmatrix} = \begin{bmatrix} 0 & \frac{1}{L\_1} \\ -\frac{1}{\mathbf{C}\_1} & -\frac{1}{\mathbf{C}\_1 \mathbf{R}\_{\text{TH}}} \end{bmatrix} \begin{bmatrix} I\_{\text{O1}} \\ V\_{\text{IN}} \end{bmatrix} + \begin{bmatrix} -\frac{1}{L\_1} & 0 \\ 0 & \frac{1}{\mathbf{C}\_1 \mathbf{R}\_{\text{TH}}} \end{bmatrix} \begin{bmatrix} V\_{\text{OUT}} \\ V\_{\text{TH}} \end{bmatrix} \tag{38}
$$

$$
\frac{d}{dt} \begin{bmatrix} I\_{\text{O1}} \\ V\_{\text{IN}} \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & -\frac{1}{\mathbf{C\_1 R\_{\text{TH}}}} \end{bmatrix} \begin{bmatrix} I\_{\text{O1}} \\ V\_{\text{IN}} \end{bmatrix} + \begin{bmatrix} -\frac{1}{L\_1} & 0 \\ 0 & \frac{1}{\mathbf{C\_1 R\_{\text{TH}}}} \end{bmatrix} \begin{bmatrix} V\_{\text{OUT}} \\ V\_{\text{TH}} \end{bmatrix} \tag{39}
$$

The Eq. (38), has the form x\_ ¼ A1x þ B1u, and Eq. (39) has the form x\_ ¼ A2x þ B2u. The averaged state space equation can be expressed in a similar form x\_ ¼ Ax þ Bu, where the A and B matrices are calculated based on the duty cycle d, of the switch SH1, the matrices A1, B1, A2, B<sup>2</sup> and are given as A ¼ d � A<sup>1</sup> þ ð1 � dÞ � A<sup>2</sup> and B ¼ d � B<sup>1</sup> þ ð1 � dÞ � B2. The vector x, in the averaged state space equation contains the average values of the state variables <IO1> and <VIN> and the vector u, contains the input variables VOUT and VTH that are considered as DC values.

When all transients in the PWM DC-DC converter have stabilized and steady state operation is achieved, then x\_ ¼ 0, thus the averaged state space equation can be used to express the DC steady state equations given as Eqs. (40) and (41).

$$
\dot{X} = 0 = AX + B\mathcal{U} \tag{40}
$$

$$X = \begin{bmatrix} I\_{\text{O1}} \\ V\_{\text{IN}} \end{bmatrix} = -A^{-1}BU = -\frac{L\_1 \mathbb{C}\_1}{D^2} \begin{bmatrix} -\frac{1}{\mathbb{C}\_1 \mathbb{R}\_{\text{TH}}} & -\frac{D}{L\_1} \\ \frac{D}{\mathbb{C}\_1} & 0 \end{bmatrix} \begin{bmatrix} -\frac{1}{L\_1} & 0 \\ 0 & \frac{1}{\mathbb{C}\_1 \mathbb{R}\_{\text{TH}}} \end{bmatrix} \begin{bmatrix} V\_{\text{OUT}} \\ V\_{\text{TH}} \end{bmatrix} \tag{41}$$

Calculating out Eq. (41) yields the DC value results given in Eqs. (42) and (43) for IO1 and VIN.

$$I\_{\rm O1} = \frac{V\_{\rm TH}}{R\_{\rm TH}D} - \frac{V\_{\rm OUT}}{R\_{\rm TH}D^2} \tag{42}$$

$$V\_{\rm IN} = \frac{V\_{\rm OUT}}{D} \tag{43}$$

The result in Eq. (43) allows to calculate the DC value duty cycle D, needed to maintain the input voltage VIN, of the PWM DC-DC converter that is equal to the output voltage of the solar tower VST ¼ VIN ¼ 450 V. Thus, the duty cycle D ¼ VOUT / VIN ¼ 124 V / 450 V ¼ 0.28.

The linear model of the open loop transfer function GV(s), for the voltage step down PWM DC-DC converter at the operating point that can be used to evaluate small signal variations in the duty cycle control variable d used to control the voltage VIN, can be developed by adding a small signal AC perturbation to the DC value D, of the duty cycle. Since a positive increase of the duty cycle d, causes a decrease in VIN as confirmed by Eq. (43), it becomes convenient for modeling purposes to express the duty cycle in terms of a new variable <sup>d</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> – <sup>d</sup> <sup>¼</sup> <sup>D</sup><sup>0</sup> <sup>þ</sup> ^d<sup>0</sup> , where D<sup>0</sup> is the DC value of d<sup>0</sup> and ^d<sup>0</sup> is the small signal AC perturbation. The linear model of the open loop transfer function for the PWM DC-DC converter provides the response to the small signal AC perturbations ^d<sup>0</sup> around the DC value D<sup>0</sup> . The averaged state space equation can be formulated in terms of the new variable d<sup>0</sup> according to Eq. (44).

$$\dot{\mathbf{x}} = \left( (1 - d')A\_1 + (d')A\_2 \right) \mathbf{x} + \left( (1 - d')B\_1 + (d')B\_2 \right) \mathbf{u} \tag{44}$$

In Eq. (44), the vector x ¼ X þ x^ and vector u ¼ U þ u^, where each vector comprises the DC value and a small signal perturbation. The Eq. (44) can be expanded as shown in Eq. (45).

$$\frac{d(\hat{X} + \hat{\boldsymbol{x}})}{dt} = \left( (1 - D^{'} - \hat{d}^{'}) A\_{1} + (D^{'} + \hat{d}^{'}) A\_{2} \right) (\hat{X} + \hat{\boldsymbol{x}}) + \left( (1 - D^{'} - \hat{d}^{'}) B\_{1} + (D^{'} + \hat{d}^{'}) B\_{2} \right) (\boldsymbol{U} + \hat{\boldsymbol{u}})$$

$$= \boldsymbol{X} A\_{1} + \boldsymbol{X} D^{'} (-A\_{1} + A\_{2}) + \hat{\boldsymbol{x}} \hat{d}^{'} (-A\_{1} + A\_{2}) + \hat{\boldsymbol{x}} A\_{1} + \hat{\boldsymbol{x}} D^{'} (-A\_{1} + A\_{2}) + \hat{\boldsymbol{x}} \hat{d}^{'} (-A\_{1} + A\_{2})$$

$$+ \boldsymbol{U} B\_{1} + \boldsymbol{U} D^{'} (-B\_{1} + B\_{2}) + \boldsymbol{U} \hat{d}^{'} (-B\_{1} + B\_{2}) + \hat{\boldsymbol{u}} B\_{1} + \hat{\boldsymbol{u}} \hat{D}^{'} (-B\_{1} + B\_{2}) + \hat{\boldsymbol{u}} \hat{d}^{'} (-B\_{1} + B\_{2}) \tag{45}$$

In Eq. (45), discarding the DC and nonlinear terms yields Eq. (46).

L1 dIO1

> dt <sup>¼</sup> <sup>V</sup>TH � <sup>V</sup>IN RTH

The state space form of the above four differential equations describing the PWM DC-DC converter having just a single phase φ1, is given as Eqs. (38) and (39) for SH1 being closed and

> IO1 VIN � �

The Eq. (38), has the form x\_ ¼ A1x þ B1u, and Eq. (39) has the form x\_ ¼ A2x þ B2u. The averaged state space equation can be expressed in a similar form x\_ ¼ Ax þ Bu, where the A and B matrices are calculated based on the duty cycle d, of the switch SH1, the matrices A1, B1, A2, B<sup>2</sup> and are given as A ¼ d � A<sup>1</sup> þ ð1 � dÞ � A<sup>2</sup> and B ¼ d � B<sup>1</sup> þ ð1 � dÞ � B2. The vector x, in the averaged state space equation contains the average values of the state variables <IO1> and <VIN> and the vector

When all transients in the PWM DC-DC converter have stabilized and steady state operation is achieved, then x\_ ¼ 0, thus the averaged state space equation can be used to express the DC

> � <sup>1</sup> C1RTH

<sup>I</sup>O1 <sup>¼</sup> <sup>V</sup>TH

tower VST ¼ VIN ¼ 450 V. Thus, the duty cycle D ¼ VOUT / VIN ¼ 124 V / 450 V ¼ 0.28.

D C1

Calculating out Eq. (41) yields the DC value results given in Eqs. (42) and (43) for IO1 and VIN.

<sup>V</sup>IN <sup>¼</sup> <sup>V</sup>OUT

The result in Eq. (43) allows to calculate the DC value duty cycle D, needed to maintain the input voltage VIN, of the PWM DC-DC converter that is equal to the output voltage of the solar

<sup>R</sup>TH<sup>D</sup> � <sup>V</sup>OUT

� D L1

� 1 L1

0

þ

2 6 4

þ

2 6 4

> � 1 L1

� 1 L1

3 7 5

3 5 IO1 VIN � �

C1 dVIN

<sup>0</sup> <sup>1</sup>

0 0 <sup>0</sup> � <sup>1</sup> C1RTH

L1

� <sup>1</sup> C1RTH

u, contains the input variables VOUT and VTH that are considered as DC values.

open, respectively.

<sup>X</sup> <sup>¼</sup> <sup>I</sup>O1 VIN " #

d dt

> d dt

IO1 VIN � �

> IO1 VIN � �

¼

170 Recent Improvements of Power Plants Management and Technology

2 6 4

¼

steady state equations given as Eqs. (40) and (41).

¼ �A�<sup>1</sup>

BU ¼ � <sup>L</sup>1C<sup>1</sup> D2

� 1 C1

2 4 dt ¼ �VOUT (36)

0

3 7 5

3 7 5

VOUT VTH � �

VOUT VTH � �

<sup>0</sup> <sup>1</sup> C1RTH

<sup>0</sup> <sup>1</sup> C1RTH

0

x\_ ¼ 0 ¼ AX þ BU (40)

<sup>0</sup> <sup>1</sup> C1RTH

0

<sup>R</sup>THD<sup>2</sup> (42)

<sup>D</sup> (43)

VOUT VTH " # (37)

(38)

(39)

(41)

$$\begin{split} \frac{d\hat{\mathbf{x}}}{dt} &= \hat{\mathbf{x}}A\_1 + \hat{\mathbf{x}}D'(-A\_1 + A\_2) + \hat{d}'\left((-A\_1 + A\_2)\mathbf{X} + (-B\_1 + B\_2)\mathbf{U}\right) \\\\ &= \hat{\mathbf{x}}A + \hat{d}'\left((-A\_1 + A\_2)\mathbf{X} + (-B\_1 + B\_2)\mathbf{U}\right) \end{split} \tag{46}$$

Taking the Laplace transform of the averaged state space equation which has the form x\_ ¼ Ax þ Bu, yields the result in Eq. (47).

$$\mathbf{x}\mathbf{X}(\mathbf{s}) - \mathbf{x}(\mathbf{0}) = A\mathbf{X}(\mathbf{s}) + B\mathbf{U}(\mathbf{s})\tag{47}$$

In Eq. (47), x(0) represents the initial value of the state vector in the time domain. For calculating the transfer function it is assumed that x(0) ¼ 0. Solving Eq. (47) for X(s) yields the result in Eq. (48).

$$\mathbf{x}(\mathbf{s}) = (\mathbf{s}I\_n - A)^{-1}\mathbf{x}(\mathbf{0}) + (\mathbf{s}I\_n - A)^{-1}B\mathbf{J}\mathbf{1}(\mathbf{s})\tag{48}$$

Applying the Laplace transform to Eq. (46), yields the result given in Eq. (49).

$$\hat{\mathbf{s}}\hat{\mathbf{x}}(\mathbf{s}) = \hat{\mathbf{x}}(\mathbf{s})A + \hat{d}'(\mathbf{s})\left((-A\_1 + A\_2)\mathbf{X} + (-B\_1 + B\_2)\mathbf{U}\right) \tag{49}$$

Solving Eq. (49) for x^ðsÞ yields the result in Eq. (50).

$$\hat{\mathbf{x}}(\mathbf{s}) = (\mathbf{s} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - A)^{-1} \hat{d}'(\mathbf{s}) \left( (-A\_1 + A\_2)X + (-B\_1 + B\_2)U \right) \tag{50}$$

The open loop transfer function given as GVðsÞ ¼ <sup>x</sup>^ðsÞ=^d<sup>0</sup> ðsÞ of the PWM DC-DC converter can be obtained by calculating out Eq. (50) to yield the result in Eq. (51).

$$\mathbf{G}\_{\rm V}(\mathbf{s}) = \frac{R\_{\rm TH}(V\_{\rm IN}D + \mathbf{s}L\_1I\_{\rm O1})}{\mathbf{s}^2 L\_1 \mathbf{C}\_1 R\_{\rm TH} + \mathbf{s}L\_1 + D^2 R\_{\rm TH}} \tag{51}$$

The analysis to yield the open loop transfer function for the multiphase voltage step down PWM DC-DC converter with N<sup>φ</sup> ¼ 32 phases requires creating a map that contains all the state space equations describing all possible states of the 64 IGBT switches during a period of duration TP, corresponding to a cycle of operation of the PWM DC-DC converter. For N<sup>φ</sup> ¼ 32 phases, there will be in effect 32 switching events occurring per cycle of operation with a phase shift φShift ¼ 11.25�. Thus, there will exist 32 state space equations of the form given in Eq. (52).

$$
\dot{\mathbf{x}} = A\_i \mathbf{x} + B\_i \boldsymbol{\mu} \tag{52}
$$

In Eq. (52), the matrices Ai and Bi describe the PWM DC-DC converter in the subinterval of duration ti between switchings of the parallel phases where <sup>P</sup><sup>N</sup><sup>φ</sup> <sup>i</sup>¼<sup>1</sup> ti <sup>¼</sup> <sup>T</sup>P. The solutions from Eq. (52) can be stacked to yield a discrete time equation valid over an entire period T<sup>P</sup> that effectively represents a map for the multiphase PWM DC-DC converter [100]. The state space averaging method used to obtain the open loop transfer function of the PWM DC-DC converter with a single phase given in Eq. (51), can also be applied to calculate the open loop transfer function for the PWM DC-DC converter with N<sup>φ</sup> ¼ 32 phases in parallel shown in Figure 7 [101]. An analysis of the present N<sup>φ</sup> ¼ 32 phase, closed loop multiphase PWM DC-DC converter including stability analysis, is highly complex especially when considering parametric variations among the parallel voltage step down converter circuits, and will be described in future work. It is also possible, to operate the multiphase PWM DC-DC converter in an open loop control mode, where a plant operator monitors and controls the electrical resistance of the electrolytic cells, and manually adjusts the duty cycle of the PWM DC-DC converter to deliver maximum power to the electrolytic cells. The plant operator balances the currents IO1–IO32, flowing in each phase of the PWM DC-DC converter by manually adjusting the amplitudes of the PWM control signals supplied to the gate terminals of the IGBT switches.

## 4. Sodium metal production plant operating characteristics

The solar cycle described in Section 2.2, the electrical design of the solar tower described in Section 3.1, and the electrical design of the sodium hydroxide (NaOH) electrolysis plant described in Section 3.2, entail that the voltage step down pulse width modulated (PWM) DC-DC converter supplying electricity from the solar tower PV device panel array directly to the NaOH electrolytic cells has to be operated according to a precise protocol.

Prior to sunrise occurring, the sodium hydroxide (NaOH) electrolytic cells are replenished to capacity with concentrated aqueous NaOH(aq) solution from the storage tanks shown in Figure 2, that are located adjacent to the Q-type metal building that houses electrical switch gear, voltage step down (PWM) DC-DC power converter units, the sodium hydroxide (NaOH) electrolytic cells, sodium (Na) metal packaging unit and chlorine (Cl2) gas separation and bottling unit. As sunrise commences, the solar tower photovoltaic (PV) device panel array begins generating electric power. The voltage step down PWM DC-DC converter supplies the electric power from the solar tower PV device panel array to the electrolytic cells at a fixed voltage given as VBAT ¼ VOUT ¼ 124 V. Before NaOH electrolysis and Na metal production can begin, the liquid aqueous NaOH(aq) has to be heated to evaporate all of the water (H2O(l)) content and fuse or melt the remaining anhydrous solid NaOH(s). The presence of water or moisture in the NaOH being electrolyzed can significantly reduce Na metal yield due to side reactions occurring between the Na metal and residual H2O. The sodium hydroxide melts into a liquid state at T<sup>f</sup> ¼ 594 � 2 K [102]. Therefore, electric power delivered by the solar tower initially at sunrise, has to be transmitted to electric heating elements for evaporating the water (H2O(l)) from the aqueous NaOH(aq) and to raise the temperature of the fused anhydrous NaOH(l) to the proper temperature for electrolysis. Once the proper temperature of the fused anhydrous NaOH(l) is reached, current from the solar tower can be transmitted to the cathode and anode electrodes of the 25 electrolytic cells electrically connected in series, to begin production of Na metal according to Eqs. (2) and (3).

Solving Eq. (49) for x^ðsÞ yields the result in Eq. (50).

172 Recent Improvements of Power Plants Management and Technology

1 0 0 1 � �

The open loop transfer function given as GVðsÞ ¼ <sup>x</sup>^ðsÞ=^d<sup>0</sup>

� AÞ �<sup>1</sup>^d<sup>0</sup> ðsÞ �

<sup>G</sup>VðsÞ ¼ <sup>R</sup>THðVIN<sup>D</sup> <sup>þ</sup> sL1IO1<sup>Þ</sup> <sup>s</sup><sup>2</sup>L1C1RTH <sup>þ</sup> sL<sup>1</sup> <sup>þ</sup> <sup>D</sup><sup>2</sup>

The analysis to yield the open loop transfer function for the multiphase voltage step down PWM DC-DC converter with N<sup>φ</sup> ¼ 32 phases requires creating a map that contains all the state space equations describing all possible states of the 64 IGBT switches during a period of duration TP, corresponding to a cycle of operation of the PWM DC-DC converter. For N<sup>φ</sup> ¼ 32 phases, there will be in effect 32 switching events occurring per cycle of operation with a phase shift φShift ¼ 11.25�. Thus, there will exist 32 state space equations of the form given in Eq. (52).

In Eq. (52), the matrices Ai and Bi describe the PWM DC-DC converter in the subinterval of

Eq. (52) can be stacked to yield a discrete time equation valid over an entire period T<sup>P</sup> that effectively represents a map for the multiphase PWM DC-DC converter [100]. The state space averaging method used to obtain the open loop transfer function of the PWM DC-DC converter with a single phase given in Eq. (51), can also be applied to calculate the open loop transfer function for the PWM DC-DC converter with N<sup>φ</sup> ¼ 32 phases in parallel shown in Figure 7 [101]. An analysis of the present N<sup>φ</sup> ¼ 32 phase, closed loop multiphase PWM DC-DC converter including stability analysis, is highly complex especially when considering parametric variations among the parallel voltage step down converter circuits, and will be described in future work. It is also possible, to operate the multiphase PWM DC-DC converter in an open loop control mode, where a plant operator monitors and controls the electrical resistance of the electrolytic cells, and manually adjusts the duty cycle of the PWM DC-DC converter to deliver maximum power to the electrolytic cells. The plant operator balances the currents IO1–IO32, flowing in each phase of the PWM DC-DC converter by manually adjusting the amplitudes of

The solar cycle described in Section 2.2, the electrical design of the solar tower described in Section 3.1, and the electrical design of the sodium hydroxide (NaOH) electrolysis plant described in Section 3.2, entail that the voltage step down pulse width modulated (PWM) DC-DC converter supplying electricity from the solar tower PV device panel array directly to

be obtained by calculating out Eq. (50) to yield the result in Eq. (51).

duration ti between switchings of the parallel phases where <sup>P</sup><sup>N</sup><sup>φ</sup>

the PWM control signals supplied to the gate terminals of the IGBT switches.

4. Sodium metal production plant operating characteristics

the NaOH electrolytic cells has to be operated according to a precise protocol.

ð�A<sup>1</sup> þ A2ÞX þ ð�B<sup>1</sup> þ B2ÞU

RTH

x\_ ¼ Aix þ Biu (52)

�

ðsÞ of the PWM DC-DC converter can

<sup>i</sup>¼<sup>1</sup> ti <sup>¼</sup> <sup>T</sup>P. The solutions from

(50)

(51)

x^ðsÞ¼ðs

The electric current is supplied to the 25 electrolytic cells electrically connected in series, by the synchronous parallel multiphase voltage step down PWM DC-DC converter at a fixed output voltage set by the utility scale battery given as VBAT ¼ VOUT ¼ 124 V. The current IOUT ¼ ICELL, supplied by the PWM DC-DC converter from the solar tower to the electrolytic cells is not constant throughout the day, increasing gradually from sunrise until the sun reaches its zenith, and subsequently decreasing gradually as the sun tracks west and eventually sets. To use the electric power generated by the solar tower PV device panel array most efficiently for Na metal production, it is necessary to configure the electrolytic cells as a variable electric load, wherein their electrical resistance can decrease gradually as the sun arcs toward zenith allowing ICELL to increase as more electric power is generated by the solar tower for electrolysis and subsequently, the electrical resistance of the electrolytic cells can begin to increase gradually as the sun passes beyond the zenith toward sunset, when ICELL must decrease as less electric power is generated by the solar tower for electrolysis.

The sodium (Na) metal production plant can effectively be controlled using only two adjustable parameters, including the set point reference voltage VSET, that controls the input voltage VIN, of the PWM DC-DC converter that is equal to the output voltage of the solar tower VIN ¼ VST, and the electrical resistance of the NaOH electrolytic cells. The set point reference voltage VSET can function either as a coarse or fine adjustment for the current supplied by the solar tower PV device panel array to the electrolytic cells at the fixed voltage VBAT ¼ VOUT ¼ 124 V. The electrical resistance of the electrolytic cells naturally increases slowly over time as the fused NaOH(l) reactant is decomposed by the current flowing between the electrodes of the cell according to Eqs. (2) and (3). The electrolytic cells can be designed for example, with a controllable electrical resistance that can remain relatively constant (or increase or decrease as needed) even as the fused NaOH(l) reactant is consumed, by varying the spacing between the anode and cathode electrodes, where the electrodes can be moved closer together to compensate the loss of reactant volume as it is consumed in the cell.

It is possible to calculate the quantity of Na metal produced throughout the year by the selfcontained sodium (Na) metal production plant sited in the different geographic locations given in Table 2, based on the hours of daylight and the prevailing air mass conditions. It is not necessary to specify a detailed design for the NaOH electrolytic cell to generate an accurate daily estimate of Na metal production yield throughout the year, if it is assumed that maximum electric power available from the solar tower PV device panel array can always be supplied to the electrolytic cells by appropriately controlling the electrical resistance of the NaOH electrolytic cells together with the set point reference voltage VSET of the PWM DC-DC converter. It is assumed that electrolysis of NaOH, and thereby Na metal production can only occur if the available current supplied by the solar tower PV device panel array has a minimum threshold value of IST ¼ 3,000 A. In Figure 10, the production yield of Na metal is calculated for each day of the hypothetical year 2015, for electrolysis of pure NaOH according to Eq. (2), for the four geographic locations listed in Table 2, using the solar position algorithm (SPA) described in Solar position algorithm for solar radiation applications written by Reda & Andreas in 2004, that is a refined algorithm based on the book, The Astronomical Algorithms written by Meeus in 1998, and is presently regarded as the most accurate [73, 74]. The SPA allows the solar zenith θsz, and azimuth γs, angles to be calculated with uncertainties of �0.0003� in the range between -2000 to 6000 years.

Figure 10. Calculated sodium (Na) metal daily production yields throughout the hypothetical year 2015, for El Paso, Texas (thick solid), Alice Springs, Australia (thin solid), Bangkok, Thailand (thick dash) and Mbandaka, DRC (thin dash).

The calculation in Figure 10 provides the expected daily sodium (Na) metal production yield under the assumption that the energy conversion efficiency ηPV ¼ 90% for the solar tower PV device panel array and furthermore, current is only transmitted to the electrolytic cells when the solar tower PV device panel array receives sufficient solar irradiance to produce the minimum threshold value of current IST ¼ 3,000 A. The calculation in Figure 10, uses the SPA algorithm to determine the solar zenith angle θsz, throughout the day from sunrise to sunset for each day of the hypothetical year 2015, and calculates the air mass (AM) using Eq. (18). The direct normal total (spectrally integrated) solar irradiance incident onto the solar tower PV device panel array is calculated in turn, using Eq. (19) in a real time manner, as the solar position and air mass change throughout the day for each day of the year. The results from Figure 10, show that consistent quantities of Na metal are produced throughout the year when the self-contained sodium (Na) metal production plant is located as near as possible to the equator where the length of the day is the most uniform. Further away from the equator, the variability in the length of the day increases however, even at a latitude ϕ ¼ þ31.8�, in El Paso, Texas, the variability in the length of the day is not so significant as to render Na metal production uneconomical during the winter months when the daylight interval becomes reduced. The mean daily sodium (Na) metal production yields for the hypothetical year 2015 are calculated as 41,998 kg/day for El Paso, 41,884 kg/day for Alice Springs, 40,281 kg/day for Bangkok and 40,947 kg/day for Mbandaka, assuming a mean annual aerosol optical thickness value k<sup>a</sup> ¼ 0.1 reflecting clear days and uninterrupted Na production for all 365 days of the year. Notwithstanding the variability in daily Na metal production throughout the year, El Paso, Texas has the highest elevation above mean sea level when compared with Alice Springs, Bangkok and Mbandaka as shown in Table 2, resulting in an increased solar irradiance incident onto the PV device panel array and thus, the highest mean daily Na metal production. A conservative estimate for the true mean daily sodium (Na) metal production yield for the year might be mNa ¼ 30,000 kg/day of Na metal, as a consequence of nonproductive days due to inclement weather and required plant maintenance. The results in Figure 10, clearly demonstrate that the scalable, self-contained solar powered electrolytic sodium (Na) metal production plant can be constructed almost anywhere on earth and especially in the southwestern region of the U.S.A., to achieve a hydrogen (H2(g)) fuel, sustainable, closed clean energy cycle.
