Thermo-Economic Optimization of an Idealized Solar Tower Power Plant Combined with MED System

Based on the reversible heat engine model, theoretical analysis is carried out for economic performance of a solar tower power plant (STPP) combined with multi-effect desalination (MED). Taking total revenue of the output power and the fresh water yield per unit investment cost as the economic objective function, the most economical working condition of the system is given by analyzing the influence of the system investment composition, the receiver operating temperature, the concentration ratio, the efficiency of the endoreversible heat engine, and the relative water price on the economic parameters of the system. The variation curves of the economic objective function are given out when the main parameter is changed. The results show that the ratio of water price to electricity price, or relative price index, has a significant impact on system economy. When the water price is relatively low, with the effect numbers of the desalination system increasing, and the economic efficiency of the overall system worsens. Only when the price of fresh water rises to a certain value does it make sense to increase the effect. Additionally, the threshold of the fresh water price to the electricity price ratio is 0.22. Under the conditions of the current price index and the heliostat (or reflector), the cost ratio and the system economy can be maximized by selecting the optimum receiver temperature, the endoreversible heat engine efficiency, and the optimum concentration ratio. Given the receiver surface temperature and the endoreversible heat engine efficiency, increasing the system concentration ratio of the heliostat will be in favor of the system economy.


Introduction
The solar tower power plant (STPP) is considered one of the most promising solar power generation technologies. There are already dozens of STPPs in operation or construction in the world. The largest of them can reach up to 100 MW [1]. In 2050, the installed capacity of 120 GW, 405 GW, or even 1000 GW could be reached globally, in which the concentrated solar power (CSP) will meet 13-15% of global electricity demand [2]. STPP technology offers the greatest potential for high efficiency. However, at the current stage, it is not economically comparable to conventional power plants. Further efforts should be made to increase the efficiency and reduce the cost [3]. Huge amounts of waste heat are released during STPP operation. Utilizing the waste heat to realize seawater desalination is an effective method to enhance the system economy. As the power plant itself also demands a large amount of fresh water, seawater desalination using its own waste heat can address the issue of water. Therefore, it is of great significance to analyze the process and the economy of the STPP combined with a multi-effect desalination (MED) system.
In general, solar collecting systems and heat engine power generation systems need to be well matched to produce the best economic benefits. How to achieve the best match between the two In the system shown in Figure 1, it is assumed that the solar irradiance input to the heliostat field of the STPP system is denoted by I s , the concentration ratio of the system is denoted by C, the light entrance area of the receiver is A R , the absorptivity of the receiver is α R , the surface operating temperature of the receiver is T R , the ambient temperature is T 0 , and the radiation heat transfer from the environment to the receiver surface is negligible. The useful heat that can be obtained by the receiver can be written as [11]: where ε R is emissivity of the receiver surface, h c is convective heat transfer coefficient between receiver and environment, and σ is the Stefan-Boltzmann constant. Reference [11] gives a thermodynamic analysis for an idealized solar tower power plant, and proposes a relation between power output and system operating or structural parameters: where Q is the sum of heat gained by the system. η is the endoreversible heat engine efficiency. R is the total thermal resistance of the system, R = R H + R L , where R H is the total thermal resistance of the front end of the power system, and R L is total thermal resistance of the rear end of the power system, including the thermal resistance of the exchangers of the desalination system.

Thermo-Economic Model
Economic analysis is utilized to analyze input and output for a specific system. The input of the system refers to the sum of the construction costs and the maintenance costs. The output refers to the sum of the gains from the power output of the system and the gains from the fresh water production. Therefore, the economic efficiency of the system can be described by the ratio of output to input.
Assuming that the total amount of electricity generated during the lifetime of the solar tower thermal power plant is denoted by W, total amount of fresh water production is denoted by m e , and overall investment of the system is denoted by M, then the investment benefit of the power plant can be expressed as: where β is electricity price, RMB/kWh (RMB approximately equals 0.15 US dollars), γ is fresh water price, RMB/kg. These prices have considered the operational cost of the system. The smaller the F, the worse the system economy. Thus, the maximum value of F should be pursued. In Figure 1, the waste heat (without considering any other use value) released from the condenser of the steam power system is denoted by Q L . In this paper, a set of low-temperature multi-effect seawater desalination system is adopted to substitute the condenser to achieve the dual-purpose power plant. Based on the endoreversible heat engine model in Reference [11], the heat released from the system is calculated as: For an ideal low-temperature MED system, under ideal conditions, the heat obtained for each effect is approximately equal, and the heat transfer area per effect is also nearly the same. Assuming there are n effects, then, in the heat release process of the steam power cycle, that is, the seawater desalination process, the total thermal resistance is: In Equation (5), A represents heat exchange area, and U represents heat transfer coefficient. If the sensible heat consumption and the heat loss of seawater are not taken into account, the relationship between water production and Q L is: Here, h f g,T is water latent heat of water vaporization at temperature T. Assuming that 100% of the heat energy is used to evaporate seawater, substituting Equations (4) and (6) into Equation (3): On the other side, the system investment is directly proportional to the solar concentrator (heliostat) area, receiver area, and heat exchanger area, and can be written as: Entropy 2018, 20, 822

of 14
Here, ρ is the price of the receiver per unit area, A M is the heat exchange area of the heat exchanger 1, A H is the heat exchange area of the heat exchanger 2, and A L is the heat exchange area of the first effect seawater desalination. δ is the price of the exchanger per unit area.
If we only analyze the relationship between investment benefits and system parameters, substituting Equations (2) and (8) into Equation (7) yields the following relationship: Assuming that the heat exchangers have the same area and the same heat transfer coefficient, then Hence, Reference [18] gives the convection heat transfer coefficient of the tower power plant receiver to the environment: Substituting the above equation into Equation (1), and substituting it into Equation (11) yields: Since η is a function of T R , Equation (12) can finally get the relationship between F and T R . On the other hand, it can be obtained from Equation (2) that Substitute it into Equation (11), then Define the proportional parameter x of the solar concentrator (heliostat) cost relative to the total investment costs: Then, Equation (14) can be written as: Let θ = γ βh f g,T be a relative price index of fresh water price versus electricity price, a new dimensionless economic objective function f can be defined as: Apparently, f can reflect economic benefit of the dual-purpose power plant system. The larger f is, the better the system economic benefit is. Equation (16) also shows that there are multiple factors that affect the system's economic benefit. Hence, various parameters of the system should be reasonably selected to maximize the total economic benefits of the system.

Impacts of Relative Price Index and MED Effect Numbers on Economic Objective Function
Obviously, the economic benefits of a solar tower power plant system strongly depend on the price of water γ, that is, it strongly depends on the relative price index θ. Meanwhile, the number of MEDs has a strong influence on the economic objective function. To evaluate this effect, assume x = 0.5, T R = 900 K, T 0 = 300 K, η = 0.4, and substitute them into Equation (16): This equation shows the effect of n and θ on the economic objective function. Figure 2 shows the three-dimensional surface calculated by this equation. In order to see the influence of the price index on the economic objective function more intuitively, the two-dimensional diagrams of the influence of n on the economic objective function when θ is equal to 0.03, 0.22, and 0.3 is plotted, as shown in Figure 3. Here, = 0.03 describes the situation when water price is 10 RMB/ton and electricity price is 0.5 RMB/kWh, which is close to fresh water price and electricity price in cities of China currently.  In order to see the influence of the price index on the economic objective function more intuitively, the two-dimensional diagrams of the influence of n on the economic objective function when θ is equal to 0.03, 0.22, and 0.3 is plotted, as shown in Figure 3. Here, θ = 0.03 describes the situation when water price is 10 RMB/ton and electricity price is 0.5 RMB/kWh, which is close to fresh water price and electricity price in cities of China currently.
From Figures 2 and 3, it can be observed that when relative price index θ is relatively small, no matter how the number of effects n increases, economic efficiency of the system always decreases. This shows that when the fresh water price is relatively low, or when the electricity price is relatively high, the establishment of a dual-purpose power plant has no economic benefit. Only when θ ≥ 0.22, adding a MED system will not hurt economic benefits of the total system, and be economically meaningful. When θ is larger, it is more economical to add the seawater desalination system. In water-scarce areas, the price of fresh water soars up, making dual-purpose power plants become very meaningful. The larger θ, the more economic benefit we can get by adding a desalination system. In order to see the influence of the price index on the economic objective function more intuitively, the two-dimensional diagrams of the influence of n on the economic objective function when θ is equal to 0.03, 0.22, and 0.3 is plotted, as shown in Figure 3. Here, = 0.03 describes the situation when water price is 10 RMB/ton and electricity price is 0.5 RMB/kWh, which is close to fresh water price and electricity price in cities of China currently.  From Figures 2 and 3, it can be observed that when relative price index is relatively small, no matter how the number of effects increases, economic efficiency of the system always decreases. This shows that when the fresh water price is relatively low, or when the electricity price is relatively high, the establishment of a dual-purpose power plant has no economic benefit. Only when ≥ 0.22, adding a MED system will not hurt economic benefits of the total system, and be economically meaningful. When is larger, it is more economical to add the seawater desalination system. In water-scarce areas, the price of fresh water soars up, making dual-purpose power plants become very meaningful. The larger , the more economic benefit we can get by adding a desalination system.

The Impact of Endoreversible Heat Engine Efficiency on Economic Objective Function
Efficiency of the endoreversible heat engine, which can strongly affect economic objective function, is an important parameter of the system. Assuming x = 0.4 (in practical system, the solar concentrator cost relative to the total investment cost is in the range of 0.35 to 0.55), T R = 900 K, T 0 = 300 K, n = 3, 5, 7, θ = 0.03, 0.07, 0.14, and substituting in Equation (16), the efficiency of how the endoreversible heat engine influences the objective function can be shown in Figure 4.
Based on Figure 4, when relative price index θ is small, the maximum value of the objective function decreases as number of effects of desalination system increases. But at this time, the objective function always has a maximum with the change of the heat engine efficiency. However, when θ becomes larger, objective function increases faster at less endoreversible heat engine efficiency. In addition, objective function increases more slowly at a range of higher endoreversible heat engine efficiency, causing the economic objective function to eventually change into a monotonic function with the change of the endoreversible heat engine efficiency. There will no longer be a maximum value, but the total economic benefits will still increase, as Figure 4 shows. However, it is critical to note that choice of the number of MED effects, limited by T L and T 0 , cannot be increased without a limit.

The Impact of Endoreversible Heat Engine Efficiency on Economic Objective Function
Efficiency of the endoreversible heat engine, which can strongly affect economic objective function, is an important parameter of the system. Assuming = 0.4 (in practical system, the solar concentrator cost relative to the total investment cost is in the range of 0.35 to 0.55), = 900 K, = 300 K, = 3, 5, 7, = 0.03, 0.07, 0.14，and substituting in Equation (16), the efficiency of how the endoreversible heat engine influences the objective function can be shown in Figure 4.
Based on Figure 4, when relative price index is small, the maximum value of the objective function decreases as number of effects of desalination system increases. But at this time, the objective function always has a maximum with the change of the heat engine efficiency. However, when becomes larger, objective function increases faster at less endoreversible heat engine efficiency. In addition, objective function increases more slowly at a range of higher endoreversible heat engine efficiency, causing the economic objective function to eventually change into a monotonic function with the change of the endoreversible heat engine efficiency. There will no longer be a maximum value, but the total economic benefits will still increase, as Figure 4 shows. However, it is critical to note that choice of the number of MED effects, limited by TL and T0, cannot be increased without a limit.

Optimum Endoreversible Heat Engine Efficiency
From Figure 4, it can be seen that when the relative price index is small, the economic objective function has a maximum with the change of heat engine efficiency. In order to obtain the maximum

Optimum Endoreversible Heat Engine Efficiency
From Figure 4, it can be seen that when the relative price index is small, the economic objective function has a maximum with the change of heat engine efficiency. In order to obtain the maximum value of the economic objective function and optimize the parameters, for Equation (16), can obtain the equation with the optimum efficiency: Apparently, the optimum efficiency is related to receiver temperature T R , the relative price index θ, and number of effects n of the desalination system. When n is taken as 5, the variation of η opt with T R and θ is shown in Figure 5. It can be found that when T R = 1500 K and θ = 0.03, the optimum efficiency is maximum, namely η opt = 0.5149; when T R = 500 K and θ = 0.08, the optimum efficiency is maximum, namely η opt = 0. This means that there is neither the optimum engine efficiency for the objective function nor the work output. Substituting Equation (18) into Equation (16), the maximum economic objective function can be written as: Equation (19) shows that there are several parameters that could affect the maximum objective function. When n, , , and are given, the effect of on the maximum objective function can be obtained. Let = 3, 5, 7; = 0.03; = 0.5; = 300 K, to gain the effect of on the maximum economic objective function, as shown in Figure 6. In this figure, maximum objective function increases with the receiver temperature. However, it decreases as number of effects of the MED system increases.
Therefore, choosing to operate at a higher receiver temperature will benefit the system economics. The number of effects of the MED system should be reasonably selected. When the relative price index is low, the number of effects should not be excessive. Substituting Equation (18) into Equation (16), the maximum economic objective function can be written as: Equation (19) shows that there are several parameters that could affect the maximum objective function. When n, θ, x, and T 0 are given, the effect of T R on the maximum objective function can be obtained. Let n = 3, 5, 7; θ = 0.03; x = 0.5; T 0 = 300 K, to gain the effect of T R on the maximum economic objective function, as shown in Figure 6. In this figure, maximum objective function increases with the receiver temperature. However, it decreases as number of effects of the MED system increases. Therefore, choosing to operate at a higher receiver temperature will benefit the system economics. The number of effects of the MED system should be reasonably selected. When the relative price index is low, the number of effects should not be excessive.
Equation (19) shows that there are several parameters that could affect the maximum objective function. When n, , , and are given, the effect of on the maximum objective function can be obtained. Let = 3, 5, 7; = 0.03; = 0.5; = 300 K, to gain the effect of on the maximum economic objective function, as shown in Figure 6. In this figure, maximum objective function increases with the receiver temperature. However, it decreases as number of effects of the MED system increases.
Therefore, choosing to operate at a higher receiver temperature will benefit the system economics. The number of effects of the MED system should be reasonably selected. When the relative price index is low, the number of effects should not be excessive.  Figure 6. Effect of receiver temperature on maximum objective function. Figure 6. Effect of receiver temperature on maximum objective function.

Effect of Receiver Temperature on the System Economy
Operating temperature of the receiver surface is also an important parameter that will have an impact on the system economy. To figure out the effect of T R on f , transform Equation (2) into Then, substitute Equation (20) into Equation (16): Additionally, substitute . With h c = T R 60 + 5 3 , we can get: Assuming α R = ε R = 1, I R = 1000 W/m 2 , U = 15,000 W/m 2 ·K, x = 0.5, T 0 = 300 K, θ = 0.03, and n = 5, the effect of solar receiver temperature on the objective function is calculated and plotted as shown in Figure 7 under different concentration ratio conditions. It can be seen from Figure 6 that when θ = 0.03, there is always an optimum receiver temperature that makes the objective function maximum under different concentration ratio conditions. It indicates that selecting the optimum receiver temperature under different concentrating conditions is very important.
( ) + . Assuming = = 1, = 1000 W/m , = 15,000 W/m • K, = 0.5, = 300 K, = 0.03, and = 5, the effect of solar receiver temperature on the objective function is calculated and plotted as shown in Figure 7 under different concentration ratio conditions. It can be seen from Figure 6 that when = 0.03, there is always an optimum receiver temperature that makes the objective function maximum under different concentration ratio conditions. It indicates that selecting the optimum receiver temperature under different concentrating conditions is very important. Objective function/f C=2000 C=1000 C=500 The temperature of the receiver T R /K C=100 Figure 7. Effect of receiver temperature on economic objective function under different concentration ratios.
In addition, Figure 8 shows the effect of receiver temperature on economic objective function when the concentration ratio is 1000, and number of effects of the MED system is 5. As can be seen from Figure 8, when is relatively small, f has a maximum value with the change of . As increases, maximum value of f increases. However, the receiver temperature corresponding to it becomes lower, resulting in f having no maximum when increases to a certain value.

Figure 7.
Effect of receiver temperature on economic objective function under different concentration ratios.
In addition, Figure 8 shows the effect of receiver temperature on economic objective function when the concentration ratio is 1000, and number of effects of the MED system is 5. As can be seen from Figure 8, when θ is relatively small, f has a maximum value with the change of T R . As θ increases, maximum value of f increases. However, the receiver temperature corresponding to it becomes lower, resulting in f having no maximum when θ increases to a certain value. Understanding this trend will be of great significance to the design and implementation of a solar dual-purpose power plant. Understanding this trend will be of great significance to the design and implementation of a solar dual-purpose power plant. Objective function/f C=1000; n=5; x=0.5 θ=0.3 The temperature of the receiver T R /K

Effect of Concentration Ratio on System Economics
It is important to select a concentration ratio for the system with a given operating temperature of the receiver. According to Equation (22), assuming = = 1 , = 300 K , = 1000 W/m , = 15,000 W/m • K, = 0.5, = 900, 1200 K and 1500 K, = 0.03 and = 5, the effect of a concentration ratio on system economics under different receiver temperatures is plotted in Figure 9. It can be seen from Figure 8 that, under different receiver temperatures, there is always an optimum concentration ratio that can maximize the economic objective function. For instance, when receiver temperature is 1200 K, the optimum concentration ratio is about 1400. They are close

Effect of Concentration Ratio on System Economics
It is important to select a concentration ratio for the system with a given operating temperature of the receiver. According to Equation (22), assuming α R = ε R = 1, T o = 300 K, I R = 1000 W/m 2 , U = 15,000 W/m 2 ·K, x = 0.5, T R = 900, 1200 K and 1500 K, θ = 0.03 and n = 5, the effect of a concentration ratio on system economics under different receiver temperatures is plotted in Figure 9. It can be seen from Figure 8 that, under different receiver temperatures, there is always an optimum concentration ratio that can maximize the economic objective function. For instance, when receiver temperature is 1200 K, the optimum concentration ratio is about 1400. They are close to the current working conditions of the actual system. To obtain the relationship between the optimum concentration ratio and the receiver temperature, find ∂ f ∂C = 0, then get the relationship between T R and C, which can be shown in Figure 10. Figure 10 shows that the higher the receiver temperature, the greater the optimum concentration ratio. However, the stronger the solar radiation is, the smaller the optimum concentration ratio is.
As Figure 11 shows, when θ is relatively small, objective function exists at a maximum along with the concentration ratio variation. Additionally, the maximum objective function and the corresponding optimum concentration ratio increases as θ increases. When θ increases above a certain value, objective function becomes a monotonic function. Under this situation, a maximum value will not appear. It implies that the system has better economic performance under the condition of a larger concentration ratio.  e function/f The temperature of the receiver T R /K Objective function/f Concentration ratio/C θ=0.15 Figure 11. Effect of concentration ratio on economic object function under different relative price indexes. Figure 11. Effect of concentration ratio on economic object function under different relative price indexes.

Conclusions
After conducting comprehensive cost-benefit analysis of a STPP combined with a MED system, the objective function of the system is given. Based on the objective function, the effect of operating parameters of the system on the economic objective function is analyzed. The conclusions are described below.
(1) When water-to-electricity relative price index is small, it is meaningless to add the number of effects of the MED system. Increasing number of effects is meaningful only after water price increases above the critical value, which is θ = 0.22. (2) When the price index is relatively low, objective function can reach maximum as a function of endoreversible heat engine efficiency. However, as the price index increases, within the range of relatively low endoreversible heat engine efficiency, objective function increases faster; within the range of relatively high endoreversible heat engine efficiency, objective function increases more slowly. This makes economic objective function vary with endoreversible heat engine efficiency as a monotonic function. It means the total economic benefit will keep growing, without maximum value. (3) When the price index is relatively low, the maximum objective function increases with receiver temperature, but decreases with the number of effects of the MED system. (4) When the price index is relatively low, objective function can reach maximum as a function of receiver temperature. When the price index increases, maximum value of economic objective function increases, but the corresponding receiver temperature is reduced until economic objective function has no maximum value. (5) The curve of the effect of the concentration ratio on the economic objective function is different when the relative price index is given different values. When the relative price index is small, economic objective function always has a maximum value with the change of concentration ratio. With the increase of the price index, economic objective function increases, and optimum concentration ratio increases as well. When the relative price index increases above a certain value, the economic objective function varies with concentration ratio as a monotonic increasing function, without maximum value.