Analysis of Gas-Steam CHP Plants Without and with Heat Accumulator and HTGR Reactor

Abstract

This study analyzes the thermodynamic and economic viability of modified high-temperature gas-cooled reactor (HTGR) gas-steam combined heat and power (CHP) systems compared to conventional CHP plants. The research addresses the critical need for efficient and sustainable energy production methods. Using comprehensive thermodynamic modeling and economic analysis, the study evaluates system performance under various operating conditions. Key findings reveal that modified CHP plants with HTGR and turboexpanders (TEs) demonstrate significantly higher efficiency and lower heat generation costs compared to conventional gas turbine (GT) CHP plants, despite higher initial capital investments. The modified systems achieve electricity generation efficiencies up to 48%, surpassing traditional nuclear power plants. The absence of CO₂ emissions and lower fuel costs in HTGR systems contribute to their economic advantage. This research provides novel insights into the potential of HTGR technology in CHP applications, offering a promising solution for future energy systems. The study’s originality lies in its comprehensive comparison of conventional and modified CHP systems, considering both thermodynamic and economic aspects, which has not been extensively explored in existing literature.

1. Introduction

The paper focuses on analyzing and comparing gas-steam combined heat and power (CHP) plants in two configurations: conventional CHP plants with gas turbines (GTs) and modified CHP plants where the GT is replaced with a high-temperature gas-cooled reactor (HTGR) and turboexpander (TE). The analysis covers both thermodynamic and economic aspects of these systems, including plants with and without heat accumulators.

The main novelty of the work lies in the comprehensive comparison between conventional gas-steam CHP plants and modified versions using HTGR and TEs, an area that has not been extensively covered in existing literature. The paper provides new insights into the potential benefits of replacing GTs with HTGR and TEs in CHP systems, particularly in terms of economic viability and fuel efficiency. The study offers a comprehensive analysis of how factors like pressure ratios, temperatures, and reactor thermal power affect the performance and economics of the proposed modified CHP systems.

The paper [1] presents a thermodynamic and economic analysis of a gas-steam plant with a modified configuration incorporating a nuclear HTGR and a TE. This study conducts a comparative analysis of CHP plants, where the GT is replaced with a nuclear HTGR reactor and a TE. The CHP plants with and without a heat accumulator are presented in Figure 1 and Figure 2, while modified CHP plants are shown in Figure 3 and Figure 4. The comparison of conventional gas-steam CHP plants with modified CHP plants is significant, as it provides additional thermodynamic and, importantly, economic insights. Such analyses have been absent from the literature until now.

This research is particularly relevant given the planned future construction of power plants operating on the Clausius–Rankine cycle with electrical outputs up to 300 MW, based on water-cooled small modular reactors (SMRs). The first such power plant is slated for construction in Canada. However, one must question whether this will actually happen and, if so, when. Water-cooled SMRs are still in the early conceptual stage. While simply reducing the geometric dimensions of large water-cooled nuclear reactors is not problematic, in our opinion, scaling up the laws of nature is impossible.

Newly built reactors must be inherently safe and self-adjusting using natural laws. This means that phenomena leading to safe reactor shutdown in emergencies must occur spontaneously, cooling the reactor sufficiently to remove decay heat and prevent hydrogen explosions. Such explosions can result from hydrogen produced by steam water radiolysis due to inadequate reactor core cooling. The natural circulation of cooling water driven by hydrostatic buoyancy forces is an example of such a safety mechanism. In contrast, reactors cooled by mechanically driven pumps and fans are vulnerable to power supply failures, which can lead to explosions.

Therefore, developing and experimentally testing unconventional technical solutions that allow buoyancy forces to operate effectively in significantly scaled-down reactors is crucial. In contrast, small helium-cooled HTGRs with helium outlet temperatures of approximately 1300 K already exist and are operational. This reactor type is particularly suitable for CHP generation.

Importantly, hierarchical gas-steam technology (Figure 4) demonstrates the highest net efficiency of electricity generation compared to other energy technologies, exceeding 60%. This is in stark contrast to pressurized water reactor power stations using only the Clausius–Rankine cycle, which have less than half this efficiency. This lower efficiency is due to the need to keep water (serving as both moderator and coolant) below its critical point of 374.2 °C to prevent evaporation and maintain reactor cooling.

It would be valuable to compare the thermodynamic and economic efficiency of power and CHP plants with SMRs against hierarchical double-cycle power and CHP plants with HTGRs. The efficiency of HTGR systems is certainly higher due to their greater electricity generation efficiency and lower unit capital costs. However, quantifying this efficiency difference remains an important question.

The efficiency of gas-steam power plants is even higher than that of fuel cells when accounting for the energy efficiency of producing the hydrogen used as fuel, which is essential. While fuel cells can achieve higher theoretical efficiencies than conventional power plants by converting chemical energy directly into electricity, they are still bound by thermodynamic limitations. The maximum theoretical efficiency of fuel cells is determined by the change in Gibbs free energy of the reaction, not the total chemical energy of the fuel. Power plants with mechanical engines, such as turbines, have efficiencies significantly below 100% in accordance with the second law of thermodynamics. Although it might appear that fuel cells have a significant advantage over gas and steam plants, this is not necessarily the case in practice. Both technologies face thermodynamic constraints and practical challenges that limit their real-world efficiencies. Modern combined-cycle GT plants can achieve efficiencies above 60%, narrowing the gap with fuel cells. Furthermore, when considering exergy analysis, both fuel cells and heat engines are subject to similar thermodynamic limitations. Consequently, while fuel cells offer certain advantages, they do not hold an overwhelming efficiency advantage over advanced conventional power plants.

In the modern world, electricity is indispensable and is the most valuable form of energy. It should be generated with the highest possible efficiency using naturally occurring fuel resources to meet global energy needs. Given these considerations, hierarchical gas-steam technology is crucial. Although the efficiency of electricity generation in gas-steam CHP plants is slightly lower than that of gas-steam power plants, the CHP generation enables significant savings in chemical and nuclear energy from fossil fuels. This is accompanied by a reduction in harmful emissions. In contrast, separate heat generation in district heating plants and electricity generation in gas-steam power plants consumes more fuel in total, despite the higher electrical efficiency.

This underscores the importance of CHP and gas-steam CHP plants. An additional significant feature is the replacement of the GT with an HTGR and a TE. Nuclear fuel is abundant on Earth, with supplies projected to last billions of years, while natural gas deposits are expected to be depleted relatively quickly.

In the works of other authors, studies related to the process of improving CHP systems can be found. For instance, a novel scheme consisting of a plasma gasifier, solid oxide fuel cells (SOFC), GT, and supercritical CO₂ cycle has been developed for power and heat cogeneration [2]. Another study [3] examines possible locations where heat recovery can occur in a triple-pressure cogeneration combined cycle power plant and calculates the amount of recoverable waste heat. In [4], researchers propose improving the accuracy of forecasting energy carrier production in CHP plants by extending the existing computer system for supervising operations with an additional calculation module. An interesting study regarding the optimization of CHP system operation is presented in [5], where composite indicators are proposed to provide a more comprehensive assessment of plant performance for the design and optimization phase, as well as a possible approach to support mechanisms.

Another research [6] presents an optimization model based on energy consumption. The thermal-electric load distribution method is proposed based on the energy consumption characteristics of CHP plants. Many available publications also deal with the use of biomass instead of fossil fuel. Some of these address small-scale CHP plants, where technical, environmental, and economic analyses have been performed [7,8,9], while others focus on large-scale facilities [10,11,12]. Study [5] proposes composite indicators to provide a more comprehensive assessment of CHP plant performance. The authors develop an optimization model for CHP plant sizing and operational strategy for district heating applications. In [6], the researchers propose a thermal-electric load distribution method based on the energy consumption characteristics of CHP plants. They develop an optimization model for large-scale CHP plants, which aligns with the manuscript’s focus on improving CHP system efficiency and operation. The paper [4] presents an approach to improve forecasting accuracy for energy carrier production in CHP plants by extending existing computer supervision systems with additional calculation modules. This research is relevant to the manuscript’s discussion on improving CHP plant performance and efficiency. This study [2] presents a novel CHP system scheme combining plasma gasification, solid oxide fuel cells, GT, and supercritical CO₂ cycles for power and heat cogeneration. While the specific technologies differ, this paper is relevant to presented manuscript’s exploration of innovative CHP configurations. The research [3] examines potential locations for heat recovery in a triple-pressure cogeneration combined cycle power plant and calculates the amount of recoverable waste heat. The study’s focus on improving CHP plant efficiency through heat recovery aligns with themes in presented manuscript.

This article [13] analyzes the impact of plant schemes and boundary conditions on the optimization of trigeneration systems. The authors use mathematical programming to model various system configurations, which is relevant in the context of CHP system efficiency optimization discussed in the manuscript. The study [14] focuses on the production of synthetic natural gas from refuse-derived fuel gasification for use in polygeneration district heating and cooling systems. This work may be useful in discussing alternative fuel sources for CHP systems. The authors [15] conduct an energy and economic analysis of small-scale distributed cogeneration systems using an Italian residential building as a case study. The results of this work can be used to compare the efficiency of various scales of CHP systems.

This article [16] presents an energetic analysis of a GT with regenerative heat exchangers using various blade cooling systems. This study may be relevant in the context of discussing the energy efficiency of CHP system components.

In [17], the authors analyze the potential of biomass-based district heating systems in rural areas. This work can provide valuable information on the application of alternative energy sources in CHP and district heating systems.

The CHP power plants in the new layout presented in this study (Figure 3 and Figure 4) are supplied with the same driving heat flux as conventional CHP plants (Figure 1 and Figure 2). Additionally, these CHP plants supply district heat to consumers according to the same annual district heat demand (Figure 5). The amounts of these measurements must be identical as a prerequisite for any kind of comparison, thermodynamic as well as economic, between these CHP plants or any other plants. Moreover, the modified CHP plants use the same dual-pressure HRSGs as in conventional CHP plants, with identical heat exchanger arrangements. (Triple-pressure HRSGs used in gas-steam CHP plants are not justified thermodynamically and, above all, economically [18]). Furthermore, to ensure the modified CHP plants are also equivalent to conventional CHP plants in all other respects (apart from the fuel, of course), the analysis was carried out for the same values of high- and low-pressure steam.

However, although the recovery and steam pressure generators are the same, the quantities and other thermal and caloric parameters of the steam produced are obviously not the same. In conventional CHP plants, these parameters are determined by the flue gases resulting from the combustion of natural gas in a GT, while in converted CHP plants, we deal with helium. Helium has a very significant advantage over flue gas. Its specific heat capacity (at constant pressure) is approximately $c_{p,He}\approx 5.0\hspace{0.33em}\mathrm{kJ}/\left(\mathrm{kgK}\right)$, whereas that of flue gases is only $c_{p,fg}\approx 1.0\hspace{0.33em}\mathrm{kJ}/\left(\mathrm{kgK}\right)$.

2. Thermodynamic Analysis of a CHP Plant Using Gas and Steam Technology with a HTGR Reactor

2.1. Analysis of the Gas Part

The Joule cycle (high-temperature) and Clausius–Rankine cycles (low-temperature) are used in a hierarchical system (Figure 6) in a gas-steam CHP plant. The Joule cycle with single-stage expansion and single-stage compression of helium is implemented in the gas part of the CHP plant at HTGR and TE (Figure 3, Figure 4, Figure 6 and Figure 7).

Optimal levels of thermal values of the Joule cycle, i.e., amounts that provide its highest energy efficiency ${\eta }_{TE,\max }$ (Figure 8 and Figure 9) and its highest power $N_{\max }^{TE}$ (Figure 10 and Figure 11), are estimated following the differentiation of the cycle efficiency equation [19]:

$${\eta }_{TE}={\displaystyle \frac{{\eta }_m^{TE}(T_2-T_3)-\frac{1}{{\eta }_m^S}(T_1-T_0)}{T_2-T_1}}={\displaystyle \frac{{\eta }_m^{TE}{\eta }_i^{TE}{\eta }_m^S{\eta }_i^S{T}_2\left(1-\frac{1}{z_{TE}}\right)-T_0(z_{TE}-1)}{{\eta }_m^S{\eta }_i^S(T_2-T_0)-{\eta }_m^S{T}_0(z_{TE}-1)}}\to \mathrm{m}\mathrm{a}\mathrm{x}$$

(1)

and applying the term:

$${\displaystyle \frac{d{\eta }_{TE}}{d{z}_{TE}}}=0$$

(2)

The value $z_{TE,opt}$ is being calculated. By applying them to Equation (1), the calculation ${\eta }_{TE,\max }$ and $N_{\max }^{TE}$ are evaluated.

The various amounts in Equation (1) are expressed by the formulae:

$$z_{TE}={\displaystyle \frac{T_{1s}}{T_0}}={\displaystyle \frac{T_2}{T_{3s}}}={\left({\displaystyle \frac{p_1}{p_0}}\right)}^{{\displaystyle \frac{\kappa -1}{\kappa }}}$$

(3)

$$T_1=T_0+{\displaystyle \frac{1}{{\eta }_i^S}}(T_{1s}-T_0)=T_0+{\displaystyle \frac{T_0}{{\eta }_i^S}}(z_{TE}-1)$$

(4)

$$T_3=T_2-{\eta }_i^{TE}(T_2-T_{3s})=T_2-{\eta }_i^{TE}{T}_2\left(1-{\displaystyle \frac{1}{z_{TE}}}\right)$$

(5)

${\eta }_m^{TE}$, ${\eta }_m^S$: mechanical efficiencies of the TE and the compressor.

${\eta }_i^S$, ${\eta }_i^{TE}$: internal efficiencies of the TE and the compressor.

In the above Formulas (1), (3)–(5), the input quantities are, of course, the temperatures $T_0$ and $T_2$ and the pressure $p_0$. An extended analysis of the optimal selection of thermal values at different parts of the Joule cycle is provided in [19]. The higher the temperature $T_2$, the higher values are reached, according to the second law of thermodynamics, by the efficiency ${\eta }_{TE,\max }$ and power $N_{\max }^{TE}$ (power as opposed to efficiency). (Formula (1) depends, of course, on the helium mass flow rate, Formula (44) (Figure 8, Figure 9, Figure 10 and Figure 11)).

For the numbers used in this study for making calculations $T_0$ = 300 K, $T_2$ = 1300 K, $p_0$ = 1 bar, ${\eta }_m^S={\eta }_m^{TE}=0.97$, ${\eta }_i^S=0.85$, ${\eta }_i^{TE}=0.87$, the optimal amounts of thermal values are $z_{TE,opt}={[{(p_1/p_0)}^{\kappa -1/\kappa }]}_{opt}$= 2.0685 (isentropic exponent for helium $\kappa$ = 1.66), ${(p_1/p_0)}_{opt}=6.215$, $T_{1\hspace{0.33em}opt}$ = 677.1 K, $T_{3\hspace{0.33em}opt}$ = 709.1 K (436 °C), ${\eta }_{TE,\mathrm{m}\mathrm{a}\mathrm{x}}=0.2961$ (Figure 9 and Figure 10).

2.2. Analysis of the Steam Part

The optimum temperature for the Joule cycle $T_{3\hspace{0.33em}opt}$ = 709.1 K does not guarantee the maximum efficiency of electricity generation in the steam part of the CHP plant, i.e., in the Clausius–Rankine cycle (Figure 6). The higher the temperature $T_3$ of the helium supplied to the HRSG over the value $T_{3\hspace{0.33em}opt}$ = 709.1 K (436 °C), the higher will be the efficiency of electricity generation in the Clausius–Rankine cycle; thus, the higher will be the electrical efficiency of the CHP plant (Formula (45)) (Figure 8 and Figure 9). This is because, with increasing temperature $T_3$, the values of mass flow rate ${\dot{m}}_h$ and of the temperatures of the superheated high- and low-pressure steam $t_h$ and $t_l$ supplied to the steam turbine will be higher. Increasing the temperature $T_3$ of the outlet helium from the TE above the value $T_{3\hspace{0.33em}opt}=709.1\hspace{0.33em}\mathrm{K}$ is possible by decreasing the compression ratio $p_1/p_0$ (Figure 12, Figure 13 and Figure 14). In the case of conventional gas-steam power plants and CHP plants, the temperature of the exhaust gas from the GT supplied to the HRSG is a parameter over which we have no influence. To be precise, it should not be influenced by reducing the opening degree of the guide vanes in the axial air compressor, which is an integral part of the GT unit. The reason is that this would result in a significant reduction in efficiency and power output. Efficiency characteristics show that the power output of the GT decreases sharply in such cases, which consequently significantly compromises the economic performance of the CHP plant’s operation. Changing the turbine load below the rated value is therefore extremely inefficient in terms of energy and economy. In summary, the GT unit must (or should) operate at its rated load (Figure 1 and Figure 2). The changes in the values of pressures $p_1,p_2$ and temperatures $T_1,T_3$ for the Joule cycle for = 1300 K as a function of the parameter $z_{TE}={(p_1/p_0)}^{\kappa -1/\kappa }$ are presented in Figure 12, Figure 13 and Figure 14. All calculations presented in the other Figure 3 and Figure 4, Figure 8, Figure 9, Figure 10, Figure 11, and Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20 were also carried out for the temperature $T_2$ = 1300 K.

Reducing the compression ratio $p_1/p_0$ and thus increasing the temperature $T_3$ results in a major efficiency gains of electricity generation at the gas and steam CHP plant ${\eta }_{el}^{G-S}$ and its output $N_{el}^{G-S}=N_{el}^{ST}+N_{el}^{TE}$ (Figure 8, Figure 9, Figure 10 and Figure 11). This occurs despite the fact that the energy efficiency of the Joule cycle itself is significantly reduced simultaneously. The increase in efficiency and performance of the Clausius–Rankine cycle more than compensates for the decrease in efficiency and performance of the Joule cycle.

It should be noted that the increase in temperature $T_3$ is not only thermodynamically advantageous but is also beneficial for economic reasons (Section 3.1).

Mass flow rates of helium (Formula (14)) in the Joule cycle of the HTGR and TE for reactor thermal powers ${\dot{Q}}_{HTGR}=530\hspace{0.17em}\mathrm{MW}$ and ${\dot{Q}}_{HTGR}=850\hspace{0.17em}\mathrm{MW}$ are presented in Figure 15 and Figure 16.

The power output of a CHP plant and its steam turbine $N^{ST}$ depends not only on temperature $T_3$ but also on the pressures $p_h$and${ p}_l$ of the high- and low-pressure steam produced in the HRSG.

Figure 17 and Figure 18 illustrate the change in power output of the steam turbine (ST) for a relatively wide range of variations in these pressure values.

The results of multivariate thermodynamic calculations of the power of the steam turbine presented in Figure 17 and Figure 18 were made for temperature $T_3$ = 944.97 K (671.82 °C, $z_{TE}$ = 1.45), i.e., for a temperature much higher than $T_{3\hspace{0.33em}opt}=709.1\hspace{0.33em}\mathrm{K}$ (436 °C, $z_{TE,opt}=2.0685$). When $T_3$ is equal to 709.1 K (Figure 13 and Figure 14), it is only the Joule cycle of the TE that reaches maximum efficiency${\eta }_{TE,\mathrm{m}\mathrm{a}\mathrm{x}}=0.2961$, as already indicated above (Figure 8 and Figure 9), and maximum power${ N}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{TE}$. This power for reactor thermal power ${\dot{Q}}_{HTGR}=530\hspace{0.33em}\mathrm{MW}$ is$N_{\mathrm{m}\mathrm{a}\mathrm{x}}^{TE}=153.8\hspace{0.33em}\mathrm{MW}$, for power${\dot{ Q}}_{HTGR}=850\hspace{0.33em}\mathrm{MW}$, $N_{\mathrm{m}\mathrm{a}\mathrm{x}}^{TE}=246.6\hspace{0.33em}\mathrm{MW}$ (Figure 10 and Figure 11). The efficiency ${\eta }_{el}^{G-S}$ (Formula (45)) and power $N_{el}^{G-S}=N_{el}^{ST}+N_{el}^{TE}$ then reach relatively small values. On the other hand, when the temperature $T_3$ equals 944.97 K, then the electrical efficiency ${\eta }_{el}^{G-S}$ (Figure 8 and Figure 9) and the electrical output of the CHP installation $N_{el}^{G-S}$ (Figure 10 and Figure 11) reach considerably higher values. Therefore, the multivariate thermodynamic calculations for modified CHP plants were carried out for this particular temperature. The findings are presented in Figure 3 and Figure 4. Further reducing the value $z_{TE}$ below 1.45 to, for example, the value $z_{TE}$ = 1.125, and thus raising in temperature $T_3$ to a value of $T_3$ = 1173.15 K (900 °C) does not result in an impressive increase in the efficiency ${\eta }_{el}^{G-S}$ and power output $N_{el}^{G-S}$ of the CHP plant (Figure 8, Figure 9, Figure 10 and Figure 11). Moreover, according to the economical study outlined in Section 3.1, this could result in less economical effectiveness of its exploitation. This is due to the fact that the capital investment in the CHP unit will then be greater due to the high increase in the output of the steam part $N^{ST}$ compared to the output of the TE $N^{TE}$ (Figure 10 and Figure 11). The steam part is much more costly than the TE.

As can be observed from the diagrams shown in Figure 17 and Figure 18, the power outputs of the steam turbine unit $N_{el,\mathrm{m}\mathrm{i}\mathrm{n}}^{ST},N_{el,sum}^{ST}$ change very little with the change in pressures $p_h$ and $p_l$. They are slightly higher for pressures $p_h=5.5\hspace{0.33em}\mathrm{MPa}$ and$p_l=0.5\hspace{0.33em}\mathrm{MPa}$. However, in order to ensure the possibility of heating mains water in the HRSG to the highest possible temperature, close to 135 °C at the peak of the maximum heat demand (Figure 3), thermodynamic calculations were carried out for pressures $p_h=8.5\hspace{0.33em}\mathrm{MPa}$ and${ p}_l=0.9\hspace{0.33em}\mathrm{MPa}$, which guarantees a temperature of the inlet helium supplied to the mains water heater in the HRSG of above 135 °C, while for pressures $p_h=5.5\hspace{0.33em}\mathrm{MPa}$ and$p_l=0.5\hspace{0.33em}\mathrm{MPa}$, this temperature is a few degrees below 130 °C. Equally importantly, for $p_h=8.5\hspace{0.33em}\mathrm{MPa}$ and $p_l=0.9\hspace{0.33em}\mathrm{MPa}$, the temperature of the low-pressure superheated steam is more than 20 °C higher than for pressures $p_h=5.5\hspace{0.33em}\mathrm{MPa}$ and$p_l=0.5\hspace{0.33em}\mathrm{MPa}$, which is beneficial for steam turbine operation.

The article does not provide any analysis of the optimization effect of the pressures $p_h$ and $p_l$ on the steam turbine electrical output, especially that the effect of these pressures on the power output is very small (Figure 17 and Figure 18). What is more, the aim of the paper is only to compare the economic efficiency of the operation of gas-steam CHP plants with HTGR and a TE with that of gas-steam CHP plants with a GT. An optimization analysis of the pressures $p_h$ and $p_l$ could be carried out using the exergy balance of the HRSG distinguishing between internal and external exergy losses. It should be noted that the degree of irreversibility in the processes occurring in upstream components (the HRSG) influences the exergy losses in downstream components (in this case, power loss in the steam turbine). This topic is extensive in itself, which is another reason why it is not addressed in this paper. Presented below are the formulae used to conduct the multivariate thermodynamic calculations. For the modified CHP plants, the calculations, as already indicated several times, were carried out for the following input data values: temperature $T_2$ = 1300 K, $z_{TE}=1.45$ pressures, $p_h=8.5\hspace{0.33em}\mathrm{MPa},$ and${ p}_l=0.9\hspace{0.33em}\mathrm{MPa}$. The results of the calculations for the conventional CHP plants with GT units operating at their rated loads are presented in Figure 1 and Figure 2 and for the modified ones in Figure 3 and Figure 4. These figures present the results of the calculations for both the heating (winter) and non-heating (summer) seasons.

• The variability of the demand for district heating capacity ${\dot{Q}}_h\left(\tau \right)$ (Figure 5) in the case of its qualitative adjustment is represented by the following formula (this formula assumes that the capacities for domestic hot water (DHW) heating in the heating and non-heating seasons are equal to each other, ${\dot{Q}}_{w\hspace{0.33em}hw}={\dot{Q}}_{s\hspace{0.33em}hw}={\dot{Q}}_{hw}$):

  $${\dot{Q}}_h(\tau )={\dot{Q}}_t^{max}{\displaystyle \frac{t_h\left(\tau \right)-t_r\left(\tau \right)}{t_{h\mathrm{m}\mathrm{a}\mathrm{x}}-t_{r\mathrm{m}\mathrm{a}\mathrm{x}}}}+{\dot{Q}}_{hw}$$

  (6)

  where the following equation obviously occurs:

  $${\displaystyle \frac{t_h\left(\tau \right)-t_r\left(\tau \right)}{t_{h\mathrm{m}\mathrm{a}\mathrm{x}}-t_{r\mathrm{m}\mathrm{a}\mathrm{x}}}}={\displaystyle \frac{t_h-t_{amb}\left(\tau \right)}{t_h-t_{amb\hspace{0.33em}\mathrm{m}\mathrm{i}\mathrm{n}}}}$$

  (7)

  where:

${\dot{Q}}_h\left(\tau \right)$: current total thermal power of the CHP plant during the heating season for central heating (CH) and DHW.

${\dot{Q}}_t^{\mathrm{m}\mathrm{a}\mathrm{x}}$: maximum thermal power of the CHP plant for heating, ventilation, and air-conditioning during the heating season.

${\dot{Q}}_{hw}:$thermal power of the CHP plant during the heating season (regardless of the ambient temperature) for DHW.

$t_h\left(\tau \right),t_r\left(\tau \right)$: current temperatures of hot mains water and return water.

$t_{h\mathrm{m}\mathrm{a}\mathrm{x}}$, $t_{r\mathrm{m}\mathrm{a}\mathrm{x}}$: temperatures of hot mains water and return water at peak demand for district heating capacity (the calculations assumed $t_{g,\mathrm{m}\mathrm{a}\mathrm{x}}$= 135 °C, $t_{p,\mathrm{m}\mathrm{a}\mathrm{x}}$= 70 °C).

$t_h$, $t_{amb}\left(\tau \right)$, $t_{amb\hspace{0.33em}\mathrm{m}\mathrm{i}\mathrm{n}}$: internal temperature of the heated space and ambient temperatures, current and minimum, τ − time (0 ≤ τ ≤${\tau }_z$).

Power ${\dot{Q}}_{h\hspace{0.33em}w\hspace{0.33em}ave}$ (Figure 5) is the total average of the current power ${\dot{Q}}_h\left(\tau \right)$ over the time interval $〈0;{\tau }_w〉$, where the time ${\tau }_w$ expressed in hours is the duration of the heating (winter) season [18]. In the economic calculations, the time ${\tau }_z$ was assumed equal to 5.064 h/yr. (Typically, ${\tau }_w$varies from 5040 ÷ 5400 h, i.e., 210 ÷ 225 days); the duration of the non-heating season (summer) was assumed ${\tau }_s$ = 3360 h/year (annual operating time of the CHP plant ${\tau }_A={\tau }_w+{\tau }_s$ = 8.424 h/yr; a two-week period of standstill of the CHP plant was assumed for maintenance and technical inspection of the equipment). In practice, most commonly, ${\dot{Q}}_{h\hspace{0.33em}w\hspace{0.33em}ave}\approx 0.5{\dot{Q}}_h^{\mathrm{m}\mathrm{a}\mathrm{x}}$. The annual heating demand $Q_A$ is expressed by the formula (see Formulas (6) and (7)):

$$Q_A={\dot{Q}}_t^{\mathrm{m}\mathrm{a}\mathrm{x}}{\int }_o^{{\tau }_w}{\displaystyle \frac{t_h-t_{amb}\left(\tau \right)}{t_h-t_{amb\mathrm{m}\mathrm{i}\mathrm{n}}}}d\tau +{\dot{Q}}_{hw}{\tau }_A={\dot{Q}}_{h\hspace{0.33em}w\hspace{0.33em}ave}{\tau }_w+{\dot{Q}}_{hw}{\tau }_s$$

(8)

As noted above, Formula (8) assumes that the capacities ${\dot{Q}}_{hw}$ for DHW generation in the heating and non-heating seasons are equal.

In the absence of an accurate ordered graph of the ambient temperature $t_{ot}\left(\tau \right)$ for a particular area (climatic zone), by means of which the duration of the heating season can be determined for a set temperature at the beginning of the season, approximate analyses can use a unified reduced graph for the heating period [18,20,21]:

$${\displaystyle \frac{t_{pg}-t_{ot}\left(\tau \right)}{t_{pg}-t_{ot\mathrm{m}\mathrm{i}\mathrm{n}}}}=1-\sqrt[3]{{\displaystyle \frac{\tau }{{\tau }_w}}}+{\left({\displaystyle \frac{\tau }{{\tau }_w}}\right)}^2\left(1-\sqrt{{\displaystyle \frac{\tau }{{\tau }_w}}}\right)$$

(9)

where$t_{sh} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e}$temperature at the start of the heating season ($t_{sh}$= +12 °C).

• The mass flow rate ${\dot{m}}_h$ of the high-pressure steam is determined using the energy balances of the high-pressure part of the HRSG. (In the equations for a traditional CHP station, the helium heat capacity flow rate, Equation (14), is to be substituted for the flue gas heat capacity rate provided to the steam generator, Equation (15)).

  $${\dot{C}}_{hel}[t_3-(t_s^h+\mathsf{\Delta }{T}_{\mathrm{m}\mathrm{i}\mathrm{n}}^h\left)\right]={\dot{m}}_h(h_h-h_h^{\prime })$$

  (10)

  where:

$h_h$: specific enthalpy of superheated high-pressure steam at pressure $p_h$ and temperature $t_h$ supplied to the turbine. (h_h is determined for a temperature of $t_h=t_3-20$ K).

$h_h^{\prime }$: specific enthalpy of high-pressure steam at the bubble point (steam quality x = 0).

$t_s^h$: saturation temperature of high-pressure steam.

$t_3$: temperature of outlet helium from the TE supplied to HRSG.

$\mathsf{\Delta }{T}_{\mathrm{m}\mathrm{i}\mathrm{n}}^h$: pinch point values (in the calculations, it was assumed that $\mathsf{\Delta }{T}_{\mathrm{m}\mathrm{i}\mathrm{n}}^h$= 10 K).

• The mass flow rate ${\dot{m}}_l$ of the low-pressure steam is calculated from the energy balance of the low-pressure part of the HRSG:

  $${\dot{C}}_{hel}[t_{out}^h-(t_s^l+\mathsf{\Delta }{T}_{\mathrm{m}\mathrm{i}\mathrm{n}}^l\left)\right]={\dot{m}}_l(h_l-h_l^{\prime })$$

  (11)

$h_l$: specific enthalpy of low-pressure steam with pressure $p_l$ superheated in the superheater to $t_l=t_{out}^h-10\hspace{0.33em}\mathrm{K}$ supplied to the low-pressure part of the turbine.

$h_l^{\prime }$: specific enthalpy of low-pressure steam at the bubble point (steam quality x = 0).

$t_s^l$: saturation temperature of low-pressure steam.

$\mathsf{\Delta }{T}_{\mathrm{m}\mathrm{i}\mathrm{n}}^l$: pinch point values (in the calculations, it was assumed that $\Delta T_{\mathrm{m}\mathrm{i}\mathrm{n}}^l$= 10 K).

• The temperature of helium exiting the high-pressure part of the steam generator (which is also the temperature of helium after the high-pressure economizer and, consequently, the temperature of helium entering the low-pressure part of the steam generator) is expressed by the following equation:

  $$t_{out}^h\equiv t_{out}^{eco,h}\equiv t_{inl}^l=t_3-{\displaystyle \frac{{\dot{m}}_h(h_h-h_l^{\prime })}{{\dot{C}}_{hel}}}$$

  (12)

• The temperature of the helium $t_{out}^l$ downstream of the low-pressure part of the steam generator (which is also the temperature of the helium after the low-pressure economizer and, consequently, the temperature of the helium entering the mains water heater integrated into the HRSG) is calculated from the equation representing the energy balance of the high- and low-pressure parts:

  $$t_{out}^l\equiv t_{out}^{eco,l}=t_3-{\displaystyle \frac{{\dot{m}}_h(h_h-h_{fw})+{\dot{m}}_l(h_l-h_{fw})}{{\dot{C}}_{hel}}}$$

  (13)

  where $h_{fw}$ is the specific enthalpy of feed water supplied to the HRSG.

In the above formulae, ${\dot{C}}_{hel}$refers to the heat capacity flow rate of helium, which is calculated using the energy balance of the HTGR with a set thermal power ${\dot{Q}}_{HTGR}$ and a set outlet temperature $T_2$ of helium cooling the reactor core:

$${\dot{C}}_{hel}={\dot{m}}_{hel}{c}_{p,He}={\displaystyle \frac{{\dot{Q}}_{HTGR}}{T_2-T_1}}$$

(14)

In the calculations, the specific heat capacity of helium at constant pressure was assumed equal to$c_{p,He}=5.234\hspace{0.17em}\mathrm{kJ}/\left(\mathrm{kgK}\right)$.

For a conventional CHP plant, the heat capacity flow rate of flue gas is calculated from the energy balance of the GT unit:

$${\dot{C}}_{fg}={\dot{m}}_{fg}{\overline{c}}_{p,fg}^{GT}{|}_{t_{amb}}^{t_{out}^{GT}}={\displaystyle \frac{\dot{P}{W}_d{|}_{t_{amb}}-N_{el}^{GT}}{t_{out}^{GT}-t_{amb}}}={\displaystyle \frac{N_{el}^{GT}(1-{\eta }_{GT})}{{\eta }_{GT}(t_{out}^{GT}-t_{amb})}}\hspace{0.17em}$$

(15)

where the chemical energy flow rate of the natural gas fed into the combustion chamber of the turbine is:

$$\dot{P}{W}_d{|}_{t_{amb}}={\displaystyle \frac{N_{el}^{GT}}{{\eta }_{GT}}}$$

(16)

where $W_d{|}_{t_{amb}}$ is the calorific value of the natural gas at ambient temperature. (Power output of the GT unit and its efficiency are usually specified by manufacturers for an ambient temperature of t_amb = 15 °C; therefore, as the calorific value of the gas at ambient temperature is slightly higher than the calorific value at normal temperature t_n = 25 °C (by only approx. 0.01%), in the formulae presented, it is possible, without making a big error, to use (W_d)_n instead of (W_d)_amb in the energy balance).

• The temperature of helium $t_{out}^{sup,h}$ downstream of the high-pressure steam superheater (in a conventional system, the heat capacity flow rate of helium${\dot{ C}}_{hel}$, Formula (14), should be replaced with the heat capacity flow rate of flue gas${ \dot{C}}_{fg}$, Formula (15); also in all the other formulae below, in the case of a conventional CHP plant, ${\dot{C}}_{hel}$should be replaced by${\dot{ C}}_{fg}$) is calculated from the equation:

  $${\dot{C}}_{hel}(t_3-t_{out}^{sup,h})={\dot{Q}}_{sup}^h$$

  (17)

  where thermal power of the high-pressure steam superheater is:

  $${\dot{Q}}_{sup}^h={\dot{m}}_h(h_h-h_h^{″})$$

  (18)

$h_h^{″}$ is the specific enthalpy of high-pressure dry saturated steam (steam quality x = 1).

• Helium temperature downstream of the high-pressure evaporator:

  $${\dot{C}}_{hel}(t_3-t_{out}^{eva,h})={\dot{Q}}_{sup}^h+{\dot{Q}}_{eva}^h$$

  (19)

  where thermal power of the high-pressure evaporator is:

  $${\dot{Q}}_{eva}^h={\dot{m}}_h(h_h^{″}-h_h^{\prime })$$

  (20)

• Helium temperature $t_{out}^{eco,h}$ after the high-pressure economizer (also the helium temperature before the low-pressure steam superheater):

  $${\dot{C}}_{hel}(t_3-t_{out}^{eco,h})={\dot{Q}}_{sup}^h+{\dot{Q}}_{eva}^h+{\dot{Q}}_{eco}^h$$

  (21)

  where thermal power of the high-pressure economizer is:

  $${\dot{Q}}_{eco}^h={\dot{m}}_h(h_h^{\prime }-h_l^{\prime })$$

  (22)

• Helium temperature after the low-pressure steam superheater:

  $${\dot{C}}_{hel}(t_3-t_{out}^{sup,l})={\dot{Q}}_{sup}^h+{\dot{Q}}_{eva}^h+{\dot{Q}}_{eco}^h+{\dot{Q}}_{sup}^l$$

  (23)

  where thermal power of the low-pressure steam superheater is:

  $${\dot{Q}}_{sup}^l={\dot{m}}_l(h_l-h_l^{″})$$

  (24)

  where $h_l^{″}$ represents the specific enthalpy of the low-pressure dry saturated steam (steam quality x = 1).

• Helium temperature after the low-pressure evaporator:

  $${\dot{C}}_{hel}(t_{out}^{sup,l}-t_{out}^{eva,l})={\dot{Q}}_{eva}^l$$

  (25)

  where the thermal power of the low-pressure evaporator is:

  $${\dot{Q}}_{eva}^l={\dot{m}}_l(h_l^{″}-h_l^{\prime })$$

  (26)

• Helium temperature after the low-pressure economizer:

  $${\dot{C}}_{hel}(t_{out}^{eva,l}-t_{out}^{eco,l})={\dot{Q}}_{eco}^l$$

  (27)

  $${\dot{Q}}_{eco}^l=({\dot{m}}_h+{\dot{m}}_l)(h_l^{\prime }-h_{fw})$$

  (28)

During the heating season (winter), the heat capacity of helium (or flue gas) downstream of the low-pressure economizer is used exclusively to heat mains water for central heating and DHW. Consequently, the condensate heater located in the steam generator, which is typically the last heat exchange surface, is not used during the winter season (Figure 1, Figure 2, Figure 3 and Figure 4). The temperature of helium exiting the boiler and entering the helium cooler (or flue gas exiting to the stack in a conventional system) is calculated using the following equation:

$${\dot{C}}_{hel}(t_{out}^l-t_{out,winter}^{HRSG})={\dot{Q}}_{mwh \mathrm{winter}}^{HRSG}$$

(29)

The power output of the mains water heater ${\dot{Q}}_{mwh}^{HRSG}$ (Formula (29)) at the peak demand for district heating capacity was assumed to be:

$${\dot{Q}}_{mwh\hspace{0.33em}winter}^{HRSG}={\dot{C}}_{hel}(t_{h\mathrm{m}\mathrm{a}\mathrm{x}}-t_{r\mathrm{m}\mathrm{a}\mathrm{x}})$$

(30)

The remaining thermal power, according to the graph of the total demand for district heating (Figure 5), is therefore supplied to the district heating network by means of heat exchangers $W_I$ and $W_{II}$ (Figure 1, Figure 2, Figure 3 and Figure 4):

$${\dot{Q}}_h^{W_I}+{\dot{Q}}_h^{W_{II}}={\dot{Q}}_h^{\mathrm{m}\mathrm{a}\mathrm{x}}-{\dot{Q}}_{mwh\hspace{0.33em}winter}^{HRSG}$$

(31)

The exchangers $W_I$ and $W_{II}$ are supplied with adjustable bleed steam from the steam turbine. At peak demand, i.e., when ${\dot{Q}}_h$ = ${\dot{Q}}_h^{\mathrm{m}\mathrm{a}\mathrm{x}}={\dot{Q}}_t^{max}+{\dot{Q}}_{hw}$ = 220 MW (Formula (6), Figure 3), the thermal output of these exchangers is:

$${\dot{Q}}_h^{W_I}={\dot{m}}_{1u}(h_{1u}^{\prime }-h_{W_I,W_{II}})+{\dot{m}}_{2u}(h_{2u}-h_{W_I,W_{II}})$$

(32)

$${\dot{Q}}_h^{W_{II}}=\left({\dot{Q}}_h^{\mathrm{m}\mathrm{a}\mathrm{x}}-{\dot{Q}}_{mwh\hspace{0.33em}winter}^{HRSG}\right){\displaystyle \frac{t_{h\mathrm{m}\mathrm{a}\mathrm{x}}-{t^{W_I}}_{out\hspace{0.33em}mw}}{t_{h\mathrm{m}\mathrm{a}\mathrm{x}}-t_{r\mathrm{m}\mathrm{a}\mathrm{x}}}}={\dot{m}}_{1u}(h_{1u}-h_{1u}^{\prime })$$

(33)

Obviously, the following equation occurs:

$${\dot{Q}}_h^{\mathrm{m}\mathrm{a}\mathrm{x}}-{\dot{Q}}_{mwh\hspace{0.33em}winter}^{RB}={\dot{m}}_{1u}(h_{1u}-h_{W_I,W_{II}})+{\dot{m}}_{2u}(h_{2u}-h_{W_I,W_{II}})$$

(34)

where:

$h_{1u},h_{2u},h_{W_I,W_{II}}$ represents the specific enthalpies of the heating bleed steam from the steam turbine and the specific enthalpy of its condensate after the exchangers $W_I$ and $W_{II}$ (specific enthalpy $h_{1u}$ is determined for a pressure of 346 kPa; enthalpy $h_{2u}$ for a pressure of 161 kPa; specific enthalpy of the condensate assumed in the calculations is $h_{W_I,W_{II}}$ = 301.5 kJ/kg, see Figure 1, Figure 2, Figure 3 and Figure 4):

$h_{1u}^{\prime }$ is the specific enthalpy at the bubble point (steam quality x = 0);

${\dot{m}}_{1u},{\dot{m}}_{2u}$ represents the mass flow rates of heating bleed steam from the steam turbine;

${t^{W_I}}_{out\hspace{0.33em}mw}$ is the mains water temperature after the exchanger $W_I$ (this temperature was assumed to be 3.5 °C lower than the saturation temperature of bleed steam with a pressure of 161 kPa, ${t^{W_I}}_{out\hspace{0.33em}mw}={t^s}_{2u}$ − 3.5 °C = 110 °C).

During the non-heating season (summer), the mains water heater in the HRSG operates at a relatively low thermal output. This is because it only operates for the purpose of generating DHW. In the case under consideration, this output power is ${\dot{Q}}_{mwh,summer}^{\mathrm{HRSG}}$ = 15 MW (Figure 5). In this situation, the condensate heater located in the steam generator, as the last surface that is heated in it, is also used in the non-heating season (Figure 1, Figure 2, Figure 3 and Figure 4). The helium temperature after the mains water heater is then calculated using the equation (cf. Equation (29)):

$${\dot{C}}_{hel}(t_{out}^n-t_{out,summer}^{mwh})={\dot{Q}}_{mwh,summer}^{\mathrm{HRSG}}$$

(35)

while helium temperature after the condensate heater located in the steam generator as the last surface that is heated in it is (this is the temperature of the outlet helium from the HRSG supplied to the helium cooler; in the case of a conventional system, this is the temperature of outlet flue gas to the stack) is expressed by the equation:

$${\dot{C}}_{hel}(t_{out,summer}^{mwh}-t_{out,summer}^{HRSG})={\dot{Q}}_{con,summer}^{HRSG}$$

(36)

with

$${\dot{Q}}_{con,summer}^{HRSG}={\dot{m}}_{\mathrm{con}\hspace{0.33em}summer}(h_{dea,summer}-h_{con})$$

(37)

where:

$h_{dea,summer}$: specific enthalpy of outlet condensate from the heater located in the HRSG (this enthalpy is determined for a temperature $t=t_{out,summer}^{mwh}$ − 33.5 °C and for a pressure in the feed water tank of 120 kPa).

$h_{con}$: specific enthalpy of condensate after the condenser (the calculations assume supercooling of the water after the condenser by one degree below the saturation temperature of the steam $t_s^{con}$ in the condenser, $t_{con}=t_s^{con}-1\hspace{0.17em}\mathrm{K}$; the pressure in the condenser was assumed to be 6 kPa; steam quality was assumed to be x = 0.9).

The mass flow rate of outlet condensate from the condenser in summer is:

$${\dot{m}}_{con\hspace{0.33em}summer}={\dot{m}}_h+{\dot{m}}_l-{\dot{m}}_{dea\hspace{0.33em}summer}$$

(38)

The mass flow rate of steam supplied to the deaerator ${\dot{m}}_{dea\hspace{0.17em}summer}$ is calculated from its energy balance and that of the feed water tank:

$$({\dot{m}}_h+{\dot{m}}_l-{\dot{m}}_{dea\hspace{0.33em}summer})h_{dea,summer}+{\dot{m}}_{dea\hspace{0.33em}summer}\hspace{0.17em}{h}_{2u}=({\dot{m}}_h+{\dot{m}}_l)h_{fw}$$

(39)

Obviously, in the heating season (winter), the mass flows to the condenser and deaerator are different, and the specific enthalpy of the steam $h_{dea}$ supplied to the deaerator is lower due to the fact that the heat exchangers $W_I$ and $W_{II}$ are supplied with adjustable bleed steam from the steam turbine. The enthalpy $h_{dea}$ results from the energy balance of the node after the exchangers $W_I$ and $W_{II}$:

$$({\dot{m}}_{1u}+{\dot{m}}_{2u})h_{W_I,W_{II}}+({\dot{m}}_h+{\dot{m}}_l-{\dot{m}}_{1u}-{\dot{m}}_{2u}-{\dot{m}}_{dea})h_{con}=({\dot{m}}_h+{\dot{m}}_l-{\dot{m}}_{dea})h_{dea}$$

(40)

The power output of the steam turbine is expressed by the following formulae:

• the steam turbine power output during the heating season (winter) at the peak of demand for district heating (this is the minimum power of the steam turbine due to the maximum extraction of steam to the heat exchangers $W_I$ and $W_{II}$):

  $$N_{el,\mathrm{m}\mathrm{i}\mathrm{n}}^{ST}=\left[{\dot{m}}_h\right(h_h-h_{cond})+{\dot{m}}_l(h_l-h_{cond})-({\dot{m}}_{2u}+{\dot{m}}_{dea}\left)\right(h_{2u}-h_{cond})-{\dot{m}}_{1u}(h_{1u}-h_{cond}\left)\right]{\eta }_{em}$$

  (41)

• the steam turbine power output during the non-heating season (summer: the power of the steam turbine then reaches its maximum value), i.e., when the CHP plant is operating exclusively to meet the demand for DHW preparation (the heat exchangers $W_I$ and $W_{II}$ are not used at that time; ${\dot{m}}_{1u}={\dot{m}}_{2u}=0$, and only the mains water heater in the HRSG is in operation with an output power of ${\dot{Q}}_{mwh,summer}^{\mathrm{HRSG}}$ = 15 MW (Figure 5)):

  $$N_{el,summer}^{ST}=\left[{\dot{m}}_h\right(h_h-h_{cond})+{\dot{m}}_l(h_l-h_{cond})-{\dot{m}}_{dea,summer}(h_{2u}-h_{cond}\left)\right]{\eta }_{em}$$

  (42)

  where:

$h_{cond}$: specific enthalpy of the steam in the condenser (the pressure in the condenser was assumed to be 6 kPa; steam quality was assumed to be x = 0.9).

${\eta }_{em}$: electromechanical efficiency of the steam turbine (in the calculations, it was assumed that${\eta }_{em}=0.98$).

The average power output $N_{el\hspace{0.33em}w\hspace{0.33em}ave}^{ST}$ of the steam turbine during the heating season (winter), i.e., when the CHP plant is operating at thermal output ${\dot{Q}}_{h\hspace{0.33em}w\hspace{0.33em}ave}$ = 110 MW (Figure 3), is determined from Formula (41) by substituting zero value ${\dot{m}}_{1u}=0$ for the flow rate${\dot{ m}}_{1u}$. This is because, in this situation, only the mains water heater in the HRSG and the primary heat exchanger $W_I$ (Figure 1, Figure 2, Figure 3 and Figure 4) are used. In Formula (41), the mass flow rates ${\dot{m}}_{2u\hspace{0.33em}w\hspace{0.33em}ave}$ and${\dot{ m}}_{dea\hspace{0.33em}w\hspace{0.33em}ave}$ should also be substituted for mass flow rates ${\dot{m}}_{2u}$ and${\dot{ m}}_{dea}$.

However, in the economic calculations of the unit costs of heat generation${ k}_h$ and${ k}_{h,amort}$, (Formulas (47) and (48)), it can be assumed without making a significant error that the power output $N_{el\hspace{0.33em}w\hspace{0.33em}ave}^{ST}$ is equal to the arithmetic mean of the power outputs $N_{el,\mathrm{m}\mathrm{i}\mathrm{n}}^{ST}$ and ${ N}_{el,summer}^{ST}$:

$$N_{el\hspace{0.33em}w\hspace{0.33em}ave}^{ST}\approx 0.5(N_{el,\mathrm{m}\mathrm{i}\mathrm{n}}^{ST}+N_{el,summer}^{ST})$$

(43)

This is because, even when the power output $N_{el\hspace{0.33em}w\hspace{0.33em}ave}^{ST}$ changes by a value $\pm 10\hspace{0.17em}\%\hspace{0.17em}{N}_{el\hspace{0.33em}w\hspace{0.33em}ave}^{ST}$, the change in value $k_h$ is only approx.$\mp 6\hspace{0.17em}\text{‰}\hspace{0.17em}{k}_h$.

• The power output of the turbo expander is expressed by the formula:

  $$N_{el}^{TE}={\dot{m}}_{hel}{c}_{p,hel}\left[{\eta }_m^{TE}(T_2-T_3)-{\displaystyle \frac{1}{{\eta }_m^S}}(T_1-T_0)\right]{\eta }_G$$

  (44)

  where ${\eta }_G$ is the efficiency of the generator. (The calculation assumes ${\eta }_G=0.98$).

The efficiency of electricity generation in the modified CHP plant (Figure 17 and Figure 18) is determined from the equation:

$${\eta }_{el}^{G-S}={\displaystyle \frac{N_{el}^{TP}+N_{el}^{TE}}{{\dot{Q}}_{HTGR}}}$$

(45)

Obviously, this efficiency reaches its highest value in the non-heating season (summer) and its lowest value in the heating season when the CHP plant is operating at its maximum heat output at peak demand for district heating ${\dot{Q}}_h^{\mathrm{m}\mathrm{a}\mathrm{x}}$ = 220 MW. Obviously, the overall energy efficiency of the CHP plant is highest during the heating season at peak demand for heating:

$${\eta }_h^{G-S}={\displaystyle \frac{N_{el}^{ST}+N_{el}^{TE}+{\dot{Q}}_h}{{\dot{Q}}_{HTGR}}}$$

(46)

because the value ${\dot{Q}}_h^{\mathrm{m}\mathrm{a}\mathrm{x}}$ = 220 MW must then be substituted for ${\dot{Q}}_h$ in the numerator of Formula (46).

To summarize Section 2, it should be explicitly stated that thermodynamic analysis, while crucial and necessary, only allows for the identification of opportunities to improve technological processes and design solutions for machinery and equipment. In a market economy, economic considerations, profit motives, and profit maximization ultimately determine the feasibility of applying a particular technical solution; cost-effectiveness analysis forms the basis for investment decisions. Indeed, economic considerations often take precedence over technical ones. However, it should be strongly emphasized that economic analysis is only possible after a thorough thermodynamic analysis. The results of the thermodynamic analysis serve as input for the economic analysis.

It should also be noted that regarding economic considerations, power plants using hierarchical gas-gas technology are as significant as gas-steam nuclear power plants with HTGRs [19].

3. Economic Analysis of Unit Costs of Heat Generation in Modified and Conventional CHP Plants

3.1. Unit Cost of Heat Generation in a Gas-Steam Combined Heat and Power Plant with a High-Temperature Nuclear Reactor

Unit costs of heat generation in any CHP plant are expressed by the formulae:

• Prior to depreciation of the CHP plant:

  $$k_h={\displaystyle \frac{K_e+K_{cap}-E_{el,A}^{EC}{e}_{el}}{Q_A}}={\displaystyle \frac{K_e+Jz\rho -E_{el,A}^{EC}{e}_{el}}{Q_A}}\equiv {\displaystyle \frac{K_e+A+F_{ave}-E_{el,A}^{EC}{e}_{el}}{Q_A}}$$

  (47)

• After depreciation of the CHP plant:

  $$k_{k,amort}={\displaystyle \frac{K_e-E_{el,A}^{EC}{e}_{el}}{Q_A}}$$

  (48)

  with the component $E_{el,A}^{EC}{e}_{el}$ in the above formulae being the revenue from the sale of the electricity produced in the CHP plant. For a gas-steam CHP plant with an HTGR and a TE, it is expressed by the formula:

  $$E_{el,A}^{EC}{e}_{el}=[N_{el}^{TE}{\tau }_A+N_{el\hspace{0.33em}w\hspace{0.33em}ave}^{ST}({\tau }_A-{\tau }_s)+N_{el,summer}^{ST}{\tau }_s](1-{\epsilon }_{el})e_{el}$$

  (49)

  where:

$e_{el}$: electricity selling price.

$E_{el,A}^{EC}$: net annual production of electricity in the CHP plant.

$K_e$: annual cost of operating the CHP plant (cost of fuel + payroll costs + cost of maintenance and repairs + taxes, fees, and insurance + environmental fees + cost of non-energy raw materials and auxiliary materials).

$K_{cap}$: annual capital cost of the CHP plant (the annual capital cost is the sum of depreciation and financing costs, i.e., costs intended to recoup the capital expenditure incurred $J$ plus interest thereon).

$N_{el\hspace{0.33em}w\hspace{0.33em}ave}^{ST}$, ${\tau }_w$: the average electrical output of the CHP plant during the heating season (winter) and output of the CHP plant during the non-heating season (summer).

$Q_A$: annual heat generation in the CHP plant (Formula (8)).

${\epsilon }_{el}$: electrical self-consumption ratio of the CHP plant (calculations assumed ${\epsilon }_{el}$ = 7%).

${\tau }_A$, ${\tau }_s$: annual operating times of the CHP plant (calculations assumed annual time of ${\tau }_A$ = 8424 h/a; time in the non-heating season ${\tau }_s$ = 3360 h/a).

Formula (49) denotes the avoided cost of heat generation at the CHP plant. Hence, in Formulas (47) and (48), there is a minus sign in front of it. An analysis of Formulas (47)–(49) shows that the higher the production of electricity $E_{el,A}^{EC}$ and the higher its selling price$e_{el}$, the lower the costs $k_h$ and${ k}_{h,amort}$. Where the avoided cost is higher than the annual operating costs of the CHP plant, i.e., if${ E}_{el,A}^{EC}{e}_{el}>K_A=K_e+K_{cap}$, then the price of heat is then, obviously, negative (Figure 19 and Figure 20). The heat could then be supplied to consumers even for free, and each year, the CHP plant would still make a gross profit with $Z_A$ the difference between the revenue from the sale of electricity produced in the CHP plant and its annual operating costs:

$$Z_A=E_{el,A}^{EC}{e}_{el}-K_A$$

(50)

The rate of depreciation and average annual financial cost are expressed by the formulae [22]:

$$A={\displaystyle \frac{zJ}{T}}$$

(51)

$$F_{ave}={\displaystyle \frac{\sum _{t=1}^T{F}_t}{T}}={\displaystyle \frac{rzJ(T+1)}{2T}}$$

(52)

$$F_t=r[zJ-(t-1\left)A\right], \mathrm{for} t=1,\dots ,T$$

(53)

where $F_t$ is the interest (cost of investment funds borrowed in the following fiscal year t ∈ 〈1; T〉, tax-deductible).

In practice, interest is paid on a monthly basis, and then Formula (52) is as follows [22]:

$${\sum }_{t=1}^{12T}{F}_t^{12}={\displaystyle \frac{\left(\sqrt[12]{r+1}-1\right)zJ(12T+1)}{2}}$$

(54)

Annual capital cost${ K}_{cap}$:

$$K_{cap}=z\rho J\equiv A+F_{ave}$$

(55)

is the sum of the costs of depreciation and financing, i.e., costs intended to recoup the capital expenditure incurred $J$ plus interest thereon where:

z: investment capital-freeze index $J$ during the construction of the CHP plant, $z>1$ [22]:

$$z={\displaystyle \frac{{(1+r)}^{b+1}-1}{(b+1)r}}$$

(56)

ρ: interest-bearing depreciation rate [22]:

$$\rho ={\displaystyle \frac{r{(1+r)}^T}{{(1+r)}^T-1}}$$

(57)

where:

b: the duration of construction of the CHP plant (the calculations assumed that for a modified CHP plant, b = 5 years).

r: interest rate on capital expenditure (the calculations assumed r = 3%).

T: duration of operation of the modified CHP plant (the calculations assumed T = 60 years).

The capital expenditure for the construction of the CHP plant is expressed by the formula:

$$J=N_{el}^{G-S}i$$

(58)

with the unit capital expenditure (per unit of installed electrical capacity) estimated at:

$$i=60\hspace{0.33em}{\left(N_{el}^{G-S}\right)}^{-0.223} \text{[PLNm/MW]}$$

(59)

where power $N_{el}^{G-S}$ is expressed in MW and expenditures in Polish currency, PLN.

As noted above, the sum $K_e+Jz\rho \equiv K_e+A+F_{ave}$ in Formula (47) represents the total annual operating costs $K_A$ of the CHP plant, while the numerator in this formula represents the annual costs of $K_{Rc}$ heat generation in it:

$$K_{Rc}=K_e+Jz\rho -E_{el,A}^{EC}{e}_{el}$$

(60)

Figure 19 and Figure 20 show the results of calculations of the unit costs of heat generation $k_h$ and $k_{h,amort}$ in modified gas-steam CHP plants as a function of unit capital expenditure i for the thermal reactor powers ${\dot{Q}}_{HTGR}=530\hspace{0.17em}\mathrm{MW}$ and${\dot{Q}}_{HTGR}=850\hspace{0.17em}\mathrm{MW}$. The calculations obviously use the results of the thermodynamic calculations presented in Figure 1, Figure 2, Figure 17 and Figure 18, which were carried out for a temperature value $T_2$ = 1300 K, $z_{TE}=1.45,$ and high- and low-pressure steam values $p_h=8.5\hspace{0.33em}\mathrm{MPa}$ and$p_l=0.9\hspace{0.33em}\mathrm{MPa}$. Importantly, if the economic calculations were carried out for pressures $p_h=5.5\hspace{0.33em}\mathrm{MPa}$ and $p_l=0.5\hspace{0.33em}\mathrm{MPa}$, for which the electrical capacity of the steam turbine unit is higher in comparison with pressures $p_h=8.5\hspace{0.33em}\mathrm{MPa}$ and $p_l=0.9\hspace{0.33em}\mathrm{MPa}$ (of course, with a change in the values of pressures $p_h$ and $p_l$, the power of the TE does not change), the unit costs $k_h$ and $k_{h,amort}$ (Formulae (47) and (48)) would decrease by only a few per mil. This is due to the minimal increase in the electrical capacity of the steam turbine unit, also by only a few per mil (Figure 17 and Figure 18). If, on the other hand, the temperature $T_2$ were significantly increased above 1300 K, all the efficiencies and powers of the TE and the steam turbine would increase significantly. This would also result in a significant reduction in the unit costs of heat generation $k_h$ and$k_{h,amort}$. Conversely, decreasing the temperature $T_2$ would decrease the efficiencies and capacities and thus increase the costs. For example, increasing the temperature $T_2$ by 100 K to $T_2$ = 1400 K would reduce the costs $k_h$ and $k_{h,amort}$ almost by half, while decreasing $T_2$ by 100 K to $T_2$ = 1200 K would cause an approximately twofold increase.

As can be seen from the analysis of the unit cost values $k_h$ and $k_{h,amort}$ shown in Figure 19 and Figure 20, when the price of electricity is low, e.g., if it is only $e_{el}$ = PLN 100/MWh, then a CHP plant with a reactor thermal power ${\dot{Q}}_{HTGR}=530\hspace{0.17em}\mathrm{MW}$ is more advantageous. This is obvious. The general rule is that the greater the financial resources involved in an economic undertaking and the product manufactured is very cheap, the more the investment does not make economic sense. However, already for a price $e_{el}$ = PLN 200/MWh, a CHP plant with a reactor thermal power ${\dot{Q}}_{HTGR}=850\hspace{0.17em}\mathrm{MW}$ is definitely more advantageous (Figure 20). This is because the avoided cost of generating heat in it, i.e., the revenue from the sale of a significantly higher amount of electricity produced in it (Formula (49)), is high; thus, the heat is definitely cheaper, and such a CHP plant is more cost-effective. In addition, the economic efficiency will then be increased by building a heat accumulator in the CHP plant, the economic viability of which also depends on electricity prices. More precisely, it depends on the difference between electricity prices at the peak and valley demand. When the peak price is only slightly higher than the valley price, the heat accumulator makes no economic sense [23].

3.2. Specific Cost of Heat Generation

Table 1. Technical and economic data of conventional gas-steam CHP plants.

Gas-Steam Combined Heat and Power Plant			Units	Technical and Economic Data	Technical and Economic Data
Estimated capital expenditure			PLNm	752.40	969.94
Gas turbine unit	Type			SGT6-500F	M701G
	Electric power output		MWe	202.00	334.00
	Compression ratio		-	17.4	21.0
	Flow rate of flue gas		kg/s	508.03	737.10
	Inlet flue gas temperature		°C	1415	1500
	Outlet flue gas temperature		°C	578	587
	Electrical efficiency		%	38.10	39.50
Operation during the heating season (winter)	Mass flow rate of gas fuel	Gas turbine	MWt	530.184	845.570
			kg/s	10.98	17.52
			Nm3/h	54,533	86,973
	Average thermal output for central heating and DHW		MWt	110.0	110.0
	Average gross electrical output of the steam turbine generator set		MWe	76.92	131.36
	Total gross electric power		MWe	278.92	465.36
	Energy efficiency		%	73.4	68.0
	Electrical efficiency		%	52.6	55.0
	Duration of operation		h p.a.	5064	5064
	Mass flow rate of gas fuel	Gas turbine	MWt	530.184	845.570
			kg/s	10.98	17.52
			Nm3/h	54,533	86,973
Operation during the non-heating season (summer)	Thermal output for central heating and DHW		MWt	15.0	15.0
	Gross electrical output of the steam turbine unit		MWe	91.57	144.28
	Total gross electric power		MWe	293.57	478.28
	Energy efficiency		%	58.2	58.3
	Electrical efficiency		%	55.4	56.6
	Duration of operation		h p.a.	3360	3360
Total gas consumption			Mln Nm3/a	459.39	732.66
Total net annual electricity production			MWh p.a.	2302,905	3805,053
Total annual heat generation			GJ p.a.	2186,784	2186,784
Annual cogeneration factor			-	3.8	6.3
Annualized energy efficiency			%	65.2	61.9
Annual costs:
fixed	Capital cost (return on capital)		PLNm p.a.	52.87	68.16
	Overhauls and maintenance		PLNm p.a.	22.57	29.10
	Payroll costs		PLNm p.a.	0.35	0.35
variable	Cost of fuel		PLNm p.a.	1212.78	1934.22
	Cost of CO2 emission permits		PLNm p.a.	353.73	564.15
	Environmental fees		PLNm p.a.	1.00	1.00
	Avoided cost (revenue from electricity)		PLNm p.a.	−460.58	−761.01
Cost of heat generation			PLNm p.a.	1182.72	1835.96
Unit cost of heat generation kh			PLN/GJ	540.85	839.57
Unit cost of heat generation kh,amort			PLN/GJ	516.67	808.40
Assumptions made:	interest rate on investment capital		% p.a.	3.0	3.0
	duration of operation assumed in calculations		years	20	20
	duration of construction		years	2	2
	calorific value of natural gas		MJ/kg	48.28	48.28
			MJ/Nm3	35.00	35.00
	gas density		kg/Nm3	0.725	0.725
	price of natural gas		PLN/Nm3	2.64	2.64
			USD/Nm3	0.600	0.600
	unit price of gas		PLN/GJ	75.43	75.43
	CO2 emissions from gas combustion		kg/GJ	55	55
	unit purchase price of CO2 emission permits		PLN/Mg	400	400
	price of electricity		PLN/MWh	200.0	200.0
	electrical self-consumption power		% of total output of the system	4.00	4.00
	US dollar exchange rate		USD/PLN	4.40	4.40
	annual operating time		h p.a.	8424	8424

summarizes the technical and economic data for conventional gas-steam CHP plants (1 and 2) that are necessary for an analysis of the unit costs $k_h$ and $k_{h,amort}$ of heat generation in them (Formulae (47) and (48)). The results of multivariate calculations of these costs are presented in Figure 21, Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26.

As can be seen from the economic values presented in Table 1, the factors that have a decisive influence on the unit cost of heat generation in a conventional gas-steam CHP plant are the price of natural gas, the price of electricity, and the unit cost of purchasing CO₂ emission permits. The very low share of the capital cost in the annual cost of heat generation causes the cost $k_{h,amort}$ is only slightly lower than the cost${ k}_h$. Figure 21, Figure 22 and Figure 23 show the unit costs of heat generation $k_h$ and $k_{h,amort}$ in a CHP plant with a GT SGT6-500F (Figure 1); Figure 24, Figure 25 and Figure 26 show the cost $k_h$ and $k_{h,amort}$ in a CHP plant with a GT M701G and a heat accumulator (Figure 2).

An analysis of the unit costs of heat generation presented in Figure 21, Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26 shows that at the current prices of gas, electricity and CO₂ emission permits (Table 1) are, colloquially speaking, simply exorbitant. These CHP plants are therefore completely economically unviable at the current prices. Only for the price of electricity above approx. PLN 600/MWh would the price of heat supplied to consumers at $e_{CO2}$ = PLN 400/MgCO₂ be relatively acceptable. It would then amount to about PLN 120–150/GJ, i.e., it would only be about twice as high as the unit price of natural gas of PLN 75.43/GJ (Table 1). Such a price of heat, and lower, is currently offered to consumers by heating plants with gas boilers, which are cheap in terms of capital expenditure. Significantly cheaper heat, at about PLN 80–100/GJ, is supplied by hard-coal-fired heating plants.

It should also be expressly stated that the price of heat, as well as electricity produced in power plants, is hugely increased by the tax on CO₂ emissions imposed by the EU. For example, this tax in a CHP plant with an M701G GT amounts to PLN 564.15 in the annual heat production costs of PLN 1835.96, which means it accounts for as much as 31% of these costs (Table 1; this tax is just as high in a system with an SGT6-500F turbine). In total, therefore, the cost of gas and CO₂ emissions allowances account for almost 100% of the annual heat production costs.

4. Summary and Final Conclusions

The study employed a comprehensive analytical approach combining thermodynamic analysis, economic analysis, comparative analysis, and sensitivity analysis to evaluate conventional and modified CHP systems. This multifaceted methodology involved developing detailed mathematical models, conducting multivariate calculations to optimize system parameters, and analyzing efficiency and power output under various operating conditions.

Through economic analysis, formulas were developed to calculate the unit costs of heat generation, taking into account factors such as capital costs, fuel costs, and CO₂ emission allowances. The comparative analysis directly compared conventional and modified CHP systems in terms of thermodynamic efficiency and economic viability, examining the impact of key parameters such as reactor thermal power, electricity prices, and natural gas prices.

The study found that the costs in modified gas-steam power plants are significantly lower than in conventional CHP plants, despite higher specific capital investments in CHP plants with HTGRs and TEs. This cost advantage is primarily attributed to two factors: the relatively low cost of nuclear fuel compared to gas and the absence of CO₂ emissions in the modified CHP system. In conventional gas-steam power plants, the share of gas costs in annual expenses is substantial, while the capital component is minimal. Additionally, the cost of CO₂ emissions represents a significant portion of the annual heat generation costs in traditional plants, a factor likely to increase as the EU seeks higher prices for CO₂ emission permits.

The study also highlighted that electricity generation efficiency in a hierarchical gas-steam CHP unit with a nuclear HTGR and TE can be significantly higher than in conventional nuclear power plants, surpassing the efficiency of pressurized water reactor (PWR) power plants.

In conclusion, the development of novel dual-cycle thermal power plants using hierarchical gas-steam generation technology with high-temperature helium-cooled reactors (HTGR) and TEs is economically and thermodynamically justified. Future research could focus on conducting thermodynamic and economic analyses of power plants and CHP plants with small modular water reactors (SMRs) using only the Clausius–Rankine cycle and comparing the results with those for hierarchical gas-steam power plants and CHP plants with high-temperature helium reactors (HTGRs).