Multicriterial Heuristic Optimization of Cogeneration Supercritical Steam Cycles

Abstract

Heuristic optimization is used to find sustainable cogeneration steam power plants with steam reheat and supercritical main steam parameters. Design solutions are analyzed for steam consumer (SC) pressures of 3.6 and 40 bar and a heat flow rate of 40% of the fuel heat flow rate. The objective functions consisted in simultaneous maximization of global and exergetic efficiencies, power-to-heat ratio in full cogeneration mode, and specific investment minimization. For 3.6 bar, the indicators improve with the increase in the ratio between reheating and main steam pressure. The increase in SC pressure worsens the performance indicators. For an SC steam pressure of 40 bar and 9 feed water preheaters, the ratio between reheating and main steam pressure should be over 0.186 for maximum exergetic efficiency and between 0.10 and 0.16 for maximizing both global efficiency and power-to-heat ratio in full cogeneration mode. The average global efficiency for an SC requiring steam at 3.6 bar is 4.4 percentage points higher than in the case with 40 bar, the average specific investment being 10% lower. The Pareto solutions found in this study are useful in the design of sustainable cogeneration supercritical power plants.

1. Introduction

The simultaneous sustainable production of heat and electricity in combined heat and power (CHP) units has an important potential for saving energy, the high-efficiency cogeneration being promoted by the energy efficiency Directive 2012/27/EU [1]. Compared to the separate production of heat and electricity, cogeneration is more environmentally friendly because it has lower consumption of primary energy and generates lower emissions. To meet the requirements of the last revision of the energy efficiency directive (11.7% savings by 2030) [2], different solutions were investigated, such as heat recovery from engine waste [3], the use of energy storage [4,5], integration of superheating regenerative Rankine cycles with renewable energy sources (biomass, solar) [6], and cogeneration using steam turbines [7,8]. The assessment of cogeneration design solutions and their performances is important to maximize their benefits, mainly to obtain a maximum decrease in the energy costs and CO₂ emissions [9]. In addition, the flexibility of steam power plant operation was studied. For example, in [10,11,12], the flexibility and the constraints that limit the transients of steam boilers are analyzed.

In cogeneration units, the steam consumer (urban or industrial) imposes both the quality and the quantity of the heat to be supplied. Thus, the requirements of the steam consumer determine the electricity generated by the unit, as the portion of steam delivered to the steam consumer (specifically, its parameters and flow rate) is extracted earlier from the turbine and does not produce electricity [13]. As the quality of the heat that is supplied to the steam consumer is shown by the delivered exergy, exergetic efficiency is one of the performance indicators of a CHP plant alongside global efficiency. Further, the ratio between the electricity produced and the heat flow rate to the steam consumer (the power-to-heat ratio in full cogeneration mode) is also used to assess the performance of a sustainable CHP plant.

As shown in scientific literature, the cycles of cogeneration power plants that use steam turbines (simple backpressure or extraction-condensing turbines) do not have steam reheating usually. Besides urban consumption, industrial steam consumers that use cogeneration may be from the pulp and paper industry [14], the sugarcane and alcohol industry [15], the ethylene industry [16], etc. The investigation of chemical recovery of black liquor and soda [14] gave the energy and exergy efficiencies for the cogeneration system of 53.7% and 32.09%, respectively. Also, in [15] different thermodynamic scenarios are analyzed to improve the technical performance of cogeneration schemes by incorporating reheating and regeneration systems in sugar-ethanol plants. By using data collected from an existing ethylene plant, the study carried out by [16] illustrates the performance of a new deterministic optimization model used for multi-type energy systems.

The optimum schemes of such power plants are studied by exergy–economic–environmental analysis [17] or by energy–exergy analysis [18]. In such cases, optimization is used to identify thermal energy storage locations and solutions with higher efficiency and higher flexibility [19,20]. There are also studies that focus on improving the existing optimization algorithms. For example, in [21], a novel optimization algorithm that finds better solutions for high-dimensional traveling salesman problems was proposed. Ref. [22] tested a prototype of a micro-CHP system with a straw-fired boiler and modified Rankine cycle. The results of the retrofitting optimization for the steam–condensate circuit showed that at the optimal operating conditions, the studied system may reach an energy utilization factor of 97.9%. For steam, nuclear, and combined cycle power plants connected to conventional desalination technologies, ref. [23] developed numerical models based on heat and mass balances and performed detailed thermo-economic analysis. The MATLAB software (version 2018) was used to investigate a cogeneration system based on a gas turbine, integrated with a Rankine cycle and an absorption refrigeration cycle [24]. The two-objective optimization (considering exergy efficiency and carbon dioxide emission index) reveals that the higher contribution to the system irreversibility is given by the gasifier (46.7%) and combustion chamber (22.9%). Also, by using MATLAB and a genetic algorithm, ref. [25] identified the optimum operation parameters for a desalinization plant in Saudi Arabia.

To study cogeneration units, lots of studies use dedicated software, such as IPSEpro [26], TRNSYS [27], and EBSILON [28]. Such dedicated commercial software is highly used to study different aspects of cogeneration units. Thus, IPSEpro was used in [29] to analyze and increase thermal efficiency by the implementation of an absorption heat pump to recover the waste heat from the power plant cooling system. In [30], IPSEpro was used to develop a methodology for the technical and economical optimization of a district heat backpressure condenser, based on a novel hybridization of differential evolution and cuckoo search algorithms. The TRNSYS simulation software was used by [31] to investigate the feasibility of replacing the feedwater heater of a conventional cogeneration power plant with a parabolic trough solar collector system and to find the useful solar energy collected. EBSILON was used in [32] to develop for a CHP plant a thermo-electric distribution method based on the characteristics of energy consumption and then to optimize the heat and power load. In [33], EBSILON was used to investigate coal-fired cogeneration systems integrated with steam ejectors at different locations, concluding that the use of a steam ejector for waste heat recovery can significantly reduce exhaust steam loss in the cogeneration system. EBSILON was also used to improve flexible operation strategies of gas turbine combined cycles [34]. The operation analysis and optimization of a cogeneration unit was made by [35] using an original mathematical model and computational program to predict relevant parameters of different operating regimes. The main advantage of using specialized commercial software consists of allowing the easy and fast simulation of different cogeneration schemes. The drawback of such software is its lower flexibility, as the scheme of the cycle is usually defined at the beginning by the user. Thus, the scheme is kept unchanged during simulation, even when different thermodynamic parameters would lead to another scheme. To overcome this inconvenience, programming platforms are used to focus on specific problems.

Thus, to allow the possibility of choosing the best scheme, the analysis presented in this paper uses the methodology given in [36] and a computer program developed under the Scilab open-source software (version 6.1.1) [37].

The objective of the study is to find the optimum design schemes and the corresponding optimum variation ranges of the parameters by using multicriterial heuristic optimization with four objective functions.

If in [36] the design optimization was a parametric one, this study builds upon and extends the previous one by using a heuristic approach. This study brings new results regarding the influence of the parameters (main steam pressure and temperature, reheat temperature, condensing pressure, and the ratio between the reheat pressure and the main steam pressure) on the objective functions and design scheme. Moreover, within the present study, optimized values of the input parameters are found. Under the Scilab platform, a genetic algorithm is used to make a multicriterial optimization of CHP plants and find their best design schemes for given requirements and limitations. To this end, different scheme designs are considered that use supercritical extraction-condensing steam turbines that involve steam extraction from high-pressure turbines. The outcome of the study consists of the Pareto design solutions that optimize simultaneously four objective functions: the global and the exergetic efficiencies, the power-to-heat ratio in full cogeneration mode, and the specific investment in equipment. Thus, the benefits of sustainable energy production are maximized.

2. Materials and Methods

2.1. Supercritical Steam Cogeneration Power Plants

The generic scheme of the thermodynamic cycle of supercritical steam cogeneration power plants is presented in Figure 1. The turbine (T) contains three main parts: the high-pressure turbine (HPT), the intermediate-pressure turbine (IPT), and the low-pressure turbine (LPT). All through the turbine, steam is extracted for the preheating of the feedwater. While the HPT has only a bleed stream, from the IPT and LPT, several steam extractions are used, depending on the size of the preheating system. Depending on the steam pressure required by the steam consumer (SC), steam to the SC is extracted from the IPT or LPT from one of the existing steam bleeds to the preheating system. The remaining steam that was not extracted along the T enters the condenser (C) and is condensed. From C, the condensate and the supply water introduced into the thermal circuit are pumped towards the preheating system using condensate pumps (CP). Entering the preheating system, water is heated by several low-pressure heaters (LPH), a deaerator (D), and several high-pressure heaters (HPH). At the exit of the D, the feedwater is sent towards the steam generator (SG) with the feedwater pumps (FWP). Within the main part of the SG (mainSG), the feedwater is transformed into steam with the required parameters. After its expansion through the HPT, the steam returns to the SG and is reheated in the steam reheater (RH). Within the electrical generator (EG), gross power is produced.

The scheme in Figure 1 is a generic one that suggests by the dashed lines that there are multiple possible schemes that can result from the optimization, with a variable number of HPHs and LPHs and an accordingly variable number of steam extractions from ST. The general configuration in Figure 1 is adaptable to the requirements imposed by the SC, whether it is an industrial, a district heating, or an internal heat consumer.

2.2. Thermodynamic Model

2.2.1. Assumptions and Restrictions

The mathematical model is based on the following main assumptions [13,36,38,39,40,41]:

• Steam generator: imposed fuel heat flow rate into SG (Q_SG), known mass flow rate (F_SG), and SG efficiency (${\eta }_{SG}$);
• Preheating line design: minimum number of preheaters (z), fewer HPHs than LPHs (z_HPH ≤ z_LPH), a maximum acceptable deaerator pressure (p_d-max), and preferably an equal temperature increase (∆t_PH) on each preheater within a maximum limit (∆t_PH-max); steam extraction that is closest to the SC is adjusted to its value;
• Steam consumer: known pressure (p_SC) and heat flow rate (Q_SC), and the option to recover the SC condensate into the deaerator or replace it by adding supply water into the condenser;
• Turbine and pumps: the isentropic efficiency of the turbine depends on the volumetric steam flow rate through each turbine zone and on the isentropic expansion of its turbine section; at the exit of the LPT, a minimum steam quality (x_outLPT-min) is required; the isentropic efficiency of the pumps is imposed;
• Specific investment in equipment (I_sEQ): costs depend mainly on fluid parameters and mass flow rates.

2.2.2. Input Data and Main Equations

The design model for the thermodynamic cycle of supercritical steam cogeneration power plants for a given known set of input data is described and validated in [36].

The main parameters used by the model include:

• The heat flow rate into the steam generator: Q_SG, in kW;
• The pressure of the main steam: p_ms, in bar;
• The temperature of the main steam: t_ms, in °C;
• The reheat temperature of the main steam: t_rh, in °C;
• The ratio between the reheating pressure (p_rh) and the main steam pressure (p_ms), dimensionless, computed as:

r_p = p_rh/p_ms

(1)

• The steam pressure at the condenser: p_c, in bar;
• The steam pressure for the SC: p_SC, in bar.

The equations that describe the main equipment within the thermodynamic cycle are as follows:

• Steam turbine (T):

$$P_T=\sum _{k=1}^{z-z_{SC}-1}{F}_{T_k}\left(h_{in{T}_k}-h_{out{T}_k}\right)+\sum _{k=z-z_{SC}-1}^{n_T}\left(F_{T_k}-F_{SC-HP}\right)\left(h_{in{T}_k}-h_{out{T}_k}\right)$$

(2)

where $P_T$ is the internal power of the T, in kW; $k$ is the current T zone; $z$ is the number of preheaters; $z_{SC}$ is the position of the steam extraction to the SC counted from the LPT exit; $n_T$ is the number of zones in the T; $F_{T_k}$ is the steam mass flow rate in zone k, in kg/s; $F_{SC}$ is the steam mass flow rate to the SC, in kg/s; $h_{inTk}$ and $h_{outTk}$ are the specific enthalpies at the input and output of zone k, in kJ/kg.

• Electrical generator (EG):

$$P_e={\eta }_m{\eta }_g{P}_T$$

(3)

where $P_e$—electrical power of the T (at EG), in kW; ${\eta }_m{\eta }_g$—mechanical and generator assembly efficiency.

• Preheaters (HPH, D, LPH) and condenser (C):

$$F_{s,i{n}_k}{h}_{s,i{n}_k}+F_{w,i{n}_k}{h}_{w,i{n}_k}+F_{c,i{n}_k}{h}_{c,i{n}_k}=F_{s,out\_k}{h}_{s,out\_k}+F_{w,out\_k}{h}_{w,out\_k}+F_{c,out\_k}{h}_{c,out\_k}$$

(4)

• Electrical pumps (CP and FWP):

$$P_{ep}=P_{CP}+P_{FWP}$$

(5)

where P_ep is the electrical motor power of the pumps, in kW [42]; P_CP and P_FWP are the electrical motor powers of CP and FWP, respectively, in kW.

• Steam generator (SG):

$$Q_{outSG}=Q_{mainSG}+Q_{RH}={\eta }_{SG}{Q}_{SG}$$

(6)

where $Q_{outSG}$ is the heat flow rate produced by SG, in kW; $Q_{SG}$ is the fuel heat flow rate into SG, in kW; $Q_{mainSG}$ and $Q_{RH}$ are the heat flow rates received in mainSG and RH, in kW; ${\eta }_{SG}$ is the SG efficiency.

In accordance with the modeling assumptions and restrictions, the best thermodynamic scheme and parameters for the given input data are recommended. To this end, a preliminary technical-economical optimization is made by finding the optimum preheating line, having a minimum number of preheaters, and thus having a minimum investment. This is made by starting from a scheme with minimum LPHs and HPHs and adding one by one additional LPHs and HPHs until simultaneously the restrictions on the division between LPHs and HPHs and the maximum values of ∆t_PH and p_d are met.

To ensure the requested SC parameters without additionally loading the scheme, the position of the preheater that is nearest to the SC is brought together to it. Thus, the equal assumption ∆t_PH is somewhat affected, but a higher number of steam extractions from the turbine is avoided. Accordingly, the model finds the turbine architecture having a minimum number of heat extractions. The reheating ratio is considered as an input value for the model, being subject to future optimization.

To find the design, several iterations are made to meet the mass and heat balance equations of the cycle. Then, the performance of the CHP plant is assessed. Four energo-exergetic and economic indicators are computed:

• The overall global efficiency of the CHP plant (η_gl), in percentages [36]:

η_gl = (P_e − P_ep + Q_SC)/Q_SG × 100

(7)

where P_e is the electrical power at EG, in kW; P_ep is the electrical motor power of the pumps, in kW; Q_SC is the heat flow rate to the SC, in kW; Q_SG is the fuel heat flow rate, in kW.

• The exergetic efficiency of the CHP plant (η_ex), in percentages [18,43,44]:

η_ex = (P_e − P_ep + Ex_SC)/Ex_f × 100

(8)

where Ex_SC is the exergy of SC, in kW, and Ex_f is the input exergy of the fuel, in kW. To compute the exergy, the reference temperature of 298.15 K was considered.

• The power-to-heat ratio in full cogeneration mode (C_CHP), dimensionless [45,46,47,48]:

C_CHP = P_eCHP/Q_SC

(9)

where P_eCHP is the power in full cogeneration mode, in kW;

• The specific investment in equipment (I_sEQ), in USD/kW:

I_sEQ = C_EQ/P_e

(10)

where C_EQ is the total cost of equipment, in USD. C_EQ2 includes the costs of the main and the electrical equipment (SGmain, HPT, RH, IPT, LPT, C, CP, LPH, D, HPH, FWP, EG), multiplied by a factor of 2.08 to consider other costs [38,39,40].

The free open-source Scilab software [37] and the XSteam physical water and steam properties [49] were used to code the model.

2.3. Thermodynamic Cycle Multicriterial Optimization

For the multicriterial optimization of the thermodynamic cycle, a two-stage approach is used:

• The choice of the SC parametric input values, specifically, the minimum and the maximum p_SC values (p_SC-min, p_SC-max);
• The use of heuristic optimization for each SC parametric pSC input value (Figure 2).

The objective function ObjF is to simultaneously maximize the overall global efficiency of the CHP plant (η_gl), maximize the exergetic efficiency of the CHP plant (η_ex), maximize the power-to-heat ratio in full cogeneration mode (C_CHP), and minimize the specific investment in equipment (I_sEQ):

ObjF = max(η_gl) & max(η_ex) & max(C_CHP) & min(I_sEQ)

(11)

subject to the restrictions imposed by the model (described in Section 2.2.1):

z_HPH ≤ z_LPH & p_d ≤ p_d-max & ∆t_PH ≤ ∆t_PH-max & x_outLPT ≤ x_outLPT-min

(12)

The cap values considered by the restrictions from Equation (12) used in the optimization are p_d-max = 12 bar, ∆t_PH-max = 32 °C, x_outLPT-min = 0.88.

The heuristic optimization (Figure 2) uses the Niched Sharing Genetic Algorithm II (NSGA-II) algorithm [39], which is among the most used algorithms for multi-objective optimization problems with constraints [50], including power plants [38,41,51]. NSGA-II uses an initial set of Pareto solutions (a first generation) that evolves further by selecting and keeping only the best solutions over a mutation-crossover process [40,41,52]. From the merged parent and child solutions, by elitism and crowding-distance strategies, the population diversity and convergence to the optimum are enhanced; only the best solutions are used for evolution. Within the model, a population of 500 individuals evolved over 500 generations, considering the mutation and crossover probabilities of 0.1 and 0.7, respectively [38,51,53].

Figure 2. Multicriterial optimization design flowchart.

Figure 2. Multicriterial optimization design flowchart.

The result of the optimization consists of the Pareto function that simultaneously optimizes the indicators in Equations (7)–(10). Thus, the objective functions considered the maximization of η_gl, the maximization of η_ex, the maximization of C_CHP, and the minimization of I_sEQ. The results given by the Pareto frontline do not dominate each other, each of them having the best value (maximum or minimum) for one of the objective functions, and thus none of them being better than another.

3. Results and Discussion

The results of the multi-objective optimization are presented comparatively in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 for p_SC-min = 3.6 bar and p_SC-max = 40 bar. In Section 3.2 there are presented the Pareto solutions that optimize η_gl, η_ex, CCHP and minimize I_sEq (Figure 3, Figure 4 and Figure 5). Also, the variation in the main parameters (t_ms, t_rh, p_ms, p_c) depending on r_p is shown in Section 3.3, emphasizing the number of preheaters (z) used in each case (Figure 6, Figure 7, Figure 8 and Figure 9). The results for the optimization of each objective function on the Pareto frontier (η_gl, η_ex, CCHP, I_sEq) are detailed in Section 3.4 (Figure 10, Figure 11, Figure 12 and Figure 13) to find the sustainable design solutions for the analyzed CHP plant.

3.1. Inputs into the Optimization Model

To find the optimum design solutions for supercritical steam cogeneration power plants, a genetic algorithm was applied separately for each of the required parametric SC values: p_SC-min = 3.6 bar and p_SC-max = 40 bar. The input parameters that were varied within the optimization were as follows:

• Main steam pressure: p_ms = 240–280 bar;
• Main steam temperature: t_ms = 550–600 °C;
• Reheat temperature: t_rh = 550–620 °C;
• Ratio between the reheat pressure (p_rh) and the main steam pressure (p_ms): r_p = 0.08–0.22;
• Condensing pressure: p_c = 0.04–0.05 bar.

Constant values were considered for the heat flow rate into the steam generator (Q_SG = 1700 MW) and the SC heat flow rate (Q_SC = 0.4Q_SG = 425 MW). The main limitations were p_d-max = 12 bar, ∆t_PH-max = 32 °C, and x_outLPT-min = 0.88.

3.2. Multi-Objective Optimization

Figure 3, Figure 4 and Figure 5 comparatively show the Pareto solutions of the multi-objective optimization for p_SC-min = 3.6 bar (Figure 3a, Figure 4a and Figure 5a) and p_SC-max = 40 bar (Figure 3b, Figure 4b and Figure 5b), emphasizing the number of preheaters used (z). Each figure also includes a subfigure showing the initial solutions (red circles) that lead to the final Pareto frontier (red filled circles).

Figure 3. Pareto frontier. Specific investment vs. global efficiency: (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 3. Pareto frontier. Specific investment vs. global efficiency: (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

There are cases when the same global efficiency can be obtained with schemes having eight or nine preheaters (Figure 3b). However, in this case the scheme with eight preheaters has a lower investment, up to 400 USD/kW. Nonetheless, the scheme with nine preheaters has higher exergetic efficiency.

Figure 4. Pareto frontier. Specific investment vs. exergetic efficiency: (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 4. Pareto frontier. Specific investment vs. exergetic efficiency: (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 5. Pareto frontier. Specific investment vs. power-to-heat ratio in full cogeneration mode (C_CHP): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 5. Pareto frontier. Specific investment vs. power-to-heat ratio in full cogeneration mode (C_CHP): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

For a steam consumer (SC) that requires steam at low pressure (3.6 bar), the increasing trend in the global efficiency (η_gl) (Figure 3a), the exergetic efficiency (η_ex) (Figure 4a), and the power-to-heat ratio in full cogeneration mode (C_CHP) (Figure 5a) of the steam power plant is due to the increase in the number of feedwater preheaters. However, it also leads to an increase in the specific investment of the plant (I_sEq) (Figure 3a, Figure 4a and Figure 5a). Compared to p_SC-min = 3.6 bar (Figure 3a, Figure 4a and Figure 5a), at p_SC-max = 40 bar (Figure 3b, Figure 4b and Figure 5b) the performance indicators η_gl, C_CHP, and I_sEq worsen, and η_ex values show minor changes. The steam mass flow rate extracted to the SC produces power only up to the extraction pressure (p_SC). The higher the p_SC values are, the less power is produced in the turbine. For this reason, the increases of p_SC from 3.6 bar to 40 bar imply the η_gl and C_CHP decrease. This worsening occurs even if the mass steam flow rate slightly decreases as a consequence of the assumption of a constant heat flow rate to the SC.

The steam cycle performance indicators (Figure 3, Figure 4 and Figure 5) are statistically analyzed (

Table 1. CHP plant performance indicator values (Q_SC = 0.4Q_SG, p_sc = 3.6 bar, and p_sc = 40 bar). Statistical analysis.

pSC [bar]	Statistical Analysis	ηgl [%]	IsEQ [USD/kW]	CCHP [-]	ηex [%]
3.6	med ± σ	74.2 ± 0.8	1985 ± 159	0.76 ± 0.03	42.6 ± 0.2
	min–max	73.2–74.9	1794–2350	0.72–0.8	42.4–43.1
40	med ± σ	69.8 ± 0.5	2208 ± 191	0.53 ± 0.03	42.6 ± 0.5
	min–max	68.9–70.7	1878–2919	0.48–0.57	41.4–43.5

) and compared for the two cases studied (p_SC = 3.6 bar and p_SC = 40 bar). If the steam consumer requires steam at only 3.6 bar, the average global efficiency of the CHP plant is 74.2%, which is 4.4 percentage points higher than in the case in which the steam consumer requires steam at 40 bar, considering Q_SC = 0.4Q_SG. Also, the average C_CHP increases from 0.53 (at 40 bar) to 0.76 (at 3.6 bar), and the average specific investment decreases by 10%. The average exegetic efficiency has no significant variations.

3.3. Optimization of Input Variables

For the Pareto solutions of the multi-objective optimization (Figure 3, Figure 4 and Figure 5), the variation in the optimum parameters (t_ms, t_rh, p_ms, p_c) depending on r_p (Figure 6, Figure 7, Figure 8 and Figure 9) is presented. These optimum parameters are highlighted differently to emphasize the number of preheaters (z). The results are given separately for p_SC-min = 3.6 bar (Figure 6a, Figure 7a, Figure 8a and Figure 9a) and p_SC-max = 40 bar (Figure 6b, Figure 7b, Figure 8b and Figure 9b), emphasizing the number of preheaters used (z).

Figure 6. Pareto frontier. Main steam temperature (t_ms) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 6. Pareto frontier. Main steam temperature (t_ms) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 7. Pareto frontier. Reheat steam temperature (t_rh) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 7. Pareto frontier. Reheat steam temperature (t_rh) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 8. Pareto frontier. Main steam pressure (p_ms) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 8. Pareto frontier. Main steam pressure (p_ms) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 9. Pareto frontier. The steam pressure at the condenser (p_c) vs. the ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 9. Pareto frontier. The steam pressure at the condenser (p_c) vs. the ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Higher values of the objective functions are usually obtained for high main steam temperatures (Figure 6a), high reheat temperatures (Figure 7a), and lower condenser pressures (Figure 9a). The main steam pressures are maintained at closer values to the upper limit of the studied range (Figure 8a).

The optimum values for the steam parameters (Figure 6, Figure 7, Figure 8 and Figure 9) are statistically analyzed in

Table 2. Optimum values for the steam parameters (Q_SC = 0.4Q_SG, p_sc = 3.6 bar and p_sc = 40 bar). Statistical analysis.

pSC [bar]	Statistical Analysis	pms [bar]	tms [°C]	trh [°C]	pc [bar]
3.6	med ± σ	275.1 ± 2.8	573.9 ± 6.4	589.3 ± 7.6	0.047 ± 0.003
	min–max	252–279.7	561.5–596.8	566.5–615.9	0.04–0.05
40	med ± σ	276.3 ± 2	578.4 ± 6.3	592.6 ± 5.7	0.045 ± 0.003
	min–max	264.4–279.5	560.7–597.3	574.2–610.7	0.04–0.05

. The average values for the main steam parameters (p_ms, t_ms, and t_rh) are slightly higher in the case with p_SC = 40 bar compared with the case with p_SC = 3.6 bar.

3.4. Optimization of the Ratio Between Reheating and Main Steam Pressure

The values of the objective functions on the Pareto frontier (η_gl, η_ex, CCHP, I_sEq) depending on r_p are shown in Figure 10, Figure 11, Figure 12 and Figure 13. As in the prior subsections, the number of preheaters (z) used in each case is indicated. In addition, the results are shown separately for p_SC-min = 3.6 bar (Figure 10a, Figure 11a, Figure 12a and Figure 13a) and p_SC-max = 40 bar (Figure 10b, Figure 11b, Figure 12b and Figure 13b).

Figure 10. Pareto frontier. Global efficiency (η_gl) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 10. Pareto frontier. Global efficiency (η_gl) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 11. Pareto frontier. Exergetic efficiency (η_ex) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 11. Pareto frontier. Exergetic efficiency (η_ex) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 12. Pareto frontier. Power-to-heat ratio in full cogeneration mode (C_CHP) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 12. Pareto frontier. Power-to-heat ratio in full cogeneration mode (C_CHP) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 13. Pareto frontier. Specific investment (I_sEQ) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

Figure 13. Pareto frontier. Specific investment (I_sEQ) vs. ratio between reheating and main steam pressure (r_p): (a) p_SC-min = 3.6 bar; (b) p_SC-max = 40 bar.

For the case p_SC = 3.6 bar, it may be observed from Figure 6a, Figure 7a, Figure 8a, Figure 9a, Figure 10a, Figure 11a, Figure 12a and Figure 13a that the optimum ratio between reheating and main steam pressure (r_p) is about 0.08 at z = 7, then rises up to r_p = 0.14 at z = 8 and tends to further increase to higher values at z = 9.

However, for an SC that requires steam at high pressure (40 bar), the multi-objective optimization shows that at z = 9 there are two optimum zones (Figure 3b, Figure 4b, Figure 5b, Figure 6b, Figure 7b, Figure 8b, Figure 9b, Figure 10b, Figure 11b, Figure 12b and Figure 13b) with different r_p intervals: between 0.10 and 0.16 and over 0.186. In this case, the higher r_p optimum interval (over 0.186) corresponds to higher exergetic efficiency of the CHP plant, while lower r_p optimum values (0.10–0.16) lead to the maximization of both η_gl and C_CHP. For p_SC = 40 bar, if lower specific investments are required to the detriment of better energy and exergy indicators, the r_p optimum intervals should be between 0.087 and 0.105. As an example, at the same specific investment, I_sEq = 2405 USD/kW, if r_p = 0.133, the results are η_gl = 70.51% and C_CHP = 0.561, while η_ex is only 40.96%. Contrariwise, at the same specific investment of I_sEq = 2405 USD/kW, if r_p = 0.217, there are obtained different values for the objective functions: η_gl = 69.35%, C_CHP = 0.496, and η_ex = 43.07%.

The steam consumer with p_SC = 40 bar requires an optimal average r_p of 0.14, lower than that obtained for the SC with p_SC = 3.6 bar of 0.17 (Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13).

3.5. Model Validation

The methodology is based on our previous work [36], where it was validated against real-world data in subsection Synthesis of the multi-objective optimizations and model validation (references [54,55,56]).

Preheating lines of supercritical steam power plants (typical parameters, operating data, technical reports, existing power plants [18,57,58,59,60,61,62]) have eight or nine preheaters, as also resulted from this optimization. The usual values of the feedwater temperature increase on a preheater are around 25–30 °C, as shown by the above references. Also, the same references indicate the deaerator pressures around 4–10 bar. Within the model, the number of preheaters is found considering a temperature increase lower than 32 °C and a deaerator pressure lower than 12 bar. Thus, the cap limit of 32 °C for the feedwater temperature increase on a preheater and 12 bar for the deaerator pressure are in line with the usual values, as shown by the above references.

As an example, Figure 14 shows the thermodynamic cycle of a supercritical steam cogeneration power plant resulting from optimization with nine preheaters and the steam consumer requiring a heat flow rate of Q_SC = 0.4Q_SG, and a steam pressure of p_SC = 3.6 bar.

Figure 14. Optimum thermodynamic cycle of supercritical steam cogeneration power plant with z = 9, Q_SC = 0.4Q_SG, p_SC = 3.6 bar.

Figure 14. Optimum thermodynamic cycle of supercritical steam cogeneration power plant with z = 9, Q_SC = 0.4Q_SG, p_SC = 3.6 bar.

The optimized thermodynamic cycle in Figure 14 can be used in one of the two optimum zones in Figure 3b, Figure 4b, Figure 5b, Figure 6b, Figure 7b, Figure 8b, Figure 9b, Figure 10b, Figure 11b, Figure 12b and Figure 13b (with r_p = 0.10–0.16 or r_p > 0.186), depending on the tradeoff between all the objective functions.

4. Conclusions

The design solutions of sustainable supercritical steam cogeneration power plants were analyzed using heuristic optimization. Schemes with reheat and different sizes of the preheating system were considered within specific ranges of the main parameters (main steam pressure and temperature, reheat temperature, condenser pressure, and ratio between reheating and main steam pressure). For the considered objective functions (η_gl, η_ex, C_CHP, I_sEq), the influence of the steam consumer pressure on the Pareto frontier was studied, showing the optimum ranges of the analyzed parameters.

While for lower SC pressure (p_SC = 3.6 bar) there is a clear increasing trend of the objective function values with the number of feedwater preheaters, for higher SC pressures the performance indicators η_gl, C_CHP, and I_sEq worsen. The Pareto solutions strongly depend on r_p, having different r_p optimum values for the maximization of each energetic and exergetic indicator and minimization of specific investment. For p_SC = 40 bar and z = 9, a higher optimum r_p interval (r_p > 0.186) corresponds to higher η_ex, while a lower r_p optimum interval (0.10–0.16) corresponds to higher η_gl and C_CHP.

Future research directions will include the analysis of optimum design solutions of supercritical steam cogeneration power plants for other SC requirements. Thus, the analysis could be extended across several steam pressure consumers, and a sensitivity analysis across a range of consumer pressures could be performed. Also, to further increase the sustainability of the CHP plant, the development of thermodynamic schemes will be considered to allow the integration of renewable energy and the use of CO₂ capture systems.