Optimization of Cogeneration Supercritical Steam Power Plant Design Based on Heat Consumer Requirements

Abstract

High-efficiency design solutions for cogeneration steam power plants are studied for different steam consumer requirements (steam pressures between 3.6 and 40 bar and heat flow rates between 10 and 40% of the fuel heat flow rate into the steam generators). Using a genetic algorithm, optimum designs for schemes with extraction-condensing steam turbines, reheat, and supercritical parameters were found considering four objective functions (high global efficiency, low specific investment in equipment, high exergetic efficiency, and high power-to-heat ratio in full cogeneration mode). A second Pareto front was computed from the prior solutions, considering the first two objective functions, resulting in the high-efficiency cogeneration schemes with a primary energy savings (PES) ratio higher than 10%. The results showed that the PES ratio depends strongly on the steam consumer requirements, rising from values under 10% for low heat flow rates and few preheaters to over 25% for a higher number of preheaters, high heat flow rates, and low steam pressures to the consumer. At the same heat flow rate to the consumer, the power-to-heat ratio in full cogeneration mode increases with the decrease in the required steam pressure to the consumer.

1. Introduction

The high interest in increasing energy efficiency production and in lowering the negative impact on the environment (decreasing the greenhouse gas emissions and lowering the fuel consumption) has led to the use of combined heat and power (CHP) plants. CHP units are considered in the 2012/27/EU Directive [1,2] by the promotion of high-efficiency cogeneration, CHP units having at least 10% primary energy savings compared to reference values of separate heat and electricity production. To obtain energy savings, several possible solutions are available: cogeneration by using gas turbines with conventional fuel [3], CHP plants by using steam turbines [4,5], and integration of biomass gasification [6], the integration of solar energy or biomass within regenerative Rankine cycles [7,8]. Also, the carbon capture technology was integrated into cogeneration power plants [9,10]. The recovery of the waste heat from engines [11] or from nuclear power plants [12] was also studied. Complex distributed cogeneration systems using solid oxide fuel cells, gas turbines, steam turbines, and heat exchangers are analyzed in [13].

In cogeneration units, the steam consumer’s quantitative and qualitative parameters depend on the steam consumer’s requirements. Usually, lower steam pressures are needed for urban consumers (heating and hot water), while possibly higher steam pressures are used by industrial steam consumers, depending on the needs of each specific industry. Examples of industries that use cogeneration units with steam turbines include the sugarcane and alcohol industry [4,14,15,16], the ethylene industry [17], the pulp and paper industry [18], water desalination [19], etc. Most cogeneration units use backpressure or extraction-condensing steam turbines without steam reheating. In case of extraction-condensing steam turbines, according to the heat requirements of urban or industrial steam consumers (steam pressure and temperature, specific steam mass flow rate), steam is extracted from the steam turbine before its complete expansion, thus reducing the electricity produced by the CHP unit [20]. The optimization of CHP units is needed and should focus on the increase in global efficiency, the increase in exergetic efficiency, and the increase in the power-to-heat ratio in full cogeneration mode. Also, according to the 2012/27/EU Directive [1], the primary energy saving compared to the separate production of heat and power is an important characteristic of CHP units that should be analyzed. Several complex energy–exergy, exergy–economic–environmental, and thermo-economic optimizations of CHP units are studied within the scientific literature [13,19,21,22]. In [15], it is shown that the integration of exergy and pinch analyses is necessary for the integration of steam turbine cogeneration systems.

Dedicated software, like AspenPlus, TRNSYS, EBSILON, SEpro, etc., was used to study and improve the performance of cogeneration power plants, along with genetic algorithms or neural networks. Thus, to predict the performance of a CHP unit, Ref. [6] used AspenPlus and a neural network model based on particle swarm optimization. For a 300 MW cogeneration unit, Ref. [10] also used AspenPlus to optimize the full heat recovery integration of a sodium-based carbon capture system. A solar-biomass driven cogeneration system in the Canary Islands was optimized in [23] by using the Multi-Objective Grasshopper Optimization Algorithm. A modified neural network combining Boolean mapping with a cutting-edge feed-forward algorithm was used by [24] to study isolated systems that include renewable energy. The TRNSYS software was used by [5] to analyze the performance of a cogeneration power plant, which was integrated with a solar system. The Modelica/DYMOLA platform was used in [25] to model a 300 MW drum-type boiler and allow efficient dynamic operation. Other examples of software used for technical and economic optimization are IPSEpro [26,27] and EBSILON [28,29]. Also, the ProSimPlus process simulator was used by [16] for the exergy analysis.

In the scientific literature, the focus is mainly on cogeneration units with steam turbines without reheat and subcritical parameters. There are a few studies that analyze cogeneration supercritical steam power plants with reheat. Also, these studies analyze specific schemes without considering the optimization of the design and the influence of the steam consumer requirements on the design solutions.

This study focuses on steam turbine cogeneration cycles with supercritical parameters and extraction-condensing steam turbines. To meet the gap in the scientific literature, this study identifies the optimum design solutions that should be considered to deliver heat at the pressure and heat flow rate required by a steam consumer. To this end, the schemes with the highest global efficiency and lowest investment were selected from a set of four objective Pareto solutions (maximized global efficiency, exergetic efficiency, power-to-heat ratio in full cogeneration mode, and minimized specific investment in equipment).

The objective of this study was to extract from the optimum schemes those schemes that meet the condition of high-efficiency cogeneration (primary energy saving over 10% [1]). The scope of this paper is to study the influence of the main parameters of the steam cycle, as well as the cycle design, on the primary energy saving ratio value.

2. Thermodynamic Cycle

The scheme of the thermodynamic cycle of supercritical steam cogeneration power plants considered in this study is shown in Figure 1.

It contains the following main components:

• The steam generator (SG), which comprises two main parts: the main part of the SG (mainSG), which generates steam with supercritical parameters, and the steam reheater (RH), where steam is reheated after its expansion in the first cylinder of the turbine.
• The steam turbine (ST) and the electrical generator (EG). The ST has three distinct cylinders through which steam passes and expands, namely the high-, intermediate-, and low-pressure turbines (HPTs, IPTs, LPTs). All these cylinders contain one or more steam extractions that provide steam to the preheating system and the steam consumer. This study considers HPTs with steam extraction. However, the number of steam extractions from IPTs and LPTs is variable, depending on the preheating system design optimization. In the electrical generator (EG), mechanical energy is transformed into electrical energy.
• The condenser (C) condenses the steam that exits the LPTs into water by using a cooling water circuit. The condensing pressure is below the atmospheric pressure to maximize the steam expansion in the ST.
• The steam consumer (SC) requires a specific heat flow rate with specific steam parameters. To ensure this requirement, the ST should have a steam extraction whose position from the ST depends on the steam pressure to the SC.
• The preheating system. Feedwater is heated through a series of low-pressure heaters (LPHs), a deaerator (D), and a series of high-pressure heaters (HPHs). Steam for the preheating system is extracted from the ST. The condensate from the HPHs is carried in cascade toward D, and the condensate from the LPHs is carried in cascade toward C.
• Electrical pumps. There are the following electrical pumps along the thermodynamic circuit: the condensate pumps (CPs), which introduce the condensate into the LPHs and send it toward D, and the feedwater pumps (FWPs), which send the feedwater through the HPHs toward the SG.

3. Materials and Methods

The design schemes of supercritical condensing steam turbines and steam reheat are analyzed and optimized by using a genetic algorithm. For given heat requirements to a steam consumer, the main parameters and the main characteristics of the thermodynamic cycle scheme that allows the highest global efficiency with the lowest specific investment in equipment are optimized. Moreover, schemes with high-efficiency cogeneration are indicated. The multi-criterion optimization is based on the methodology described in [30] and the Non-Dominated Sorting Genetic Algorithm 2 (NSGA-II).

3.1. Input Data, Assumptions, and Restrictions

The input data used in the model of the thermodynamic cycle schemes (Figure 1) are given in

Table 1. Input data into the thermodynamic cycle scheme.

Parameter	Symbol	Units
Heat flow rate to steam generator	QSG	kW
Main steam pressure	pms	bar
Main steam temperature	tms	°C
Difference between reheat and main steam temperature	Δtrh	°C
Ratio between the reheating pressure and the main steam pressure	rp	-
Steam pressure at condenser	pc	bar
Heat flow rate at SC	QSC	kW
Steam pressure at SC	pSC	bar

.

Besides the input data in Table 1, which were varied during simulation, there are also design data (equipment, connections), design computing options (if some design conditions are imposed or not), constant values, restrictions, and assumptions. The key restrictions and assumptions considered by the model are as follows [20,30,31,32]:

• For the steam generator, the use of the same known fuel, with a given heat flow rate of fuel (Q_SG); the efficiency of the SG is constant (0.95).
• For the steam turbine, there is a bleed steam in the HPT; the steam quality at the LPT exit must be higher than the minimum accepted value (0.88).
• For the electrical generator, the mechanical efficiency of the turbine and electrical generator (η_mg) depends on the internal power of the ST.
• For the condenser, the condenser steam pressure (p_c) is an input parameter (Table 1), which varies within a given interval.
• For the steam consumer, the steam pressure at the SC and the heat flow rate at the SC are input values (Table 1), modified according to the SC requirements. The SC condensate is replaced by the supply water at the condenser (C).
• For the preheating system, equal temperature increases in preheaters, lower than the maximum imposed value of 32 °C (except for preheaters next to the SC); use of the lowest number of preheaters; the number of HPHs should not exceed the number of LPHs; the deaerator pressure should be lower than the maximum imposed value (12 bar); steam extraction pressure to preheaters closest to the SC is adjusted to be equal to the SC pressure.
• For the pumps, the electrical motor overall efficiencies of the pumps are constant (0.85 for the FWP and 0.8 for the CP).

3.2. Modeling Equations

The equipment in Figure 1 (steam generator, steam turbine, pumps, preheating system, condenser) is modelled by the mass and heat balance equations. The main equations [30] that model the thermodynamic cycle are given below.

3.2.1. Steam Generator (SG, SGmain, RH)

The two parts of the SG (mainSG and RH) are calculated separately. The heat flow rate in SGmain, in kW, is given by

$${Q_{SGmain}=F}_{SG}\left(h_{ms}-h_{inSG}\right),$$

(1)

where $Q_{SGmain}$ is in kW; F_SG—steam flow rate at the SG exit, in kg/s; h_ms—main steam specific enthalpy at the SG exit, in kJ/kg; and h_inSG—feed water specific enthalpy in the SG, in kJ/kg;

The heat flow rate in the RH depends on which cylinder the SC is fed from. Thus, if the SC is fed from the HPT, $Q_{RH}$, in kW, is

$$Q_{RH}=\left(F_{SG}-F_{outHPT}-F_{bs}-F_{SC}\right)\left(h_{inRH}-h_{outRH}\right),$$

(2)

where F_outHPT—preheating steam flow rate from the HPT exit, in kg/s; F_bs—bleed steam flow rate from the HPT to the last HPH, in kg/s; F_SC—steam flow rate to the SC, in kg/s; h_inRH—specific enthalpy of steam at the RH input, in kJ/kg; and h_outRH—specific enthalpy of steam at the RH output, in kJ/kg.

If the SC is fed after the HPT (from the IPT or LPT), $Q_{RH}$, in kW, is given by

$$Q_{RH}=\left(F_{SG}-F_{outHPT}-F_{bs}\right)\left(h_{inRH}-h_{outRH}\right).$$

(3)

The heat flow rate produced in the entire SG, in kW, is obtained by summing the heat flow rate in SGmain (Equation (1)) and in the RH (Equation (2) or Equation (3)):

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

(4)

where $Q_{outSG}$—heat flow rate produced in the SG, in kW; Q_SG—fuel heat flow rate in the SG, in kW; and η_SG—SG efficiency.

3.2.2. Steam Turbine (ST)

The turbine is divided into several zones. A zone is defined at the entrance of a cylinder up to the first steam extraction, between two consecutive steam extractions, and from the last steam extraction to the exit of the cylinder. Considering that within the ST there are n_ST zones, the specific steam enthalpy at the output of ST zone k (where k varies between 1 and n_ST), in kJ/kg, is

$$h_{outSTk}=h_{inSTk}-\left(h_{inS{T}_k}-h_{t/outS{T}_k}\right){\eta }_{S{T}_k},$$

(5)

where h_inSTk—specific steam enthalpy at ST zone k input, in kJ/kg, h_outSTk—specific steam enthalpy at ST zone k output, in kJ/kg; $h_{t/outTk}$—theoretical specific enthalpy at exit of zone k, in kJ/kg; and ${\eta }_{STk}$—isentropic efficiency of the k zone.

The internal power produced by the ST zones k that are upstream of the steam extraction to the SC, in kW, is

$$P_{STk}=F_{S{T}_k}\left(h_{in{ST}_k}-h_{outS{T}_k}\right),$$

(6)

and the internal power produced by the ST zones k that are downstream of the steam extraction to the SC, in kW, is

$$P_{STk}=\left(F_{S{T}_k}-F_{SC}\right)\left(h_{in{ST}_k}-h_{outS{T}_k}\right).$$

(7)

Summing the internal power produced by all ST zones (calculated with Equation (6) or Equation (7)) yields the internal power of the ST, P_ST, in kW:

$$P_{ST}={\sum }_{k=1}^{n_{ST}}{P}_{STk}.$$

(8)

Two other important values are calculated for ST, which are subject to optimization:

• The dimensionless ratio between the reheating pressure and the main steam pressure:

  $$r_p=p_{rh}/p_{ms},$$

  (9)
• The difference between the reheat and the main steam temperature, in °C:

  $${\Delta t}_{rh}=t_{rh}-t_{ms},$$

  (10)

  where $t_{rh}$—reheat steam temperature, in °C; $t_{ms}$—main steam temperature, in °C.

3.2.3. Electrical Generator (EG)

The power at EG, in kW, is given by

$$P_{EG}={\eta }_{mg}{P}_{ST},$$

(11)

where η_mg—the mechanical-generator assembly efficiency, dimensionless.

3.2.4. Preheating System (HPH, D, LPH) and Condenser (C)

All through the preheating system, the feedwater temperature increases from the saturation temperature at the condenser ($t_c$, in °C) to $t_{inSG}$, the feedwater temperature into SG, in °C. From these, the temperature increase across the preheater, in °C, is calculated as

$${\Delta t}_p=\left(t_{inSG}-t_c\right)/z,$$

(12)

The energy balance equation on a preheater is

$$F_{s/in}{h}_{s/in}+F_{w/in}{h}_{w/in}+F_{c/in}{h}_{c/in}=F_{s/out}{h}_{s/out}+F_{w/out}{h}_{w/out}+F_{c/out}{h}_{c/out},$$

(13)

where $F_{s/in}$, $F_{s/out}$,—input/output steam mass flow rate, in kg/s; $h_{s/in}$, $h_{s/out}$—specific enthalpy of steam, in kJ/kg; $F_{w/in}$, $F_{w/out}$—input/output feedwater mass flow rate, in kg/s; $h_{w/in}$, $h_{w/out}$—feedwater specific enthalpy, in kJ/kg; $F_{c/in}$, $F_{c/out}$—input/output condensate mass flow rate, in kg/s; and $h_{c/in}$, $h_{c/out}$—condensate specific enthalpy, in kJ/kg.

The energy balance equation on a preheater is

$$F_{s/in}{h}_{s/in}+F_{w/in}{h}_{w/in}+F_{c/in}{h}_{c/in}=F_{w/out}{h}_{w/out}+F_{c/out}{h}_{c/out},$$

(14)

where $F_{s/in}$—input steam mass flow rate, in kg/s; $h_{s/in}$—steam specific enthalpy, in kJ/kg; $F_{w/in}$, $F_{w/out}$—input/output feedwater mass flow rate, in kg/s; $h_{w/in}$, $h_{w/out}$—input/output feedwater specific enthalpy, in kJ/kg; $F_{c/in}$, $F_{c/out}$—input/output condensate mass flow rate, in kg/s; and $h_{c/in}$, $h_{c/out}$—condensate specific enthalpy, in kJ/kg.

In Equation (14), the output condensate mass flow rate is

$$F_{c/out}=F_{s/in}+F_{c/in},$$

(15)

Also, in the specific case of a contact heat exchanger (the deaerator), $h_{w/out}$ equals $h_{c/out}$ in Equation (14).

3.2.5. Electrical Pumps (FWP, CP)

The electrical motor power of the EP, calculated as the sum of the electrical motor power of the CP ($P_{CP}$) and the electrical motor power of the FWP ($P_{FWP}$), in kW, is

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

(16)

The electrical motor power of the CP, in kW, is

$$P_{CP}=F_{CP}\left(h_{t/outCP}{ - h}_{inCP}\right)/{\eta }_{CP},$$

(17)

where $F_{CP}$—CP mass flow rate, in kg/s; $h_{t/outCP}$—theoretical output specific enthalpy at the CP, in kJ/kg; $h_{inCP}$—input specific enthalpy at the CP, in kJ/kg; and ${\eta }_{CP}$—CP and electrical motor overall efficiency.

Similar to Equation (17), the electrical motor power of the FWP, in kW, is

$$P_{FWP}=F_{FWP}\left(h_{t/outFWP}{ - h}_{inFWP}\right)/{\eta }_{FWP},$$

(18)

where $F_{FWP}$—FWP mass flow rate, in kg/s; $h_{inFWP}$, $h_{t/outFWP}$—input/theoretical output specific enthalpy at the FWP, in kJ/kg; and ${\eta }_{FWP}$—FWP and electrical motor overall efficiency.

The design of the cycle results from an iterative computation. Starting from an initial number of preheaters and considering the input parameters, preheaters are added, and the scheme is progressively reconfigured to meet the modeling assumptions and restrictions. Iterations stop when all the mass and heat balance equations are fulfilled simultaneously for all equipment. Thus, the best scheme is found, with the minimum number of preheaters and with the lowest investment.

3.3. Performance Indicators

Once the scheme of the thermohydraulic circuit is set up, the performance of the CHP plant is assessed by the calculation of four energetic, exergetic, and economic performance indicators (

Table 2. CHP plant performance indicators.

Performance Indicator	Symbol	Units	Equation	Legend
global efficiency of the CHP plant	ηgl	%	ηgl = (PEG − PEP + QSC)/QSG × 100	PEG—power at EG, in kW; PEP—motor power of EP; QSC—heat flow rate to SC, in kW; QSG—fuel heat flow rate in SG, in kW.
exergetic efficiency	ηex	%	ηex = (PEG − PEP + ExSC)/Exf × 100 *	PEG—power at EG, in kW; PEP—motor power of EP; ExSC—exergy of SC, in kW; Exf—input exergy of the fuel, in kW
power-to-heat ratio in full cogeneration mode	CCHP	-	CCHP = PeCHP/QSC	PeCHP—power in full cogeneration mode, in kW
specific investment in equipment	IsEQ	USD/kW	IsEQ = CEQ fm/PEG	CEQ—cost of equipment (SGmain, RH, HPT, IPT, LPT, C, HPH, D, LPH, FWP, CP, EG), in USD; fm—multiplying factor to consider other costs and construction costs (fm = 2.08) [33]; PEG—power at EG, in kW.

* The reference temperature to compute exergy was 298.15 K.

).

Additionally, the primary energy savings (PES) ratio was calculated [1]:

PES_ratio = [1 − 1/(η_hCHP/η_hREF + η_eCHP/η_eREF)] × 100,

(19)

where PES_ratio—primary energy savings ratio, in percentage; η_hCHP—heat efficiency of the CHP; η_eCHP—electrical efficiency of the CHP; η_hREF—reference efficiency for heat production; and η_eREF—reference efficiency for electricity production. Thermodynamic cycles that have PES over 10% meet the condition of high-efficiency cogeneration and are promoted by the 2012/27/EU Directive [1].

The numerical calculations to compute the model were made under Scilab software [34] by using the XSteam physical water/steam properties [35]. The model was validated in [30] against data from existing power plants in the literature [36,37,38].

3.4. Thermodynamic Cycle Optimization

The objective function consists of optimizing the four performance indicators shown in Table 2:

Objective_function_4 = {η_gl = max; η_ex = max; C_CHP = max; I_sEQ = min},

(20)

subject to the model restrictions indicated in Section 3.1.

The optimization problem with constraints in Equation (20) is solved by using the Niched Sharing Genetic Algorithm (NSGA II) algorithm [34,39]. The heuristic optimization follows the processes in the flowchart shown in Figure 2. Starting from an initial set of Pareto design solutions that were found for the first generation of population, the evolution to better generations was made by successive operations of mutation and crossover strategies. Only the Pareto solutions of each generation were chosen and saved for further evolution [40,41]. Thus, throughout the evolution from one generation to another, the set of Pareto solutions was successively improved toward the optimum. The optimization program using the NSGA II algorithm was developed under Scilab, by using an initial population of 500, which evolved over 500 generations [32,42].

In Figure 2, the selection of the best design schemes for the thermodynamic cycle was made through three successive steps:

• Heuristic optimization of the four objective functions, Objective_function_4, shown in Equation (19): the simultaneous maximization of global efficiency, exergetic efficiency, and power-to-heat ratio in full cogeneration mode, and minimization of specific investment in equipment.
• Two-objective Pareto optimization, chosen from the four objective Pareto solutions, involving simultaneous maximization of global efficiency and minimization of specific investment in equipment:

  Pareto_2/4 = {η_gl = max; η_ex = max}.

  (21)
• Selection of high-efficiency cogeneration schemes from the two objective Pareto solutions (Equation (21)).

The methodology was used for different SC consumer requirements to highlight the optimized solutions and their corresponding PES_ratio. The results are presented and discussed in Section 4.

4. Results and Discussion

The genetic algorithm was run for different combinations of heat flow rates to the SC (between 0.1Q_SG and 0.4Q_SG) and steam pressures to the SC (between 3.6 and 40 bar). This section contains the input data and the values on the Pareto frontiers (objective functions, corresponding optimum parameters, and chosen design).

4.1. Input Data

Pareto solutions that maximize global efficiency and minimize the specific investment in equipment were searched for thermodynamic cycles schemes (Figure 1) that have the fuel heat flow rate Q_SG = 1700 MW. The variable input parameters of the cycle that were used by the genetic algorithm are given in

Table 3. Input data into the optimization model.

Parameter	Units	Domain of Variation
pms	bar	240–280
tms	°C	550–600
Δtrh	°C	0–20
rp	-	0.08–0.22
pc	bar	0.04–0.05

.

The optimization model was run independently for 12 different requirements of the SC (three heat flow rate levels to the SC and four steam pressures at the SC).

The considered values for the heat flow rates to the SC were as follows:

• Low heat flow rate to the SC: Q_SC = 0.1Q_SG = 170 MW;
• Medium heat flow rate to the SC: Q_SC = 0.2Q_SG = 340 MW;
• High heat flow rate to the SC: Q_SC = 0.4Q_SG = 680 MW.

For each of these heat flow rate levels to the SC, four steam pressures to the SC were considered to cover a large range of values: p_SC = 3.6 bar, p_SC = 12 bar, p_SC = 20 bar, and p_SC = 40 bar. Hence, the genetic algorithm was run 12 times, resulting in 12 Pareto frontiers for 4 objective functions, which then gave another 12 Pareto frontiers for 2 objective functions. High-efficiency cogeneration Pareto solutions (PES_ratio over 10%) were put into evidence.

The results of the optimization are grouped into three categories according to the level of the heat flow rate to the SC (low, medium, high) and are presented in Section 4.2, Section 4.3 and Section 4.4. Each category contains all four SC steam pressures (3.6 bar, 12 bar, 20 bar, 40 bar). In Section 4.2, Section 4.3 and Section 4.4, the results on the Pareto frontier and corresponding optimum values of the ratio between reheating and main steam pressure and PES_ratio are shown. Also, the statistical analysis for the optimum input parameters and for the CHP plant performance indicators (PES_ratio, η_gl, I_sEQ, C_CHP, η_ex) are computed for the three considered heat flow rates (for Q_SC = 0.1Q_SG, Q_SC = 0.2Q_SG, and Q_SC = 0.4Q_SG).

4.2. Pareto Design Solutions for Low Heat Flow Rate to SC

Figure 3 presents the Pareto frontier for a low heat flow rate to the SC (Q_SC = 0.1Q_SG), highlighting the steam pressure to the SC with different colors: blue for p_SC = 3.6 bar, orange for p_SC = 12 bar, green for p_SC = 20 bar, and red for p_SC = 40 bar. Moreover, the number of preheaters used in each case is indicated by different markers: squares for z = 7, triangles for z = 8, and circles for z = 9. This legend was used in all the following graphs for an easier comparison of data.

In the case of a low heat consumer (Q_SC = 0.1Q_SG), η_gl varies between 47.8% and 51.1% (Figure 3), depending on the steam pressure (p_SC) of the SC and on the number of preheaters (z). For the same number of preheaters, the increase in p_SC (from 3.6 bar to 20 bar) leads to a decrease in η_gl, for the same I_sEQ. This happens because the steam flow that is sent to the SC is no longer used to produce work in the turbine. However, if the steam is extracted from the HPT exit or from the HPT bleed (for example for p_SC = 40 bar, Figure 3), it is possible that η_gl increases due to the change in conditions, as the steam flow extracted for SC is not entering the steam reheater and it is not receiving heat from the steam generator.

Figure 4, Figure 5 and Figure 6 and

Table 4. Optimum values for low heat flow rate to the SC (Q_SC = 0.1Q_SG). Statistical analysis.

pSC [bar]	Statistical Analysis	rp [-]	pms [bar]	tms [°C]	Δtrh [°C]	pc [bar]
3.6	med ± σ	0.16 ± 0.03	272.3 ± 5.4	576.8 ± 7.1	15.1 ± 3.2	0.047 ± 0.003
	min–max	0.08–0.22	246.4–279.9	558.6–591.7	5.5–19.4	0.04–0.05
12	med ± σ	0.17 ± 0.03	274.3 ± 4.6	578.8 ± 6.3	15 ± 2.3	0.045 ± 0.003
	min–max	0.09–0.22	254.6–279.2	566–596.2	4.7–19.7	0.04–0.05
20	med ± σ	0.15 ± 0.05	272.1 ± 6.8	579.4 ± 5.5	14.9 ± 3.4	0.045 ± 0.003
	min–max	0.08–0.22	244.1–278.9	559.8–591.4	4.1–19.1	0.04–0.05
40	med ± σ	0.12 ± 0.02	274.2 ± 3.7	577 ± 7	15.7 ± 1.6	0.047 ± 0.003
	min–max	0.09–0.16	263.5–279.3	557.9–595.9	6.9–18.8	0.04–0.05

and

Table 5. CHP plant performance indicators values for Q_SC = 0.1Q_SG. Statistical analysis.

pSC [bar]	Statistical Analysis	PESratio [%]	ηgl [%]	IsEQ [USD/kW]	CCHP [-]	ηex [%]
3.6	med ± σ	11 ± 1.3	50.1 ± 0.6	2005 ± 143	0.71 ± 0.04	40.3 ± 0.5
	min–max	7.7–13.2	48.4–51.1	1799–2455	0.6–0.76	38.9–41.3
12	med ± σ	10.2 ± 1.2	49.6 ± 0.6	2080 ± 167	0.56 ± 0.03	40.6 ± 0.5
	min–max	7.2–12.1	48.2–50.5	1836–2640	0.48–0.61	39.4–41.4
20	med ± σ	9.5 ± 1.1	49.2 ± 0.5	2102 ± 179	0.53 ± 0.03	40.4 ± 0.6
	min–max	7.1–11.2	48.1–50	1821–2500	0.48–0.57	39.2–41.2
40	med ± σ	8.8 ± 1.3	48.9 ± 0.6	2003 ± 135	0.48 ± 0.07	40.3 ± 0.6
	min–max	6.5–11	47.8–50	1829–2464	0.38–0.55	39.2–41.3

contain the results on the Pareto frontier corresponding to the heat flow rate to the SC of 0.1Q_SG.

The CHP plant with a low heat consumer (Q_SC = 0.1Q_SG) does not always meet the condition of a high-efficiency cogeneration power plant (PES_ratio > 10%), as can be seen in Figure 4 and Figure 6. This condition is not met in any case with z < 9.

For z = 9, Q_SC = 0.1Q_SG, and p_SC for 3.6 bar to 20 bar, the optimum r_p is over 0.14, while for z = 7 and 8, r_p is under 0.12 (Figure 5 and Figure 6). For p_SC = 40 bar, the optimum r_p is between 0.095 and 0.16, with the highest values being for the highest z, as for the lower p_SC cases.

4.3. Pareto Design Solutions for Medium Heat Flow Rate to SC

The results of the design optimization for the heat flow rate to the SC of 0.2Q_SG are presented in Figure 7, Figure 8, Figure 9 and Figure 10 and

Table 6. Optimum values for medium heat flow rate to the SC (Q_SC = 0.2Q_SG). Statistical analysis.

pSC [bar]	Statistical Analysis	rp [-]	pms [bar]	tms [°C]	Δtrh [°C]	pc [bar]
3.6	med ± σ	0.16 ± 0.04	273.4 ± 5.1	576.4 ± 5.8	15.8 ± 2.9	0.047 ± 0.003
	min–max	0.08–0.22	252.8–278.5	563.4–593.7	4.9–19.7	0.04–0.05
12	med ± σ	0.18 ± 0.03	275.2 ± 2.4	580.3 ± 8	14.7 ± 2.6	0.045 ± 0.003
	min–max	0.09–0.22	258.1–279.8	559–599.6	4.8–18	0.04–0.05
20	med ± σ	0.16 ± 0.05	274.3 ± 4.8	579.8 ± 6.2	14.8 ± 2.1	0.045 ± 0.004
	min–max	0.08–0.22	254.5–278.4	563.9–598.8	8.3–18.6	0.04–0.05
40	med ± σ	0.12 ± 0.02	273.7 ± 3.8	580 ± 7.8	15.2 ± 2.6	0.046 ± 0.003
	min–max	0.08–0.16	256–279.9	556.2–598.2	4.1–19.3	0.04–0.05

and

Table 7. CHP plant performance indicators values for Q_SC = 0.2Q_SG. Statistical analysis.

pSC [bar]	Statistical Analysis	PESratio [%]	ηgl [%]	IsEQ [USD/kW]	CCHP [-]	ηex [%]
3.6	med ± σ	16.5 ± 1.1	58.1 ± 0.6	1971 ± 144	0.74 ± 0.04	41 ± 0.5
	min–max	13.5–18.5	56.5–59.2	1767–2465	0.65–0.8	39.7–42
12	med ± σ	14.9 ± 0.9	57.1 ± 0.5	2136 ± 176	0.61 ± 0.03	41.4 ± 0.4
	min–max	11.6–16.1	55.5–57.8	1833–2489	0.52–0.65	40.1–42
20	med ± σ	13.7 ± 0.9	56.5 ± 0.4	2152 ± 204	0.56 ± 0.01	41.2 ± 0.6
	min–max	11–15	55.5–57.2	1815–2834	0.54–0.58	40–42
40	med ± σ	12.8 ± 1.2	56 ± 0.6	2085 ± 191	0.52 ± 0.04	41.1 ± 0.6
	min–max	9.9–14.4	54.5–56.9	1845–2640	0.45–0.56	39.6–41.9

. The colors and the markers used in the figures have the same significance as in the prior case.

For a medium heat consumer (Q_SC = 0.2Q_SG), η_gl varies between 54.5% and 59.1% (Figure 7). If the SC requires steam at 3.6 bar, η_gl is over 57.7% for z = 9. In this case, I_sEQ varies between 1868 USD/kW and 2465 USD/kW (Figure 7 and Figure 8). For a similar I_sEQ = 2000 USD/kW and the same z = 9, η_gl = 56% if the SC requires p_SC = 40 bar and increases by 4.3%, reaching η_gl = 58.4%, if the SC requires only p_SC = 3.6 bar (Figure 7).

Except for two Pareto points, for p_SC = 40 bar and z = 7, in all cases, PES_ratio > 10% for Q_SC = 0.2Q_SG (Figure 8 and Figure 10). However, it is observed that the PES_ratio decreases with the increase in p_SC (Figure 10, Table 7). The values of the PES_ratio are between 9.8% and 14.5% (Figure 10) for p_SC = 40 bar, and z from 7 to 9. If the scheme with z = 9 is chosen, the PES_ratio is between 12.6% and 14.5% (Figure 10).

However, for p_SC = 40 bar, r_p has a restrictive interval compared to the other cases, and it stops at r_p = 0.16 (Figure 9 and Figure 10, Table 7). This is due to the jump of the steam extraction for the SC, for p_SC = 40 bar, in the HPT. Thus, for z = 7 and z = 8, the SC is fed with the steam from the HPT extraction (the HPT bleed), and for z = 9, the steam for the SC is taken from the HPT exit.

4.4. Pareto Design Solutions for High Heat Flow Rate to SC

Figure 11, Figure 12, Figure 13 and Figure 14 and

Table 8. Optimum values for high heat flow rate to the SC (Q_SC = 0.4Q_SG). Statistical analysis.

pSC [bar]	Statistical Analysis	rp [-]	pms [bar]	tms [°C]	Δtrh [°C]	pc [bar]
3.6	med ± σ	0.16 ± 0.04	274.4 ± 3.3	572.3 ± 6.3	15.5 ± 2.6	0.047 ± 0.003
	min–max	0.08–0.22	252–279.3	561.5–594.9	4.3–19.4	0.04–0.05
12	med ± σ	0.17 ± 0.04	275.2 ± 3.1	572.7 ± 6.1	12.3 ± 2.3	0.047 ± 0.003
	min–max	0.09–0.22	254.5–279.7	557.2–591.2	5–18.3	0.04–0.05
20	med ± σ	0.15 ± 0.07	273.9 ± 4.3	578.3 ± 7.7	13.8 ± 3.4	0.044 ± 0.004
	min–max	0.08–0.22	263.8–278.7	558.2–591.9	1.4–18.2	0.04–0.05
40	med ± σ	0.12 ± 0.02	276.6 ± 1.9	578.9 ± 6.8	13.9 ± 2.5	0.045 ± 0.003
	min–max	0.09–0.2	269.3–279.5	566.3–597.3	6.5–19	0.04–0.05

and

Table 9. CHP plant performance indicators values for Q_SC = 0.4Q_SG. Statistical analysis.

pSC [bar]	Statistical analysis	PESratio [%]	ηgl [%]	IsEQ [USD/kW]	CCHP [-]	ηex [%]
3.6	med ± σ	25.4 ± 0.9	74 ± 0.7	1856 ± 132	0.76 ± 0.04	42.2 ± 0.5
	min–max	22.9–27	72.4–75.2	1680–2410	0.67–0.82	41.1–43.2
12	med ± σ	21.7 ± 1	71.4 ± 0.7	2022 ± 137	0.62 ± 0.03	42.5 ± 0.4
	min–max	19.3–23.2	69.7–72.5	1832–2469	0.53–0.67	41.3–43.3
20	med ± σ	20.76 ± 0.5	70.8 ± 0.3	2146 ± 263	0.58 ± 0.01	41.5 ± 0.8
	min–max	19.65–21.48	70–71.3	1793–2607	0.56–0.6	41.2–43.4
40	med ± σ	19.53 ± 0.78	70 ± 0.5	2139 ± 174	0.54 ± 0.03	42.3 ± 0.5
	min–max	17.81–20.59	68.9–70.7	1878–2730	0.49–0.57	41.4–43.1

present the design optimization Pareto solutions for the heat flow rate to the SC of 0.4Q_SG.

In the case of a high heat consumer, η_gl has the highest values. For Q_SC = 0.4Q_SG, η_gl varies between 68.8% and 75.2% (Figure 11). Also, in that case, PES_ratio has the highest values. For Q_SC = 0.4Q_SG, the PES_ratio varies between 18% and 27% (Figure 12). For the lowest p_SC (p_SC = 3.6 bar), the PES_ratio is over 23.6% (Figure 12 and Figure 14).

When the steam pressure at the exit of the HPT increases, ${\Delta t}_p$ (Equation (12)) also increases. This happens until the maximum imposed value of ${\Delta t}_p$ is reached. When ${\Delta t}_p$ reaches its maximum accepted value, z is increased to reduce ${\Delta t}_p$. In Figure 11, as well as in Figure 3 and Figure 7, at SC pressures of 40 bar, a discontinuity in the variation of the global efficiency of the CHP plant is observed when the number of preheaters, z, changes.

In the cases in which the steam consumer (SC) requires 40 bar, the following apply:

-

If z = 8, the SC is fed from the steam extraction somewhere along the HPT, with the pressure at the HPT exit able to vary freely between the limits imposed for $r_p$.

-

If z = 9, the SC is fed from the steam extraction at the exit of the HPT, thus fixing the pressure at the HPT exit at 40 bar, as required by the SC.

Therefore, in the case in which the SC is fed from the steam extraction at the HPT exit (z = 9), the number of possible cases is drastically reduced. For example, for a particular p_ms, there is only one possible value for $r_p$, not a range of values. Thus, there are no more continuous possible cases between the minimum $r_p$ and the one corresponding to 40 bar. Because of this, when z changes from 8 to 9, a jump is generated ($r_p$ jumps directly to the high value that corresponds to the SC pressure of 40 bar). The corresponding feedwater temperature into the SG also jumps to a higher value. Therefore, the global efficiency of the CHP plant also jumps abruptly to a higher value (Figure 3, Figure 7 and Figure 11).

Similar discontinuities in Figure 3, Figure 7 and Figure 11 can be observed at an SC pressure of 20 bar, when the steam extraction for the SC jumps from the exit of the HPT into the IPT.

The PES_ratio depends strongly on Q_SC, increasing with Q_SC (Table 5, Table 7 and Table 9). Also, the PES_ratio depends on the p_SC and the number of preheaters (z). For Q_SC = 0.4Q_SG and p_SC = 3.6 bar, if z = 9, the PES_ratio is between 25.1% and 27.0%, while for z = 7, the PES_ratio is between 23.9% and 23.2%. Also, for Q_SC = 0.4Q_SG and p_SC = 20 bar, if z = 9, the PES_ratio is between 20.8% and 21.5%, while for z = 7, the PES_ratio is between 19.7% and 20.9% (Figure 12 and Figure 14).

The optimum r_p depends strongly on the number of preheaters (z) (Figure 5, Figure 6, Figure 9, Figure 10, Figure 13 and Figure 14). For Q_SC = 0.4Q_SG and p_SC = 20 bar, if z = 9, r_p is over 0.204, while for z = 7, r_p is only 0.087. Also, for Q_SC = 0.4Q_SG and p_SC = 12 bar, if z = 9, r_p is over 0.148, while for z = 8, r_p is between 0.093 and 0.115 (Figure 13 and Figure 14).

η_ex does not have significant variations even if the power produced by the turbine varies significantly (Table 5, Table 7 and Table 9), because the decrease in power is compensated by an increase in exergy sent to the SC.

C_CHP depends strongly on p_SC. C_CHP increases if p_SC decreases, for the same Q_SC (Table 5, Table 7 and Table 9), because more power is produced in cogeneration. The steam required for the SC produces power until it is extracted from the turbine; the lower the p_SC, the greater the additional power produced by the turbine.

PES_ratio and ${\eta }_{gl}$ show best values at 3.6 bar due to the higher expansion ratio of the steam between the main steam pressure and SC at the lower pressure (3.6 bar). This leads to an increase in the mechanical work in the turbine and in the electrical power. As the fuel heat flow rate has an imposed value, ${\eta }_{gl}$ increases. Similarly, PES_ratio increases. After steam is extracted from the turbine for SC, the steam mass flow rate in the rest of the turbine (from SC pressure to the condenser pressure) decreases substantially, and therefore, the corresponding mechanical work and electrical power for that section of the turbine decreases substantially. The shorter this last turbine section is, the better the performance.

5. Conclusions

The optimum design of cogeneration with steam turbines depends on the heat consumer requirements. Steam consumers requiring steam pressures between 3.6 bar and 40 bar were analyzed, for delivered heat flow rates between 10% and 40% of the fuel heat flow rate into the steam generators. The best design solution was searched among schemes with extraction-condensing steam turbines, reheat, and supercritical parameters. Based on a design methodology, by using a genetic algorithm, a first Pareto frontier was defined. The four objective functions were global efficiency, specific investment in equipment, exergetic efficiency, and power-to-heat ratio in full cogeneration mode. From the first Pareto, a second one was computed by selecting the solutions with maximum global efficiency and minimum specific investment in equipment. High-efficiency cogeneration schemes with primary energy savings ratios higher than 10% were selected.

The main conclusions of the optimization study are as follows:

• The ratio between the reheating pressure and the main steam pressure (r_p) depends strongly on the number of preheaters (z); higher z corresponds to a higher optimal r_p.
• The PES_ratio depends strongly on SC requirements. When Q_SC increases, the PES_ratio also increases from values below 10% (for Q_SC = 0.1Q_SG and a small number of preheaters) to values above 25% (for Q_SC = 0.4Q_SG, p_SC = 3.6 bar, and z = 9).
• For a medium-heat consumer (Q_SC = 0.2Q_SG) at I_sEQ = 2000 USD/kW and z = 9, η_gl increases by 4.3% if p_SC decreases from 40 bar to 3.6 bar.
• For the same Q_SC, C_CHP increases with the decrease in p_SC due to the supplementary power produced in cogeneration.

This study showed the influences of the requirements of the steam consumer and of the main steam parameters on cogeneration design solutions. A future study can analyze the heuristic optimization of supercritical steam cogeneration power plants with two steam consumers. Also, this subject can be further investigated by studying the possible integration of CO₂ capture and renewable energy sources in the optimized schemes.