Thermodynamic–Economic Co-Optimization of Condenser Cooling Water Flow Under Time-of-Use Spot Pricing: Marginal Sensitivity and Negative-Price Superposition

Abstract

Electricity spot markets with time-of-use pricing create hour-by-hour variations in the economic value of thermal adjustments, requiring coal-fired units to adapt cold-end operation to real-time price signals. However, the nonlinear coupling between circulating water flow and condenser backpressure remains insufficiently characterized across the full operating envelope, and existing optimization strategies target steady-state heat consumption without accounting for the time-varying economic value of identical thermal adjustments under spot pricing. This study develops a quasi-steady-state thermodynamic–economic model that links real-time electricity prices with the nonlinear heat-transfer response of the circulating water system. The model enables the adaptive selection of pump combinations and blade-opening angles by balancing marginal pump power savings against marginal turbine output losses under time-of-use price signals. Using actual electricity spot market data from Zhejiang Province, simulations under different seasonal conditions show clear economic gains. The maximum hourly saving reaches 2190.79 CNY during summer negative-price periods, which is about 5.3 times higher than that in winter, while backpressure deviations remain within 12.5% of the design value. The seasonal disparity is governed by the initial heat exchange driving force, a fundamental thermodynamic property amplified by the negative-price superposition effect. The framework establishes a physical basis for market-responsive cold-end regulation across seasonal and load conditions, supporting the economic dispatch of coal-fired units in spot market environments.

1. Introduction

The large-scale integration of variable renewable energy is transforming coal-fired power plants from baseload providers into flexible resources required to perform deep peak shaving and rapid load cycling [1,2]. Under flexible dispatch, the cold-end system operates far from its design point for extended periods. For a typical 600 MW class unit, reducing load from rated capacity to 105 MW increases the specific heat consumption from approximately 7840 to 9340 kJ/(kW·h) [3], and auxiliary power consumption accounts for 5–8% of gross generation [4]. The advent of electricity spot markets adds a further dimension: time-of-use pricing causes the economic value of a given thermal adjustment to vary by an order of magnitude within a single day, with peak-to-valley price ratios exceeding 10:1 and negative prices appearing during renewable surplus periods [5]. Optimizing cold-end operation under these conditions requires coordinating condenser thermal performance with real-time market signals, yet current approaches face fundamental constraints in the continuous characterization of backpressure–flow sensitivity across the operating envelope, the quantitative coupling of time-varying price signals with the nonlinear heat-transfer response, and the identification of conditions under which flow adjustment yields a net economic gain after accounting for the backpressure-induced output penalty.

Condenser backpressure is governed by the heat balance between steam condensation and cooling water heat absorption, and its response to circulating water flow constitutes the physical foundation of cold-end regulation. Research on backpressure–flow characterization has evolved from steady-state heat balance models under design conditions [6,7], through variable-load extensions that couple condenser pressure with exhaust steam parameters during transient processes [8,9], to system-level coupled models integrating condensers, cooling towers, and pump networks to analyze the joint influence of pump combination, variable-frequency operation, and thermal conditions on cold-end performance [10,11]. In parallel, the micro-incremental power method has been applied to determine optimal backpressure ranges by balancing marginal turbine expansion work against marginal cooling water pump consumption [12]. Reported backpressure prediction deviations under quasi-steady conditions are typically within 3–5% of measured values, and optimal backpressure selection achieves coal consumption reductions of 0.5–1.5 g/(kW·h) [13,14]. However, the underlying heat-transfer physics imposes a fundamental nonlinearity that these discrete-point characterizations do not capture: condenser backpressure depends on the logarithmic mean temperature difference, which is governed by the ratio of inlet and outlet terminal temperature differences through a logarithmic function. When flow reduction drives the outlet cooling water temperature close to the steam saturation temperature, the outlet terminal difference approaches zero, and the backpressure sensitivity to further flow changes increases sharply. This sensitivity amplification arises precisely under the high water temperature and low-flow conditions characteristic of deep peak-shaving operation. Individual case studies have observed that reducing flow by 40% at an elevated water temperature causes backpressure to increase by over 100% relative to the design value [15]. The continuous backpressure–flow sensitivity across the full load–temperature operating envelope therefore remains uncharacterized, limiting the physical basis available for adaptive flow regulation under varying conditions.

Extending cold-end optimization from thermal performance characterization to real-time adaptive regulation requires resolving a more fundamental obstacle: the nonlinear coupling between time-varying external signals and the condenser’s heat-transfer response. Price-driven generation dispatch has progressed from equilibrium analyses of pilot electricity spot markets and their pricing mechanisms [16,17], through multi-signal optimal dispatch strategies demonstrating that time-of-use price signals significantly influence unit output decisions and incentivize flexible generation response [18,19], to operational-level extensions incorporating price prediction into unit maintenance scheduling and multi-stage intraday adjustment strategies [20,21]. Initial attempts to extend price signals to auxiliary equipment have shown that introducing electricity prices into pump start–stop and load distribution decisions reduces auxiliary operating costs by 3–8% [22]. Environmental parameter studies have further revealed that ambient temperature variations of 10–15 °C across seasons shift the condenser operating point from a thermally relaxed regime with large driving temperature differences to a thermally constrained regime with small driving temperature differences, altering the optimal flow rate by up to 50% [23,24]. However, existing price-driven and environment-aware studies treat the cold-end system as an aggregate power consumer with fixed or linearized input–output characteristics. The condenser, in reality, exhibits a coupled nonlinear response: circulating water flow simultaneously affects the tube-side convective heat-transfer coefficient through its dependence on flow velocity and the logarithmic mean temperature difference through the resulting change in coolant temperature rise. The backpressure change then feeds back to turbine output through the final-stage expansion process. This multi-physics coupling means that the net economic benefit of a given flow adjustment is a nonlinear function of three simultaneously varying inputs—load, water temperature, and electricity price—rather than a separable product. Neither purely thermal optimization nor unit-level price-driven dispatch resolves this coupling at the auxiliary equipment level. The quantitative conditions under which circulating water flow reduction yields a net economic gain, after accounting for the backpressure-induced output penalty, remain unexplored.

Through the summary and analysis of the above literature, the main research gaps are identified as follows:

(1) Existing cold-end system models predominantly focus on steady-state or quasi-steady-state operating conditions, while the time-varying operational characteristics of circulating water systems under deep peak-shaving conditions receive insufficient attention. Most models do not adequately consider the combined effects of load fluctuations, meteorological variations, and electricity market price signals simultaneously.

(2) Although existing research has demonstrated the guiding role of electricity price signals on macro-level generation dispatch, how to deeply couple high-frequency fluctuating time-of-use price information with the complex thermal–hydraulic characteristics of cold-end systems to achieve economically optimal dispatch of circulating water pump combinations remains an unresolved problem.

(3) Current optimization strategies for circulating water systems predominantly adopt a single objective of minimizing heat consumption, without incorporating the economic trade-off between auxiliary power costs and turbine vacuum benefits under time-varying electricity spot market conditions. A coordinated optimization framework that balances cold-end thermal performance with real-time market economics is still lacking.

To address these challenges, this study makes the following contributions:

(1) A quasi-steady-state coupled model of the cold-end system is established that simultaneously incorporates unit load, ambient meteorological conditions, and real-time electricity spot price signals. This model enables the time-varying quasi-steady characterization of circulating water system behavior under the complex time-varying operating conditions typical of deep peak-shaving scenarios.

(2) A novel pump combination optimization strategy is proposed that introduces electricity price adjustment weights to balance the trade-off between circulating water pump power consumption costs and turbine vacuum benefits. This strategy enables the adaptive regulation of pump number and blade-opening angles in response to time-of-use price fluctuations, replacing the conventional single-objective heat consumption minimization logic.

(3) The proposed optimization framework is validated using actual electricity spot market data from Zhejiang Province, demonstrating significant economic benefits across diverse load and seasonal conditions. The results provide practical reference for refined operation and the cost control of coal-fired units in electricity market environments, particularly under negative electricity price scenarios during renewable energy surplus periods.

2. Modeling Methods and Optimization Approach

2.1. Cold-End System Configuration

Figure 1 shows the structural schematic diagram of the cold-end system using a double-backpressure and double-shell condenser: the circulating cooling water first enters the low-pressure condenser, and after temperature rise, flows into the high-pressure condenser through the circulating water connecting pipe between the high-pressure and low-pressure condensers, and then exits from the outlet of the high-pressure condenser after another temperature rise. The exhaust steam from the low-pressure cylinder of the steam turbine is divided into two parts, entering the low-pressure side and high-pressure side, respectively, and the condensate is discharged after mixing in the hot well. The unit is equipped with multiple adjustable-blade circulating water pumps. The blade angle of each pump can be adjusted continuously, enabling flow regulation in response to changes in unit load and environmental conditions.

To clarify the coupling relationship among the submodels used in the proposed modeling method, the overall computational workflow is first summarized before introducing the detailed equations, as shown in Figure 2. The model takes the unit load, exhaust steam parameters, circulating water inlet temperature, time-of-use electricity price, candidate pump operating states, and maintenance-cost coefficients as inputs. For each candidate operating state, the pump hydraulic model first calculates the circulating water flow rate, pump head, pump efficiency, and pump power. The calculated flow rate is then introduced into the condenser heat-transfer model to solve the cooling water outlet temperature, steam saturation temperature, and condenser backpressure. Based on the condenser backpressure, pump auxiliary power consumption, fuel cost, and time-of-use electricity price, the original hourly thermodynamic–economic benefit is evaluated. Finally, after introducing the equivalent maintenance cost caused by pump switching and blade-angle adjustment, the original independent hourly optimization problem is transformed into a 24 h continuous-period path optimization problem with transition-cost constraints.

2.2. Simulation Modeling of Circulating Water System

The condenser heat exchange model is established based on the heat balance method. The model calculates the heat exchange rate, condenser backpressure, and cooling water outlet temperature from given boundary conditions.

The condenser operates as a shell-and-tube heat exchanger with steam condensing on the shell side and cooling water flowing through the tube side.

The backpressure of the condenser is obtained by converting the saturated steam temperature in the main condensation zone. To calculate the saturated steam temperature, equations for steam heat release, cooling water heat absorption, and condenser tube-shell surface heat exchange are first listed. The heat exchange equation set is as follows:

$$Q_h=KA\mathsf{\Delta }{T}_{lm}$$

(1)

$$Q_s=D_{pai}(h_{s,in}-h_{s,out})$$

(2)

$$Q_w=m_w{c}_p(T_{w,out}-T_{w,in})$$

(3)

where Q_h is the heat exchange amount of the condenser, W; Q_s is the heat release from steam condensation, W; Q_w is the heat absorption by cooling water, W; D_pai and m_w are the mass flow rate of steam and cooling water, kg/s; h_s, in and hs, out are the inlet and outlet enthalpy of steam, kJ/kg; c_p is the specific heat capacity of water at constant pressure, kJ/(kg·K); T_w, in and Tw, out are the inlet and outlet temperatures of condenser cooling water, K.

Equations (1)–(3) establish the energy conservation constraint: under given exhaust steam conditions, the cooling water outlet temperature is inversely proportional to the flow rate. This relation is the physical origin of the backpressure–flow coupling analyzed in Section 2.5.

Since the once-through cooling system has almost no effect from cooling water return flow heat, the cooling water inlet temperature is mainly constrained by the external environment and serves as an external disturbance variable, while changes in circulating water flow are reflected through the condenser heat balance relationship by changes in cooling water temperature rise and backpressure. According to the heat balance inside the condenser, the steam heat release, cooling water heat absorption, and condenser heat exchange amount are equal. Since changes in cooling water flow will cause temperature changes, three equations need to be solved simultaneously. First, the solution method for condenser heat exchange amount is listed, where the heat exchange area is read from the design drawings.

The heat-transfer coefficient K of the condenser is obtained from the HEI formula [25], with a coefficient of determination R² of 0.95, showing high accuracy:

$$K={\eta }_0{\eta }_c{\eta }_w{\eta }_m$$

(4)

$${\eta }_m={\displaystyle \frac{0.002362h_b^2-0.2093h_b+4.056}{h_b+3.863}}$$

(5)

$${\eta }_0=12.87{\nu }^5-145.1{\nu }^4+638.4{\nu }^3-1525{\nu }^2+3007\nu +714.7$$

(6)

$${\eta }_w=1.151\exp \left[-{\left({\displaystyle \frac{t_{in}-55.95}{75.15}}\right)}^2\right]+0.07026\exp \left[-{\left({\displaystyle \frac{t_{in}-20.01}{14.15}}\right)}^2\right]$$

(7)

where η_c is the cleanliness factor of the condenser, mainly affected by cooling water quality, generally taken as 0.8–0.85 [26]; since the unit in this case study has good water quality and daily tube bundle cleaning is performed, this parameter is taken as 0.85; η₀ is the cooling tube outer diameter correction factor, which is inversely proportional to the cooling tube outer diameter; η_w is the cooling water temperature correction factor, which is directly proportional to the cooling water temperature; η_m is the correction factor for cooling tube material and wall thickness; materials with higher thermal conductivity have larger correction factors, and materials with thicker walls and larger thermal resistance have lower correction factors; h_b is the wall thickness, mm; v is the circulating water flow velocity, m/s.

ΔT_lm is obtained by the logarithmic mean temperature difference method:

$$\mathsf{\Delta }{T}_{lm}={\displaystyle \frac{(T_s-T_{w,in})-(T_s-T_{w,out})}{\ln (\frac{T_s-T_{w,in}}{T_s-T_{w,out}})}}$$

(8)

where ΔT_lm is the logarithmic mean temperature difference, K; T_s is the saturated steam temperature, K; T_w,in and T_w,out are the inlet and outlet temperatures of condenser cooling water, K.

In this equation set, T_w,in is determined by ambient conditions, exhaust steam parameters are obtained from the turbine model, and the heat-transfer coefficient K is calculated from Equations (4)–(7). The unknowns T_w,out and T_s are solved simultaneously. The condenser backpressure P_c is then obtained from T_s via the saturation pressure relation, Equation (9).

$$P_c=9.81{({\displaystyle \frac{T_s-173.15}{57.66}})}^{7.46}$$

(9)

2.3. Adjustable-Blade Pump-Characteristic Modeling

From a system perspective, circulating water flow rate is not an independently adjustable variable but is jointly determined by circulating water pump hydraulic characteristics, blade pitch angle, and network resistance. Determining the optimum cooling water flow rate requires comprehensive consideration of both cycle efficiency and economic costs [27].Therefore, based on the condenser heat-transfer model, it is necessary to further calculate the head, efficiency, and power of circulating water pumps, providing foundations for cold-end system safety analysis and economic optimization.

The intersection points of the head curves at various blade openings of variable-pitch blade circulating water pumps with the pipeline characteristic curves are the operating points of circulating water pumps at each blade openings [28]. The intersection diagrams for single-pump and dual-pump operation are shown in Figure 3, respectively.

Based on the performance curves of the adjustable-blade pump at different blade angles, interpolation and polynomial fitting are used to calculate the operating parameters under arbitrary working conditions. Equation (10) is used to determine the head:

$$H=A+BQ+C{Q}^2$$

(10)

where coefficients A, B, C are coefficients in the head-flow polynomial, which can be obtained by fitting head and flow rate; Q is the cooling water flow rate, m³/s.

The polynomial coefficients were identified by least-squares fitting of manufacturer performance data, with R² > 0.99 for all blade angles considered.

The pump efficiency of circulating water pumps varies with cooling water flow rate. For convenience in research calculations, efficiency curves for different blade openings are all represented by a unified efficiency formula, as shown in Equation (11):

$$\eta =D+EQ+F{Q}^2$$

(11)

where coefficients D, E, F are coefficients in the efficiency-flow polynomial.

The power of circulating water pumps is mainly related to factors such as cooling water flow rate, head, and pump efficiency. The expression for pump power is as follows:

$$P=(\rho gHQ)\eta$$

(12)

where P is pump power, kW; H is head, m; ρ is fluid density; g is gravitational acceleration.

2.4. Thermodynamic–Economic Decision Framework

Integrating the previously defined physical models and boundary conditions, the proposed optimization method is a cold-end operation optimization method with the objective of maximizing unit comprehensive net benefits. This method takes the unit reference load given by upper-level dispatch as input and, under the premise of allowing unit output to vary slightly with cold-end operation modes, comprehensively considers circulating water inlet temperature and time-of-use price information to evaluate cold-end operating conditions and economic performance under different circulating water pump operation strategies.

In the optimization process, the circulating water pump operation mode serves as the core decision variable, including the number of circulating water pumps and blade pitch angle. For given loads and environmental conditions, the corresponding circulating water flow rate and head are determined based on circulating water pump-characteristic curves, and the cold-end model is invoked to calculate condenser backpressure. On this basis, combined with the backpressure-output correction model and unit coal consumption characteristics, the actual unit output, fuel consumption, and circulating water pump plant auxiliary power caused by cold-end operation mode changes are calculated.

The unit comprehensive net benefit is defined as power generation revenue minus fuel cost and circulating water pump power consumption cost, with the following expression:

$$E_{total}=E_{electric}-E_{coal}-E_{pump}$$

(13)

where E_electric is power generation revenue calculated based on time-of-use pricing; E_coal is fuel consumption cost; E_pump is circulating water pump plant auxiliary power cost.

2.5. Marginal Sensitivity Analysis

The optimization in Equation (13) can be interpreted through the marginal sensitivity of two competing physical responses. From Equations (1) and (2), the cooling water outlet temperature under constant steam-side heat-duty Q_s satisfies

$$T_{w,out}=T_{w,in}+{\displaystyle \frac{Q_s}{m_w\cdot c_p}}$$

(14)

where Q_s is the steam-side heat-duty, kW; m_w is the cooling water mass flow rate, kg/s; c_p is the specific heat capacity of water, kJ/(kg·K).

Equation (14) shows that the coolant temperature rise is inversely proportional to flow rate. The outlet terminal temperature difference therefore decreases as flow decreases. Because the logarithmic mean temperature difference in Equation (8) depends on the ratio of the inlet and outlet terminal differences through a logarithmic function, its sensitivity to the outlet terminal difference increases sharply as the latter approaches zero. This drives the condenser backpressure sensitivity to flow to increase nonlinearly in the low-flow regime.

At the optimal flow rate, the marginal pump power saving from flow reduction equals the marginal turbine output loss caused by the associated backpressure increase:

$${\left.{\displaystyle \frac{\partial W_{pump}}{\partial Q}}\right|}_{Q*}={\left.{\displaystyle \frac{\partial W_{turbine}}{\partial P_c}}\cdot {\displaystyle \frac{\partial P_c}{\partial Q}}\right|}_{Q*}$$

(15)

where Q^* is the optimal volumetric flow rate, m³/s; ∂W_pump/∂Q is the marginal pump power with respect to flow, kW/(m³/s); ∂P_c/∂Q is the marginal backpressure sensitivity to flow, kPa/(m³/s); ∂W_turbine/∂P_c is the turbine output sensitivity to backpressure, kW/kPa, determined from the backpressure-correction curve. Both sensitivities are evaluated numerically by central differencing the model outputs across the operating modes in

Table 1. Circulating water flow rate.

Blade-Opening Angle/°	Single-Pump Operating Flow/m3·s−1	Flow Rate of Dual Pumps Operating in Parallel/m3·s−1
30	5.85	9.85
60	8.53	12.86
80	10.417	14.596
100	12.284	16.278
130	13.437	17.153

, requiring no additional computation.

The net benefit change is decomposed into two physical power terms. The pump power change is

$$\mathsf{\Delta }{W}_{pump}=W_{pump,design}-W_{pump,opt}$$

(16)

where W_pump,design and W_pump,opt are the pump power under design and optimized conditions, calculated from Equation (12), kW. A positive value indicates pump power saving.

The turbine output change due to backpressure variation is

$$\mathsf{\Delta }{W}_{turbine}=k_{bp}\cdot (P_{c,opt}-P_{c,design})$$

(17)

where k_bp is the backpressure-correction coefficient, kW/kPa; P_c,opt and P_c,design are the optimized and design backpressure, kPa. When backpressure rises, ΔW_turbine < 0.

The net power change is

$$\mathsf{\Delta }{W}_{net}=\mathsf{\Delta }{W}_{pump}+\mathsf{\Delta }{W}_{turbine}$$

(18)

This quantity is purely thermodynamic and independent of price. The economic benefit is

$$\mathsf{\Delta }{E}_{net}(t)=\mathsf{\Delta }{W}_{net}·p(t)$$

(19)

where p(t) is the electricity price at time t, CNY/kWh; when p(t) < 0 and flow reduction causes a net decrease in generation ΔW_net < 0, the product becomes positive: reduced generation avoids the penalty of selling at negative prices, creating a superposition where both pump savings and output reduction contribute positively.

The algorithm traverses all feasible pump combinations under safety constraints, evaluates Equation (13) for each, and selects the combination with maximum E_total. The procedure is shown in Figure 4.

2.6. Optimization with Maintenance-Cost Consideration

In the preceding thermodynamic–economic model, the operating strategy is optimized according to the hourly net benefit. However, in actual operation, frequent pump start–stop events and blade-angle adjustments may introduce additional maintenance costs. If these costs are ignored, the calculated benefit may be overestimated, and the resulting strategy may involve excessive switching. Therefore, an equivalent maintenance cost is introduced to improve the engineering feasibility of the optimization model.

For the t-th time interval, the operating state of the circulating water pump system is defined as

$$s_t=(n_t,{\theta }_t)$$

(20)

where n_t is the number of operating circulating water pumps, and θ_t is the blade angle. Without considering maintenance costs, the original thermodynamic–economic benefit of state s_t is expressed as

$$B_t^0(s_t)=E_{electric,t}(s_t)-E_{coal,t}(s_t)-E_{pump,t}(s_t)$$

(21)

where E_electric,t, E_coal,t, and E_pump,t denote the electricity revenue, coal consumption cost, and circulating water pump power cost, respectively.

The equivalent maintenance cost consists of two parts. The first part is the pump-switching cost caused by changes in the number of operating pumps between adjacent intervals. The second part is the blade-angle adjustment cost caused by changes in blade angle. To keep the model concise, these two costs are combined as

$$C_m(t)=c_{sw}\left|n_t-n_{t-1}\right|+\lambda c_{sw}\min (n_t,n_{t-1}){\displaystyle \frac{\left|{\theta }_t-{\theta }_{t-1}\right|}{{\theta }_{max}-{\theta }_{min}}}$$

(22)

where c_sw is the equivalent cost of one pump-switching event, λ is the relative cost coefficient of blade-angle adjustment, and θ_max and θ_min are the upper and lower limits of the blade angle.

Because plant-specific maintenance records were unavailable, this study does not attempt to perform direct maintenance-cost accounting. Instead, following the common treatment of start-up and transition costs in power-system dispatch studies [29], an equivalent switching penalty is introduced only for a sensitivity-based engineering scenario. A reference equivalent switching penalty c_ref = 1955 CNY/event is assumed for a pump with reference power P_ref = 3400 kW. The representative single-pump power in the present case is P_rep = 2305.4 kW. The equivalent pump-switching cost is then scaled as

$$c_{sw}=c_{ref}({\displaystyle \frac{P_{rep}}{P_{ref}}})$$

(23)

which gives c_sw = 1325.6 CNY/event. The blade-angle adjustment coefficient is assumed as λ = 0.05, with θ_min = 30° and θ_max = 130°. The adopted transition-cost coefficients are used as assumed engineering penalty parameters for scenario evaluation, rather than as plant-specific maintenance accounting data. Since the first interval has no previous operating state, its transition cost is set to zero.

After introducing the maintenance cost, the optimization is no longer treated as an independent hourly selection problem. Instead, it is formulated as a continuous-period optimization problem over the whole day. The objective is to maximize the 24 h cumulative net benefit:

$${\max }_{s_1,s_2,\dots ,s_T}{\displaystyle \sum _{t=1}^T\left[B_t^0(s_t)-C_m(t)\right]}, T=24$$

(24)

This formulation retains the original thermodynamic–economic benefit calculation while introducing an assumed penalty for unnecessary state transitions between adjacent intervals. In the numerical calculation, all feasible combinations of pump number and blade angle are first evaluated for each hour, and then the optimal daily operating path is selected by continuous-period optimization. This allows the strategy to balance short-term economic benefit and operational stability.

2.7. Model Validation

2.7.1. Design Parameter

This study performs calculations for a 630 MW unit. As a key equipment of the unit, the main design parameters of the turbine and condenser are shown in

Table 2. Main design parameters of steam turbines.

Project	Standard
Unit Model	C630/575-16.66/2.87/537/570
Unit type	Subcritical, single reheat, single-shaft four-stage, double-backpressure condensing steam turbine
Rated Power	630 MW

and

Table 3. Key design parameters of the condenser.

Project	Standard
Thermal transfer area	33,300 m2
Design inlet temperature for circulating water	19 °C
Design flow of circulating water	16.67 m3/s
Designed backpressure	4.9 kPa
Steam turbine steam discharge flow rate	1153.094 t/h
Thickness of condenser tube wall	0.5 mm

, respectively.

2.7.2. Error Analysis

To verify the accuracy and reliability of the system steady-state model and core equipment calculation models constructed above, three typical load conditions—THA, 60%THA, and 30%THA—were selected for simulation verification. By calculating the relative deviations in simulation results relative to design reference values, the calculation accuracy of the models was quantitatively evaluated. The calculation results are shown in

Table 4. Validation of generator unit models under different operating conditions.

	100%THA			60%THA			30%THA
Project	Des.	Cal.	Δ/%	Des.	Cal.	Δ/%	Des.	Cal.	Δ/%
Power/MW	630	630	0	380	380	0	190	190	0
Main steam flow/(t/h)	1828.9	1828.5	−0.02	1042.8	1042.8	0.029	515.7	515.9	−0.03
Main steam enthalpy/(kJ/kg)	3396.2	3399.0	0.08	3437.7	3440.8	0.09	3482.1	3485.8	0.1
Main steam temperature/(°C)	537	537	0	537	537	0	537	537	0
Backpressure/kPa	4.9	4.9	0	4.9	4.9	0	4.9	4.9	0

.

The relative errors calculated in the table represent the deviation degree of simulation calculation values relative to unit design reference values, which can verify whether the calculation deviations in models under different load conditions are within allowable ranges. Comparison results show that the relative error values of the constructed models are all very small, within 0.1%. Therefore, the system steady-state model and core equipment calculation models can accurately calculate key parameters such as turbine exhaust steam flow rate and exhaust steam enthalpy, demonstrating the reliability of the established models. The validation was performed at design flow conditions. Therefore, additional validation under off-design single-pump low-flow operation is further provided in Section 2.7.3.

2.7.3. Off-Design Validation Under Single-Pump Low-Flow Operation

To verify the applicability of the proposed model under off-design low-flow conditions, field operating data under single-pump operation were selected for comparison. Since the circulating water flow rate was not directly recorded in the DCS dataset, the single-pump operating state was identified according to the currents of circulating water pumps A and B. The circulating water flow rate was then obtained by linear interpolation based on the blade-opening angle of the operating pump and the single-pump blade-angle-flow characteristic relationship. Taking the design circulating water flow rate of 16.67 m³/s as the reference value, the 50–70% design flow range, corresponding to 8.34–11.67 m³/s, was selected as the off-design low-flow validation range.

To cover different low-flow levels and cooling water temperature conditions, the 50–70% design flow range was divided into four sub-ranges: 50–55%, 55–60%, 60–65%, and 65–70%. For each sub-range, one operating point with a relatively low water temperature and one operating point with a relatively high water temperature were selected, resulting in eight representative single-pump samples. Based on these samples, the circulating water inlet temperature, circulating water flow rate, and turbine exhaust steam flow rate were used as model inputs. The circulating water outlet temperature, condenser backpressure, and unit output calculated by the model were compared with the corresponding field measurements, as shown in

Table 5. Validation results under off-design single-pump operation.

	Flow	Temp.	Tout,mea/°C	Tout,cal/°C	Δ/%	Pc,mea/kPa	Pc,cal/kPa	Δ/%	Ng,mea/MW	Ng,cal/MW	Δ/%
1	50–55%	Low	19.62	19.86	1.223	2.41	2.33	3.319	248.46	249.59	0.455
2	50–55%	High	21.74	22.02	1.288	2.72	2.64	2.941	248.46	249.53	0.431
3	55–60%	Low	24.15	23.87	1.159	2.96	2.89	2.365	517.15	519.12	0.381
4	55–60%	High	24.60	24.35	1.016	3.12	3.06	1.923	514.17	516.14	0.383
5	60–65%	Low	26.20	26.26	0.229	3.55	3.43	3.380	605.11	607.05	0.321
6	60–65%	High	36.40	36.27	0.357	6.15	6.17	0.325	560.20	557.47	0.487
7	65–70%	Low	35.30	35.22	0.227	5.88	5.91	0.510	598.11	595.78	0.390
8	65–70%	High	36.85	36.39	1.248	6.45	6.21	3.721	601.14	597.44	0.615

.

As shown in Table 5, the model provides good predictive performance for the circulating water outlet temperature, condenser backpressure, and unit output within the 50–70% design flow range. The mean absolute percentage errors of the circulating water outlet temperature, condenser backpressure, and unit output are 0.843%, 2.311%, and 0.433%, respectively. The unit output shows a relatively small deviation, indicating that the backpressure-corrected output calculation agrees well with the field operating data. The slightly larger deviation in condenser backpressure is mainly attributed to the increased cooling water temperature rise, reduced outlet terminal temperature difference, and enhanced backpressure sensitivity to flow disturbances under low-flow conditions.

To further illustrate the agreement between model predictions and field measurements, Figure 5 presents scatter comparisons between measured and calculated values for the circulating water outlet temperature, condenser backpressure, and unit output. The dashed line represents the (y = x) reference line, and points closer to this line indicate better agreement between calculation and measurement.

As shown in Figure 5 the eight sample points are generally distributed close to the (y = x) reference line, indicating that the model can reasonably capture the variation trends of the main cold-end parameters under single-pump low-flow operation. These results demonstrate that the proposed condenser–circulating water system model is applicable to off-design single-pump operation within the 50–70% design flow range and can provide a model basis for subsequent circulating water pump operation optimization.

3. Sensitivity Analysis of Factors Influencing Cold-End System Optimization

This paper takes a 630 MW coal-fired unit as the research object. The unit is a subcritical unit with once-through intermediate reheating, single-shaft four-cylinder four-exhaust, double-backpressure condensing steam turbine, equipped with two adjustable-blade axial flow circulating water pumps. This chapter verifies the economy and applicability of the optimization strategy by calculating the optimal operating combination under different loads and seawater temperatures, combined with actual time-of-use electricity price data.

3.1. Reference Scenario and Data Preparation

3.1.1. Characteristics of Zhejiang Electricity Spot Market

Figure 6 shows the time-of-use electricity price information of the Zhejiang electricity spot market. The electricity prices exhibit pronounced peak–valley variation. The peak electricity price is concentrated in the 16:00–19:00 time period, with a peak value reaching 1.5 CNY/kWh; the valley electricity price appears in the 05:00–11:00 time period, with even negative electricity prices of −0.125 CNY/kWh appearing at 09:00. The electricity price during nighttime and early morning periods is maintained at a mid-level of 0.875–1.25 CNY/kWh, with a significant peak–valley price difference.

3.1.2. Boundary Parameter Settings for Typical Operating Conditions

Under fixed dispatch reference load conditions, this paper uses typical operating loads (30%, 60%, 100%) as representatives to measure and economically evaluate the flow rates of various circulating water pump combination modes. The pump units are double-backpressure condensing steam turbines, equipped with two adjustable-blade circulating water pumps. The cooling water flow is evenly distributed, and flow regulation is achieved by adjusting the blade angle. The unit pump flow corresponding to different blade openings is determined through pipeline characteristic calculations, as shown in Table 1.

3.2. Benefit Sensitivity Analysis

3.2.1. Sensitivity Analysis of Circulating Water Inlet Temperature on Benefits

Circulating water temperature, as a core variable affecting condenser heat exchange intensity, will directly change the unit’s backpressure level and pump power loss when it fluctuates, thereby affecting the system’s economic benefits. Previous studies have demonstrated similar temperature-dependent optimization characteristics. Figure 7 shows the optimal operation strategy of circulating water pumps under different circulating water temperatures and the corresponding backpressure change characteristics. The traditional scheme configures dual-pump operation year-round with a constant blade angle of 100°. The strategy in this paper adaptively adjusts the pump operation mode based on economic principles, and in most conditions, only one pump needs to be operated to meet the cooling demand.

Single-pump operation reduces the cooling water flow through the condenser. From Equation (14), the reduced flow increases the coolant temperature rise, driving the outlet cooling water temperature closer to the steam saturation temperature and reducing the outlet terminal temperature difference. As analyzed in Section 2.5, the backpressure sensitivity to flow increases sharply in this regime. At 30% load, the optimized backpressure averages 2.88 kPa, approximately 12.5% above the design value of 2.56 kPa. Despite this penalty, the power balance decomposition (Equation (18)) confirms that the pump power saving exceeds the turbine output loss, keeping the net benefit positive under all tested conditions. The corresponding trends across the three loads are summarized in Figure 8.

The optimization increases backpressure, reducing turbine cycle efficiency, but the pump power saving exceeds the generation loss, yielding a positive ΔW_net under all tested conditions. As the circulating water temperature increases, the outlet terminal difference at design flow decreases, placing the condenser in the high-sensitivity regime where flow reductions produce larger backpressure penalties. The optimization space contracts accordingly.

The marginal sensitivity of the cold-end system under 30% THA load is quantified in

Table 6. Marginal sensitivity verification at 30% load under different cooling water temperatures.

Tw,in/°C	Q/m3·s−1	Pc/kPa	Wpump/kW	∂Wpump/∂Q/kW·s·m−3	kbp·∂Pc/∂Q /kW·s·m−3	Ratio
15	5.85	2.9477	1246.9	133.77	30.14	4.44
15	8.53	2.5614	1605.4	117.58	22.46	5.24
15	10.417	2.4212	1783.9	121.90	9.06	13.45
15	12.284	2.3287	2063	172.68	5.74	30.08
15	13.437	2.2854	2305.4	210.23	4.44	47.35
20	5.85	3.9250	1246.9	133.77	72.85	1.84
20	8.53	3.4332	1605.4	117.58	55.62	2.11
20	10.417	3.2543	1783.9	121.90	25.09	4.86
20	12.284	3.1361	2063	172.68	16.84	10.25
20	13.437	3.0807	2305.4	210.23	13.48	15.60
25	5.85	5.1773	1246.9	133.77	147.83	0.91
25	8.53	4.5565	1605.4	117.58	114.2	1.03
25	10.417	4.3299	1783.9	121.90	53.82	2.26
25	12.284	4.1802	2063	172.68	36.92	4.68
25	13.437	4.1100	2305.4	210.23	29.91	7.03

. The marginal power consumption of the pump (∂W_pump/∂Q) and the marginal power gain of the turbine (k_bp·∂P_c/∂Q) are calculated using the central difference method based on the steady-state operating points from Table 1. As shown in the results, at an inlet temperature of 25 °C, the ratio between marginal consumption and gain transitions from 0.91 to 1.03 as the blade angle increases, indicating that the theoretical optimal flow rate (where the ratio equals one) is captured within the regulation range. Conversely, at 15 °C and 20 °C, the ratios consistently exceed unity, which validates that the system reaches its boundary optimum at the minimum blade angle of 30° due to the excessive cooling capacity in low-load and low-temperature conditions.

Under high-load or high-temperature conditions, due to the surge in turbine exhaust volume, the thermal load exceeds the single-pump carrying capacity limit, and the strategy will automatically switch to dual-pump mode. Dual-pump operation can significantly increase flow rate and tube internal flow velocity, significantly increasing the heat-transfer coefficient and strengthening heat exchange, making the optimized backpressure average 0.1053 kPa lower than the design value. At this time, the system benefit will rebound, and its essence is that the heat exchange gain brought by the flow increase exceeds the pump power consumption cost. In high-load environments, insufficient single-pump flow will lead to a sharp deterioration in backpressure, while switching to dual pumps utilizes lower backpressure to significantly reduce unit power generation efficiency losses. Taking 100% load as an example, dual-pump operation makes the benefit recover from 29.5818 CNY/h to 360.0753 CNY/h.

3.2.2. Sensitivity Analysis of Unit Load on Benefits

Unit load, as a key factor affecting exhaust volume, directly determines the heat exchange demand of the cold-end system. Figure 9 shows the optimal operation strategy of circulating water pumps and benefit response patterns when the unit load changes from 30% to 100% under constant water temperature conditions of 10 °C, 15 °C, 20 °C, and 25 °C.

From the results of the four groups of comparative experiments, the benefit shows a characteristic of first monotonically decreasing and then stepping up with increasing load, and shows more significant optimization potential in the low-load range. In the low-load stage, the exhaust volume is small, single-pump operation can meet the heat exchange demand, and the benefit increase mainly comes from the significant reduction in pump power consumption; as the load increases, the circulating water flow under single-pump operation gradually approaches the upper limit of heat exchange capacity, the cold-end backpressure gradually increases, power-saving benefits are continuously offset by power generation losses, and the benefit increase value continues to decline.

The switching boundary is determined by the heat balance constraint: single-pump operation can sustain the exhaust heat-duty only while the product of maximum single-pump flow, heat exchange effectiveness, and thermal driving force exceeds the steam-side heat load. Since the heat load increases with unit load while the thermal driving force decreases with rising water temperature, the switching load shifts to lower values at higher temperatures. When the load reaches the single-pump operation limit, the system switches to dual-pump mode to ensure operation safety, and the benefit curve shows an pronounced inflection point here, and the higher the circulating water temperature, the earlier the load position where this inflection point appears. At 15 °C, the benefit inflection point is around 610 MW, while at 25 °C, it has advanced to approximately 430 MW. This is because the heat exchange temperature difference decreases under high-temperature conditions, and a larger flow is needed to maintain the same heat exchange capacity. After switching to dual pumps, due to the increase in tube internal flow velocity strengthening heat exchange, the optimized backpressure significantly falls below the design value, and the benefit correspondingly shows a significant rebound. Taking the 25 °C condition as an example, the benefit increase rebounds from the lowest value at the inflection point of 35.2331 CNY/h to approximately 207.5664 CNY/h. The mechanism of this change lies in the fact that the steam expansion work gain obtained by reducing backpressure under high load far exceeds the increase in power consumption cost brought by dual-pump operation.

3.2.3. Sensitivity Analysis of Temperature–Load Coupling on Benefits

To comprehensively reveal the distribution pattern of cold-end system benefits, this section couples the circulating water inlet temperature with the unit load for analysis. As shown in Figure 10, as the unit load and circulating water inlet temperature increase synchronously, the system benefit increase value shows pronounced nonlinear variation, forming a concave region in the middle of the benefit increase surface. When the water temperature is approximately 23 °C and the unit load is approximately 476 MW, single-pump operation is close to the upper limit of heat exchange capacity: on the one hand, the higher water temperature weakens heat exchange performance; on the other hand, the larger exhaust volume requires more cooling water flow. At this operating point, the net power change ΔW_net (Equation (18)) approaches zero: the pump power saving is almost exactly offset by the turbine output loss from backpressure rise, yielding a minimum benefit of approximately 41.79 CNY/h. This valley in the benefit surface delineates the applicability boundary of single-pump optimization; beyond this boundary, dual-pump operation becomes necessary for both safety and economic superiority.

As the operation strategy switches from single pump to dual pump, the benefit increase value in the high-load range rebounds significantly, and the benefit surface rises again. The reason is that multi-pump operation increases the circulating water flow, and strengthens the cold-end heat exchange capacity, thereby effectively suppressing backpressure increase, making the power generation gain brought by backpressure improvement exceed the power consumption increased by multi-pump operation. In summary, the coupling relationship between unit load and circulating water inlet temperature determines the optimal switching interval of the circulating water pump operation mode.

3.2.4. Sensitivity Analysis of Condenser Cleanliness Factor

The condenser cleanliness factor reflects the effect of tube fouling, scaling, and biological attachment on heat-transfer performance. In the baseline model, η_c was set to 0.85, representing a relatively clean condenser condition. Since the condenser cleanliness may vary with water quality, operating time, and maintenance conditions, a sensitivity analysis was conducted using the actual load and circulating water inlet temperature of a typical operating day. The electricity price was fixed at the daily average value, and η_c was set to 0.70, 0.75, 0.80, and 0.85 to evaluate its influence on condenser backpressure and optimization benefit.

Figure 11a shows the condenser backpressure under the design circulating water flow. Under the same operating boundaries, a higher η_c leads to a lower condenser backpressure. This is because an increase in η_c improves the overall heat-transfer coefficient of the condenser, enhances heat-transfer capacity, and reduces the saturation temperature required for steam condensation, thereby lowering the backpressure.

Figure 11b presents the optimized condenser backpressure. In general, the optimized backpressure decreases as η_c increases, but the curves are not strictly monotonic at some hours. This is because, under optimized operation, the circulating water flow is determined by the pump combination and blade angle. When η_c changes, the optimal pump strategy may switch, leading to discrete changes in circulating water flow. In addition, the objective is to maximize the comprehensive net benefit rather than simply minimize backpressure, so local convergence or the crossing of the backpressure curves may occur.

Figure 11c shows the benefit of the optimized condition relative to the design condition. The benefit remains positive for most hours under different η_c values, indicating that the circulating water optimization strategy remains economically applicable under different condenser cleanliness conditions. The differences among the curves are limited because η_c affects both the design and optimized conditions, and part of its influence is offset when the benefit is calculated as their difference.

Figure 11d shows the change in optimized benefit relative to η_c = 0.85. When η_c is lower than 0.85, the benefit change is negative, and the loss becomes larger as η_c decreases. This indicates that a lower cleanliness factor weakens condenser heat-transfer performance, increases the backpressure penalty, and reduces the economic benefit of optimized circulating water operation.

Quantitatively, relative to the η_c = 0.85 baseline, the hourly optimized-benefit reduction is approximately 0–80 CNY/h when η_c decreases to 0.80, approximately 0–160 CNY/h when η_c decreases to 0.75, and can approach 300 CNY/h in the most affected hours when η_c decreases to 0.70. Although the economic margin decreases with cleanliness deterioration, the optimized benefit remains positive for most hours within the tested η_c range of 0.70–0.85. This indicates that the proposed optimization strategy remains economically meaningful under moderate condenser fouling, while regular cleanliness monitoring and tube cleaning are important for preserving the benefit potential, especially under high cooling water-temperature conditions.

Overall, a decrease in η_c increases the backpressure under both design and optimized conditions and reduces the optimized benefit relative to the η_c = 0.85 baseline. However, within the range of η_c = 0.70–0.85, the optimized operation still provides positive benefits relative to the design condition. This indicates that the proposed circulating water optimization method does not rely on the single assumption of η_c = 0.85, and its main conclusion remains valid under different condenser cleanliness conditions. In practical operation, condenser cleanliness should be monitored and maintained to prevent fouling from reducing the economic benefit of cold-end optimization.

4. Comparison of Optimization Results Based on Time-of-Use Electricity Prices Under Typical Operating Conditions

To verify the economic adaptability of the cold-end system optimization strategy based on time-of-use electricity prices under different operating backgrounds, this chapter selects three typical operating conditions—summer high water temperature, winter low water temperature, and stable load—for comparative analysis to reveal the internal correlation mechanism among cold-end operating characteristics, time-of-use electricity prices, and economic benefits.

4.1. Comparative Analysis of Cold-End Response and Economics Under Typical Climatic Conditions

4.1.1. Summer Operating Conditions

Figure 12 shows the operating data for a typical day in August. On that day, the circulating water inlet temperature fluctuated gently, maintained between 27.98 and 29.96 °C, with an average water temperature of approximately 28.80 °C. The unit load showed obvious daily fluctuation characteristics: the high-load stage of 580–630 MW occurred during 16–24 h, while other periods were the medium–low load stage of 300–530 MW. This condition is used to analyze the influence of time-of-use electricity prices on the unit’s cold-end operation and benefit characteristics under high circulating water temperature conditions.

Figure 13 shows the time-varying adjustment pattern of circulating pump operation mode driven by time-of-use electricity prices. During the 8–12 h low electricity price period, the system switches to single-pump small opening operation; during medium–high electricity price periods, heat exchange is strengthened by increasing blade angle or switching to dual-pump operation.

Figure 14 shows that under design conditions, the circulating water flow is constant at 16.67 m³/s. During the 8–12 h low electricity price and negative electricity price intervals, the optimization strategy switches to single-pump small opening operation, reducing the circulating water flow to the daily valley value of 5.85 m³/s. At this time, the flow reduction leads to a decrease in heat exchange capacity, and the condenser backpressure rises to the daily peak value of 10.36 kPa. Although the backpressure increase deteriorates the unit heat consumption rate and reduces output, the significant reduction in pump power consumption makes the power-saving benefit significantly higher than the power generation loss. During the 9–11 h negative-price interval, the power balance decomposition (Equation (19)) reveals the mechanism: reducing flow to 5.85 m³/s produces a pump power saving (ΔW_pump > 0), while the net generation decreases (ΔW_net < 0). Multiplied by the negative price (p(t) = −0.125 CNY/kWh), the product ΔW_net·p(t) becomes positive: both pump savings and reduced output contribute to a net benefit of 2190.79 CNY/h. This superposition effect is unique to negative-price conditions. Although dual-pump high-flow operation increases pump power consumption, the reduction in backpressure increases the available enthalpy drop across the turbine final stages. With high electricity prices, the power generation gain brought by increased unit output far exceeds its power consumption cost, thereby achieving a stable net benefit of an average of 359.86 CNY/h. In summary, the power-saving benefit during low electricity price and negative electricity price periods is most significant, while medium–high electricity price periods rely on heat exchange strengthening to reduce backpressure and achieve stable gains.

To comprehensively evaluate the performance of the proposed price-adaptive optimization strategy, four representative hours are selected based on the electricity price profile, covering the flat period at 12:00, the negative-price interval at 10:00, the peak demand at 14:00, and the super-peak at 17:00.

Table 7. Power balance decomposition at selected hours under summer conditions.

Hour/h	P(t)/CNY·kWh−1	ΔWpump/kW	ΔWturbine/kW	ΔWnet/kW	ΔEnet/CNY·h−1
10	−0.125	798.36	−18,324.60	−17,526.24	2190.78
12	0.25	−814.27	335.05	1149.32	287.33
14	1.00	447.56	−150.64	296.92	247.32
17	1.50	−48.92	394.55	245.63	518.45

summarizes the specific contributions of pump power and turbine output to the net economic benefit at these selected hours. At hour 10, when the electricity price drops to −0.125 CNY/kWh, the circulation flow is reduced to achieve a pump power saving of approximately 798.36 kW. Although the resulting backpressure rise leads to a turbine output loss of 18,324.60 kW, the net power change of −17,526.24 kW actually yields a positive economic outcome. Both terms contribute favorably to the overall benefit because the pump saving avoids power consumption at a negative cost while the reduced output prevents the financial penalty of selling electricity below zero, resulting in a net economic gain of 2190.78 CNY/h. In contrast, at hour 18 with a super-peak price of 1.50 CNY/kWh, the optimization shifts toward dual-pump high-flow operation. In this scenario, the pump consumption is reduced by 40.51 kW, and this is complemented by a turbine output gain of 470.87 kW due to improved vacuum conditions. The high electricity price amplifies this net generation increment into a total economic benefit of 645.53 CNY/h.

4.1.2. Winter Operating Conditions

Under winter conditions with a cooling water inlet temperature of approximately 17.91 °C, the enhanced thermal driving force provides a substantial heat exchange margin. The optimization follows the price-response logic established for summer conditions in Figure 15, Figure 16 and Figure 17, characterized by flow reduction during low-price hours and flow increase during high-price periods. However, the magnitude of the resulting benefits differs significantly. During the negative-price interval between 9 and 11 h, the maximum benefit reaches 414.69 CNY/h, representing only 18.9% of the summer equivalent. This divergence stems from the large outlet terminal difference at low water temperatures, which situates the condenser in a low-sensitivity regime. Consequently, flow reductions produce only modest backpressure increases—peaking at 4.26 kPa compared to 10.36 kPa in summer—thereby limiting both pump power savings and the negative-price superposition effect. During medium-to-high price periods, the average net benefit is 144.05 CNY/h. The optimized backpressure deviates by a mere 0.0052 kPa from the design value, confirming that winter conditions offer a restricted yet positive optimization space.

Comparing the summer and winter decompositions in

Table 8. Power balance decomposition at selected hours under winter conditions.

Hour/h	P(t)/CNY kWh−1	ΔWpump/kW	ΔWturbine/kW	ΔWnet/kW	ΔEnet/kW
10	−0.125	796.23	−4035.99	−3239.79	404.97
15	1.00	132.16	−40.51	91.65	91.65
18	1.50	−87.34	124.81	37.47	56.21

reveals the physical origin of the pronounced seasonal benefit disparity. At the same negative price (−0.125 CNY/kWh), the winter ΔW_turbine is only approximately −4035.99 kW versus −18,324.60 kW in summer because the large outlet terminal difference (approximately 15 °C in winter versus 7 °C in summer) places the condenser in the low-sensitivity regime where ∂P_c/∂Q is small. Consequently, the winter ΔW_net remains substantially less negative (net generation loss is greatly reduced despite flow reduction), eliminating the negative-price superposition channel. The winter benefit therefore derives solely from the pump power-saving term, which alone cannot match the dual-channel summer gain.

4.2. Validation of Price-Response Strategy Under Typical Full-Day Operating Conditions

Under stable-load conditions in May, characterized by a load range of 520–630 MW and cooling water inlet temperatures between 17.8 and 20.8 °C, load fluctuations remain constrained. Nevertheless, the price signal continues to drive meaningful adjustments, as illustrated in Figure 18, Figure 19 and Figure 20. During negative-price intervals, the maximum benefit reaches 412.56 CNY/h through single-pump low-angle operation, while the backpressure rises to 4.58 kPa. For medium-to-high price periods, the net benefit averages 147.28 CNY/h. These results demonstrate that price-driven cold-end adjustment remains effective even in the absence of load-regulation flexibility. The optimization potential originates from the price differential itself rather than from load-induced changes in exhaust heat-duty, thereby confirming the universality of the price-response mechanism across diverse operating regimes.

4.3. Engineering Feasibility Evaluation Considering Maintenance Costs

The preceding results show that the time-of-use cold-end optimization strategy can improve thermodynamic–economic performance by adjusting the circulating water pump number and blade angle. In practical operation, frequent pump switching and blade-angle regulation may increase equipment wear and operational burden. Therefore, an equivalent transition-cost penalty is further introduced to examine whether the optimized strategy remains economically effective after considering switching-related constraints.

The maintenance cost is divided into pump-switching cost and blade-angle adjustment cost. The former is related to changes in the number of operating pumps between adjacent intervals, while the latter is related to the variation in blade angle. Since these costs depend on the previous operating state, a continuous-period optimization method is adopted over 24 h instead of independent hourly optimization. This allows the model to balance hourly economic benefit and transition cost.

The baseline parameters used in this scenario are c_sw = 1325.6 CNY/event, λ = 0.05, θ_min = 30° to θ_max = 130°. These values are used to test the sensitivity of the optimized path to transition penalties and should not be interpreted as plant-specific maintenance accounting data.

Table 9. Equivalent maintenance-cost scenario evaluation results.

Scenario	Benefit Without Cost	Benefit with Cost	Reduction	Switching Without Cost	Switching with Cost
Summer	8825.46 CNY/day	5854.26 CNY/day	33.67%	4	2
Winter	4592.29 CNY/day	2773.93 CNY/day	39.60%	4	1
Stable load	3332.05 CNY/day	1397.94 CNY/day	58.05%	4	0

summarizes the results of the equivalent maintenance-cost scenario. After the transition-cost penalty is introduced, the 24 h cumulative net benefit decreases in all three scenarios but remains positive. The benefit decreases from 8825.46 to 5854.26 CNY/day in summer, from 4592.29 to 2773.93 CNY/day in winter, and from 3332.05 to 1397.94 CNY/day under the stable-load condition. Meanwhile, pump-switching events are reduced from four to two, one, and zero, respectively. These results indicate that the optimized strategy is sensitive to switching penalties, but the economic value of cold-end optimization is not fully offset under the assumed scenario.

Among the three scenarios, the summer case still shows the highest net benefit after considering maintenance costs, because condenser backpressure is more sensitive to circulating water flow under high cooling water temperature. In contrast, the stable-load case shows the largest benefit reduction rate, reaching 58.05%, since the marginal benefit of frequent switching is limited under relatively steady boundary conditions.

Figure 21 presents the winter case as a representative example of the hourly operating path. After the equivalent transition-cost penalty is introduced, pump switching becomes more conservative, while blade-angle regulation is still retained. This suggests that the strategy does not simply suppress all regulation, but shifts from frequent pump switching to smoother operation with moderate blade adjustment. Although the hourly benefit may decrease during some transition intervals, the 24 h cumulative benefit remains positive.

Overall, the equivalent maintenance-cost scenario does not change the basic price-responsive mechanism of cold-end optimization. It mainly modifies the switching boundary between adjacent operating states, reduces unnecessary short-duration switching, and provides a sensitivity-based evaluation of the engineering feasibility of the optimized strategy.

4.4. Summary of Thermal Sensitivity and Price Signal Coupling Mechanisms

These three scenarios are unified through marginal sensitivity analysis and power balance decomposition. Under summer conditions, where the cooling water inlet temperature is approximately 29 °C, the small outlet terminal temperature difference at design flow places the condenser in a high-sensitivity regime. In this state, backpressure responds sharply to any changes in flow. Although flow reduction yields substantial pump power savings, the marginal turbine output loss per unit of backpressure rise remains moderate due to the already diminished expansion work at elevated initial backpressures. When combined with negative prices that convert a net generation decrease into a positive economic return, the maximum benefit reaches 2190.79 CNY/h.

Conversely, winter conditions characterized by an 18 °C cooling water inlet temperature result in a large outlet terminal difference, placing the condenser in a low-sensitivity regime. Here, flow reductions produce only modest backpressure increases, and the corresponding pump power savings are smaller, yielding a maximum benefit of 414.69 CNY/h. This inverse relationship between thermal driving force and optimization potential represents a fundamental thermodynamic property of the condenser system, independent of market design.

Across all three scenarios, the basic response to electricity price remains consistent: circulating water flow tends to be reduced during low-price or negative-price periods and increased when the electricity price is high. The intensity of this response is jointly determined by the backpressure–flow sensitivity of the operating regime and the exhaust heat-duty constraint that defines the minimum acceptable flow. After maintenance costs are introduced, this mechanism remains unchanged. The maintenance-cost constraint mainly modifies the switching boundary between adjacent operating states, suppressing short-duration pump switching when the marginal benefit is insufficient. Therefore, the optimized strategy is governed by the combined effects of thermal sensitivity, electricity price, load boundary, and transition cost.

4.5. Applicability and Extension of the Model

The proposed framework is designed for hourly or near-hourly cold-end operation planning under electricity spot market conditions. It can be used to compare feasible pump combinations and blade-opening schemes, identify periods suitable for flow reduction or flow increase, and support the selection between single-pump and two-pump operation under different load, cooling water temperature, and electricity price conditions. Therefore, the model serves as a thermodynamic–economic decision-support tool for market-oriented operation, rather than a high-frequency closed-loop control model.

The present model adopts a quasi-steady-state assumption within each hourly dispatch interval. To examine the influence of short-term variable-flow transients, a representative flow-adjustment process was selected from the winter operating cases and simulated in Dymola. In this case, the circulating water flow rate was reduced from 17.153 m³/s to 13.437 m³/s according to the optimized operating path. The non-steady period was identified according to the condenser backpressure variation rate. When the absolute value of the condenser backpressure variation rate became lower than the stability threshold of 0.01 kPa/min, the system was regarded as reaching the steady stage.

To quantify the contribution of the non-steady period, the accumulated benefit during the non-steady period was compared with the accumulated benefit over the corresponding one-hour dispatch interval:

$${\eta }_{ns}={\displaystyle \frac{E_{ns}}{E_h}}\times 100\%={\displaystyle \frac{{\displaystyle {\int }_{t_0}^{t_0+3600}{I}_{ns}(t)B(t)dt}}{{\displaystyle {\int }_{t_0}^{t_0+3600}B(t)dt}}}\times 100\%$$

(25)

where B(t) is the benefit rate obtained from the optimization results, and I_ns(t) is a non-steady-state indicator that equals one during the identified non-steady period and zero otherwise. As shown in Figure 22, the non-steady period lasts approximately 238 s. The accumulated benefit during this period is 7.11 CNY, while the accumulated benefit over the corresponding one-hour interval is 111.81 CNY. Therefore, the non-steady-period benefit accounts for only 6.36% of the hourly benefit integral. This result indicates that the variable-flow transient process occupies only a small portion of the hourly dispatch interval and has a limited influence on the hourly thermodynamic–economic benefit. Accordingly, the quasi-steady-state formulation is retained, while pump-switching frequency is considered through the transition-cost penalty.

For sub-hourly dispatch, frequent pump switching, or rapid blade-angle adjustment, transient responses may become more important because the hydraulic settling process and condenser thermal inertia occupy a larger proportion of the control interval. In practical operation, minimum operating time, switching-frequency limits, and hysteresis bands for blade-angle adjustment can be introduced to reduce unnecessary short-duration switching. These measures are consistent with the transition-cost formulation in Section 2.6, where adjacent-state changes are penalized in the 24 h operating path. If the framework is extended to primary-frequency regulation, secondary-frequency regulation, or other seconds-to-minutes control tasks, transient hydraulic simulation, condenser thermal-inertia modeling, and high-frequency DCS measurements should be incorporated.

Although the numerical results are obtained from a 630 MW subcritical unit in Zhejiang Province, the proposed framework is not limited to this specific unit or market. The numerical values of the optimal flow rate, daily benefit, and switching boundary should not be directly transferred to other cases; instead, they should be recalculated after replacing the local market and operating boundary data. For applications in other provincial electricity markets, the time-of-use or spot price sequence, load profile, circulating water inlet temperature, fuel cost, and operational constraints should be updated according to local conditions. For applications to supercritical or ultra-supercritical units, plant-specific thermodynamic and equipment parameters should also be recalibrated, including turbine exhaust flow and enthalpy, condenser heat-transfer area and tube-side parameters, cleanliness factor, pump head-flow and efficiency curves, circulating water system resistance, design flow and design backpressure, and the turbine backpressure-output correction curve. After these parameters are updated, the same optimization procedure can be used to recalculate the optimal pump combination, blade opening, circulating water flow rate, condenser backpressure, economic benefit, and switching boundary for the target unit. Therefore, the absolute numerical results are case-specific, whereas the marginal trade-off between pump power saving and turbine output change remains transferable.

5. Conclusions

This study developed a quasi-steady-state thermodynamic–economic co-optimization framework for cold-end operation under electricity spot market conditions. The framework couples the condenser heat balance with real-time time-of-use price signals and was validated on a 630 MW coal-fired unit using actual Zhejiang Province spot market data. The analysis yields the following key insights:

(1) The framework achieves effective coupling between cold-end thermal behavior and spot market price signals. Through the marginal sensitivity analysis, the optimal circulating water flow is identified as the point where the marginal pump power saving equals the marginal turbine output loss caused by backpressure increase. This marginal equilibrium principle is quantitatively validated in Table 6, governing the adaptive adjustment of pump combination and blade-opening angle.

(2) The power balance decomposition reveals the thermodynamic mechanism behind the observed seasonal differences. Under summer high-temperature conditions, the maximum benefit during negative-price periods reaches 2190.79 CNY/h, approximately 5.3 times the corresponding winter value. This disparity arises from the inverse relationship between thermal driving force and optimization potential, amplified by the negative-price superposition effect where both pump savings and output reduction contribute positively. The power decomposition results in Table 7 and Table 8 quantitatively verify this superposition mechanism and also demonstrate that the switching between single-pump and dual-pump modes follows a clear marginal condition rather than empirical rules.

(3) The hierarchical decision framework based on physical models has a clear structure and stable computation, and is suitable for hourly or quasi-steady cold-end operation decisions. Backpressure deviations remain within 12.5% of design values across all tested scenarios, confirming the operational feasibility of the optimized strategies. The equivalent maintenance-cost scenario further shows that unnecessary short-duration pump switching can be reduced while maintaining positive daily net benefits. However, the present model does not explicitly resolve transient hydraulic and thermal processes during pump start–stop or blade-angle adjustment. Although the off-design validation includes representative 50–70% single-pump low-flow samples, further field validation over wider operating conditions is still needed. When the method is extended to higher-parameter units, plant-specific exhaust, condenser, pump, and backpressure-correction parameters should be recalibrated, while the marginal trade-off logic remains applicable.