Modelling and Optimization of Petrochemical Hybrid Renewable Energy Systems Considering Energy Interchangeability, Uncertainty and Storage for Coupling Energy Supply and Utilization Sides

Abstract

Petrochemical hybrid renewable energy systems (PHRESs), integrating renewable and fossil energy sources, have garnered more and more attention for sustainable manufacturing. However, achieving concurrent optimization of energy supply reliability and carbon mitigation in these complex systems remains a critical challenge. This study proposes an innovative bilateral optimization framework coupling supply-side energy management with demand-side flexibility. On the supply side, a scenario-based two-stage stochastic programming method synergizes with energy storage systems to address renewable energy intermittency, considering a time-of-day tariff from the grid. On the utilization side, heat energy-based and shaft work-based energy interchangeability are introduced and leveraged to enable both qualitative and quantitative flexibility in process unit requirements and thus obtain energy consumption relaxation models for relaxing the design boundaries of PHRESs. These dual strategies are then coupled in a two-stage mixed-integer programming model framework for the optimal design of PHRESs. Applied to a large-scale refinery incorporating carbon taxation and dynamic electricity price, the proposed methodology demonstrates superior performance through five comparative cases. Compared to the Base Case, the Optimal Case using the proposed method can reduce the total annual cost by 14.82%, and stochastic programming reveals over a 40% probability of carbon mitigation in the uncertain space.

1. Introduction

Energy conservation and carbon mitigation of energy systems have become global initiatives for achieving the target of carbon neutrality, especially in energy-intensive industries, such as the petrochemical industry [1]. Taking a refinery as an example, the industry usually comprises a series of process units and an auxiliary energy system. The sustainable development of the energy-intensive industry faces challenges in ensuring a reliable energy supply and reducing carbon emissions, given its operational characteristics and critical constraints on energy consumption and carbon emissions. Thus, industrial hybrid renewable energy systems (PHRESs) comprising renewable energy (RE) and fossil energy (FE) have attracted increasing attention and are being intensively investigated to simultaneously reduce carbon emissions and ensure a reliable energy supply.

In a PHRES, carbon emission is closely related to energy consumption; the energy consumption levels of process units mainly determine the configuration and design parameters of the PHRES, i.e., design boundaries. However, in most mathematical-programming-method-driven research on energy systems, the relationships between energy systems and process units in energy requirement quantities and forms are regarded as rigid ones. These rigid relationships will reduce the free degrees of the optimal design of PHRESs, which will result in the optimization boundary narrowing in advance for PHRESs design, and lead to sub-optimal results.

In our previous publication [2], these rigid relationships were relaxed by considering energy interchangeability, and two kinds of energy interchangeability were investigated. As shown in Figure 1, the first is heat energy-based interchangeability between steam and fuels. For example, in the crude distillation unit, crude oil after a series of heat exchanges needs to be further heated from 285 °C to 360 °C. This gap is usually offset using a fired heater or an overheat steam stream, which leads to a rigid demand mode for both the type and quantity of hot utility. However, a combination method can also fulfill this demand: Crude oil (285 °C) can first be heated using a 3.5 MPa superheated steam stream before entering the fired heater. This approach reduces fuel consumption in the fired heater and alters the hot utility configuration, thereby relaxing both the type and quantity of hot utility demand. The second is shaft work-based energy interchangeability between steam and electricity. For example, a compressor can be driven by either electricity or steam. This interchangeability of steam versus electricity leads to different patterns of energy requirements, further affecting the optimal design of PHRESs.

In a previous publication, investigating these two kinds of interchangeability for industrial energy system optimization resulted in reductions of 10.96% in total annual cost (TAC) and 19,845 t/year in carbon emissions, respectively. However, that work considered only fossil energy. Given the urgent requirements for carbon peaking and neutrality, RE is essential for energy systems, especially energy-intensive ones.

As a carbon-free energy source, introducing RE can significantly reduce the carbon emissions of PHRESs. However, the intermittency and uncertainty of RE will significantly impact energy supply reliability. To address this challenge, uncertain optimization and energy storage systems are employed within the PHRES. Consequently, concurrently optimizing both energy supply reliability and carbon mitigation in such a complex system presents a significant challenge. In this work, the PHRES is decomposed from the perspective of energy flows into an energy supply side and an energy utilization side.

On the energy supply side, to leverage its carbon-free nature while addressing its intermittency and uncertainty, RE is introduced and coupled with FE to enhance energy supply reliability and achieve carbon mitigation. On the energy utilization side, energy interchangeability is investigated, thereby enhancing energy supply reliability and contributing to carbon mitigation. These two sides are interconnected through utility streams and energy storage systems. A mixed-integer programming (MIP) framework is then formulated and applied to the design optimization of such a complex PHRES in South China. Compared to the Base Case, the Optimal Case obtained using the proposed method reduces the TAC by 14.82%, and stochastic programming reveals a probability exceeding 40% for carbon mitigation within the uncertain space. When compared to Case 1, which employed the method from publication [2], the Optimal Case reduces TAC by 7.69% while stochastic programming reveals a probability exceeding 80% for carbon mitigation within the uncertain space.

The rest of this paper is structured as follows. First, the related work is reviewed, followed by the problem statement. Subsequently, modeling for the design of PHRESs is presented, covering four aspects: the energy supply side, the energy utilization side, energy storage systems, and balances and constraints within the total site. This is followed by the objective function and the optimization framework of PHRESs for the design optimization. Then, five case studies on a PHRES in a large-scale refinery are presented, and their results are discussed. Finally, the conclusions and outlook are drawn.

2. Literature Review

RE has been crucial for global energy transition and carbon mitigation. However, uncertainty and intermittency are always the primary restrictions to their exploitation and consumption. The research on the uncertainty of RE has been intensively investigated, wherein the methods for dealing with uncertainty mainly include Monte Carlo simulation [3], robust optimization [4], chance constraint programming [5], and stochastic programming (SP) [6]. All these methods mainly tackle uncertainty from the intrinsic perspective by considering the uncertain regions of uncertainty parameters. Among them, SP methods exhibit superior performance in addressing uncertainty, especially in long-term uncertain optimization problems for renewable energy-based energy systems, in which not all the decisions need to be determined at the beginning [7].

Among SP methods, the two-stage SP (TSSP) has been intensively applied in the optimization of HIRESs. Wang et al. [8] proposed a sustainable optimal retrofit framework for utility systems based on life cycle assessment and TSSP, considering RE uncertainty. Compared to the traditional utility system without any RE, the TAC was reduced by 4.7%, and the global warming potential was decreased by 3.9%. Qian et al. [9] proposed an MINLP based on the TSSP method for sustainable retrofit design of petrochemical energy systems, considering uncertainties in solar irradiation and wind speed. Xu et al. [10] proposed a novel TSSP framework for optimal retrofit of sustainable energy-integrated coupled production materials and steam systems, where the uncertainties of wind speed and solar radiation were considered. Yang et al. [11] proposed a TSSP model of sustainable utility systems considering the uncertainty of RE to minimize the annual total cost, maximize the renewable energy penetration rate, and minimize the grid net interaction level. The TSSP method was also employed for the reliable design optimization of petrochemical energy systems in our previous work [12]. These studies demonstrate the advantages of TSSP methods in addressing the uncertainty to optimize resource allocation and facilitate the integration of RE sources into traditional energy systems. However, in these studies, the relationships between energy systems and energy utilization side were weakened, and the energy demand models were fitted as linear ones or fixed as constant values.

Furthermore, energy storage systems are also vigorously developed and deployed to alleviate the impact of energy uncertainty from the external perspective, such as battery energy storage [13], thermal energy storage [14], air compression energy storage [15], pumped hydro energy storage [16], and their combination. Taking battery energy storage as an example, the battery can be installed to store the surplus electricity. When the energy output of RE is intermittent or in shortage, the battery can immediately release the stored electricity to offset the shortage and thus maintain regular operation. Although energy storage technologies have been intensively researched and developed rapidly, their application in energy-intensive process units still faces challenges. This is primarily due to the high temperatures and energy densities involved, which can restrict the safety, capacity, and installation space of energy storage systems [17]. Consequently, the direct deployment of these systems in such industries remains difficult. Among the various energy storage technologies, battery energy storage and thermal energy storage offer better general applicability and are integrated in this work to illustrate their roles and impact on energy management and regulation.

In the process industry, the potential for energy interchangeability among process units significantly influences the types and quantities of energy mediums, thereby affecting their energy consumption levels. To the best of our knowledge, limited research has focused on the impact of energy utilization side optimization on the energy supply of PHRESs in energy-intensive industries. Based on heat exchange and comprehensive economic and environmental indicators, Wu et al. [18] optimized the configuration and scale of power generation components and the heating pipeline layout of distributed energy systems. Zhang et al. [19] studied the collaborative optimization of multi-energy stations in distributed energy networks. Kim [20] explored the problem of energy integration between electrification and utility streams in process units. These studies attempt to realize the integration and optimization of multi-energy systems by considering energy interchange, but these energy interchanges represent the exchange and mutual supply of a single energy medium between different subsystems or sub-networks. There is a lack of research on the interchange mechanism between various energy media and its effect on the energy supply side. Numerous energy exchange possibilities and modes exist among process units. In our previous publication [2], we investigated two mechanisms of energy interchange: One is heat-based energy interchange for exchange between fuel and steam, and the other is shaft work-based energy interchange for exchange between power and steam. By implementing these two interchange mechanisms, we relaxed the rigid constraints originally existing between the energy supply side and utilization side, achieving the global integrated optimal design of the industrial energy system with significant economic benefits and carbon emission reductions. However, in that work, RE and energy storage systems were not considered.

Mathematical programming is a predominant approach in process systems engineering (PSE), particularly for optimizing PHRESs, and these models can be formulated with single or multiple objectives to determine optimal system configurations [21,22]. Driven by sustainable development requirements, an increasing number of PSE design optimization problems now employ multi-objective optimization that simultaneously considers economic, energy, and environmental criteria to achieve collaborative trade-offs [23]. Chamandoust et al. [24] proposed a tri-objective framework for optimal scheduling of smart energy hub systems, where economic performance, carbon emissions, and reliability were chosen as the objectives to minimize the operation cost, the emission pollution, and the deviation of the electrical load profile from its desired value. Li et al. [25] developed a multi-objective optimization framework for industrial energy systems to minimize the operating cost and environmental impact. Jia et al. [26] established a multi-objective optimization framework for the regional integrated energy system, where economic benefits, carbon reduction, and reliability were maximized. These studies demonstrate that optimizing energy systems considering multiple objectives is essential. Among them, environmental performance and economic performance, especially total annual cost and carbon emissions, are the most frequently utilized objectives. In this work, economic and environmental performance are concurrently evaluated. The first is the TAC, comprising annualized capital expenditure, operation and maintenance (O&M) costs, and operating costs. The second is carbon emissions, which are monetized via a carbon tax. To mitigate the computational burden of multi-objective optimization, these objectives are scalarized into a single objective by integrating the carbon tax into the TAC.

The comparison between the highly related paper mentioned in the literature and this work is summarized in

Table 1. Comparison between the highly related papers mentioned in the literature and this work.

Reference	Energy Uncertainty Considered	Energy Demand Expression	Energy Storage System Considered	Type of Model Framework	Objective	Solution Methodology
Tang et al. [2]	-	PA-based relaxation models considering energy interchangeability	-	MINLP	TAC	GAMS software
Wang et al. [8]	Solar irradiation, wind speed	Linear model	TES	MINLP	TAC, global warming potential	TSSP in Python
Qian et al. [9]	Solar irradiation, wind speed	Fixed values	TES	MINLP	TAC, carbon emission	TSSP and ε-constraint method in GAMS software
Xu et al. [10]	Solar irradiation, wind speed	Coefficient model	-	MINLP	Operating cost (with carbon tax)	TSSP in GAMS software
Yang et al. [11]	Solar irradiation, wind speed	Linear model	-	MINLP	Annual total cost (with carbon tax), renewable energy penetration rate, grid net interaction level	TSSP and ε-constraint method in Python
Tang et al. [12]	Solar irradiation, wind speed	Fixed values	-	MINLP	TAC, reliability	TSSP and ε-constraint method in GAMS software
Wu et al. [18]	-	Hourly energy demand considering energy interchange	TES	MIP	Overall annual cost, CO2 emissions	Integration in single objective function
Zhang et al. [19]	-	Predicted load	-	-	Primary energy saving ratio, annual total cost saving rate, carbon dioxide emission reduction ratio	Designer’s Simulation Toolkit
Chamandoust et al. [24]	Solar irradiation, wind speed	EMG, TMG	-	MINLP	Operation cost, emission pollution, deviation of the electrical load profile from its desired value	Augmented ε-constraint method and fuzzy approach in GAMS software
Li et al. [25]	-	Fixed values	-	MINLP	Operating cost, environmental impact	Integration in single objective function using weighted technique in GAMS software
This work	Solar irradiation, wind speed	PA-based relaxation models considering energy interchangeability	BES, TES	MIP	TAC (with carbon tax)	TSSP in GAMS software

PA—Pinch analysis; EMG—electrical main grid; TMG—thermal main grid; TES—thermal energy storage; BES—battery energy storage; TSSP—two-stage stochastic programming; MINLP—mixed-integer non-linear programming; MIP—mixed-integer programming; TAC—total annual cost; GAMS—general algebraic modeling system.

. The reviewed literature reveals significant yet often isolated advancements across three critical dimensions for optimizing PHRESs:

(a)

Uncertainty mitigation via stochastic programming to handle RE intermittency;

(b)

Operational flexibility via energy storage systems to balance temporal supply–demand mismatches;

(c)

Demand-side flexibility through mechanisms like energy interchangeability to relax rigid utility requirements.

A prevalent pattern is the focused investigation of one or two of these dimensions: for instance, applying TSSP with simplified, fixed demand models or optimizing storage in systems with inflexible demand. Crucially, there is a lack of a unified modeling and optimization framework that explicitly and simultaneously integrates the co-optimization of supply-side uncertainty, demand-side flexibility, and bi-directional energy storage. This integration is essential to fully exploit their synergies and avoid the sub-optimal designs that may result from a segmented approach.

To bridge these research gaps, this study proposes a systemic optimization framework for PHRES through synergistic energy supply-utilization integration. The principal innovations and contributions are summarized as follows:

(1)

Addressing the common segmented treatment of uncertainty, flexibility, and storage in prior studies, a novel threefold investigation is proposed where the energy uncertainty in the supply side, the energy interchangeability in the utilization side, and energy storage systems are simultaneously integrated for the optimal design of PHRESs.

(2)

In contrast to models that apply stochastic programming only to rigid systems or optimize flexibility under deterministic conditions, a novel two-stage stochastic MIP (TSSMIP) optimization framework is formulated. This framework explicitly couples the wait-and-see operational decisions under uncertainty with the here-and-now design of both supply-storage assets and demand-side flexibility mechanisms.

(3)

Moving beyond single-objective or decoupled economic-environmental assessments, both economic performance and carbon emissions are integrated into a single objective within the TSSMIP framework via a carbon tax, providing a policy-relevant cost-optimal solution.

(4)

To demonstrate the practical superiority of this integrated approach over isolated strategies, the TSSMIP framework is applied to a large-scale industrial refinery through five comparative case studies.

(5)

The results quantitatively verify that the proposed method can significantly reduce the TAC and show considerable carbon mitigation potential, thereby validating the value of the proposed integrated framework.

3. Problem Statement

Taking the refinery site shown in Figure 2 as an example, it can be divided into the energy supply side and the energy utilization side, and these two sides are coupled through energy mediums. On the energy supply side, as the energy hub, the energy system is responsible for energy conversion, dispatch, and delivery. It integrates FE and RE within a total site and converts them into diverse secondary energy to fulfill energy requirements from process units. On the energy utilization side, process units consume energy for driving the reactions, separations, and purifications to convert crude feedstocks to chemical products and liquid fuels.

In the process industry, as the bridge for coupling energy supply and utilization sides, the allocation of energy mediums is crucial for improving economic performance and reducing energy consumption levels across the site. On the one hand, on the energy supply side, with the permeation of RE, uncertainties will lead to fluctuations in energy output from RE resources and consequently affect the design boundaries of the FE subsystem. On the other hand, the energy interchangeabilities of process units can relax the original rigid constraints of energy requirements and thus relax the design boundaries of the whole PHRES.

To address these uncertain design boundary issues, the TSSP method proposed by Liu and Pistikopoulos [27] is applied to convert the uncertain optimization problem into a two-stage problem, and a mixed-integer programming (MIP) mathematical framework is formulated in this work for the optimal design of the PHRES. In this framework, the design boundaries of the PHRES are determined in the first stage, and the operating boundaries are optimized in the second stage. Moreover, two kinds of energy storage technology are also incorporated for peak shaving of heat and electricity to enhance the flexibility and reliability of the PHRES.

Consequently, the whole mathematical framework can be constructed in three parts: first, the modeling for the energy supply side, including the modeling for fossil energy subsystems and renewable energy subsystems; second, the modeling for the energy utilization side, including the energy consumption relaxation models; third, the modeling for energy storage systems, including the modeling for heat energy storage and electric energy storage. The total annual cost (TAC), comprising the capital cost, operating and maintenance cost, and carbon tax cost, is employed as the objective. Furthermore, constraints are formulated to ensure the balance of energy and materials across the total site. The detailed mathematical models for these parts are comprehensively introduced in the following sections.

It should be emphasized that the TSSMIP framework proposed in this paper is essentially universal. The selection of photovoltaic and photo-thermal (PV and PT) systems in this case aims to focus on verifying the framework’s ability to handle multi-energy flow collaboration under high volatility and homologous uncertainty. The input module and uncertainty description method of this framework can be adjusted according to specific energy forms. For instance, fluctuations in biomass energy supply and changes in the amount of industrial waste heat recovery can all be incorporated into the model by defining corresponding sets of uncertain parameters. This modular design ensures the compatibility of the framework with other renewable energy sources (such as biomass energy and geothermal energy) and industrial waste heat resources, laying the foundation for more extensive research in the future.

4. Mathematical Model

The TSSMIP mathematical framework for the optimal design of PHRESs coupling the energy supply side and energy utilization side is constructed. On the energy supply side, the mathematical models for energy conversion are formulated. On the energy utilization side, two kinds of energy interchangeability are considered, and the energy consumption relaxation models (ECRMs) around process units based on energy interchangeability are introduced. The two sides are then coupled through the interaction of energy and material flows. Given the fluctuation of hourly energy supply, energy storage systems are incorporated into the PHRES for peak shaving and stability maintenance of energy supply, and two kinds of energy storage technology are introduced and modeled. Last, the energy and material balances in a total site are constrained and formulated. In the mathematical models, the sets of scenarios, time, and steam levels are indexed by scen, h, and i, respectively.

4.1. Modeling for the Energy Supply Side

On the energy supply side, primary energy is converted into secondary energy mediums through various techniques and modules. The mathematical models for them are formulated in this section, including photovoltaic (PV) modules, photothermal (PT) modules, wind turbine modules, gas turbine systems, fuel-based boilers, and steam turbines. The energy mediums generated by the energy supply side contain middle-pressure steam (MPS), low-pressure steam (LPS), low-low-pressure steam (LLPS), and power.

4.1.1. Photovoltaic/Thermal Modules

Renewable energy has been crucial in the global energy transition, and solar energy, one of the promising substitutes for fossil energy, has been rapidly investigated. The common techniques for solar energy utilization include PV and PT techniques, and the former converts solar energy into electric energy by relying on the photoelectric effect, while the latter converts solar energy into thermal energy through heat exchange. To increase the permeability and consumption rate of solar energy in the PHRES, the PV and PT techniques are simultaneously incorporated using solar panels and solar thermal collectors (STCs) to convert solar energy into power and heat. The power generation (PPV_scen,h) of the solar panel is related to the solar irradiance (SI_scen,h) of each scenario in the uncertain scenario set, ambient temperature (TA), and the number of PV panels (NPV), and it can be represented in Equations (1)–(4) [28].

In a PV panel, the operating temperature (TC_scen,h) of the cell is determined by the ambient temperature (TA), the solar irradiance (SI_scen,h) and the nominal operating temperature (NOTC), and it can be calculated using Equation (1). The operating current (IC_scen,h) of the cell is determined by the operating temperature (TC_scen,h), short-circuit current (ISC), current temperature coefficient (KC), and solar irradiance (SI_scen,h), and it can be calculated using Equation (2). The operating voltage (VC_scen,h) of the cell is determined by the operating temperature (TC_scen,h), the open-circuit voltage (VOC), and the voltage temperature coefficient (KV), and it can be calculated using Equation (3). Lastly, the total power output by PV panels, which is tightly related to the number of panels (NPV) and the nominal power (PNOM) of the cell, can be calculated using Equation (4), and the NPV is defined as an integer variable:

$$T{C}_{scen,h}=TA+S{I}_{scen,h}·\left({\displaystyle \frac{NOTC-20}{800}}\right)$$

(1)

$$I{C}_{scen,h}=S{I}_{scen,h}·\left[{\displaystyle \frac{ISC+KC·(T{C}_{scen,h}-25)}{1000}}\right]$$

(2)

$$V{C}_{scen,h}=VOC-KV·T{C}_{scen,h}$$

(3)

$$PP{V}_{scen,h}=NPV·PNOM·{\displaystyle \frac{V{C}_{scen,h}·I{C}_{scen,h}}{VOC·ISC}}$$

(4)

Meanwhile, STCs can absorb solar energy for heat generation, and the heat (QPT_scen,h) collected by an STC is determined by solar irradiance (SI_scen,h), area of the STC (APT), nominal efficiency (EFN), inlet fluid temperature (TIN), and ambient temperature (TA). The relationship between them can be demonstrated using Equation (5) [28], where parameters ASC and BSC are the model coefficients of the STC. Then, the heat collected by the STC can be utilized for steam generation, and the relationship between the heat (QPT_scen,h) and mass flowrate of steam (FPT_scen,h,i) can be described using Equation (6), where parameter EPT is the heat efficiency for steam generation of photothermal modules and taken 0.90 in this work [12,29]. The operating boundary for steam generation by the STC is constrained in Equation (7), while Equation (8) represents a logical constraint that forces only one pressure-level steam to be generated at once. Moreover, to evaluate the cost of steam generation in the PT modules, the economic evaluation models of heat recovery steam generators (HRSGs) are employed.

$$QP{T}_{scen,h}·TMW=S{I}_{scen,h}·APT·[EFN-ASC·(TIN-TA)-BSC·{(TIN-TA)}^2]$$

(5)

$$QP{T}_{scen,h}·EPT={\displaystyle \sum _i\{FP{T}_{scen,h,i}·}[L{W}_i·(SS{P}_i-S{P}_i)+L{H}_i+L{S}_i·(EE{P}_i-TWPT)]\}$$

(6)

$$FP{T}^{\min }·P{T}_{scen,h,i}\le FP{T}_{scen,h,i}\le FP{T}^{\max }·P{T}_{scen,h,i}$$

(7)

$${\displaystyle \sum _{i}P{T}_{scen,h,i}=1}$$

(8)

4.1.2. Wind Turbine Modules

Similarly, wind energy is also intensively explored for substituting traditional energy. Wind energy is usually harvested by wind turbines, and the power generation (PWT_scen,h) of the wind turbine can be obtained using a segment function that has been represented in Equation (9) [30], where the uncertain parameter WS_scen,h is the practical wind speed for each scenario in the uncertain scenario set, parameters V_ci and V_co are the cut-in wind speed and cut-off speed, and they are also the condition boundaries for normal operation of the wind turbine. Only when the practical wind speed (WS_scen,h) is faster than the cut-in speed (V_ci) and slower than the cut-off speed (V_co) will the wind turbine work normally, and when the practical wind speed (WS_scen,h) reaches the rated wind speed (V_rate), the output power of the wind turbine will be the maximum rated output power (PWT_rate). The total power generation (TPWT_scen,h) of wind turbines is determined by the total number (NWT) of wind turbines and represented in Equation (10).

$$PW{T}_{scen,h}=\left\{\begin{array}{l}0, W{S}_{scen,h}<V_{ci} or W{S}_{scen,h}>V_{co}\\ PW{T}_{rate}·{\displaystyle \frac{W{S}_{scen,h}^3-V_{ci}^3}{V_{rate}^3-V_{ci}^3}}, V_{ci}\le W{S}_{scen,h}\le V_{rate}\\ PW{T}_{rate}, V_{rate}<W{S}_{scen,h}\le V_{co}\end{array}\right.$$

(9)

$$TPW{T}_{scen,h}=NWT·PW{T}_{scen,h}$$

(10)

4.1.3. Gas Turbine Systems

Although fossil energy introduces serious emission issues, the high reliability of energy supply makes it an essential energy conversion technology in the energy-intensive industry. Because of the higher energy utilization efficiency and combined heat and power generation, gas turbine (GT) systems are the primary choice in the PHRES. A typical GT system usually consists of an air compressor, a combustion chamber, and a turbine, and it is combined with an HRSG for heat recovery and utilization. The gas-to-power efficiency of a GT system usually ranges from 0.20 to 0.35, and the heat recovery efficiency of the flue gas exhausted from the GT is around 0.80 [31]. In this work, the set of GT systems is indexed by g, and the efficiency models of the GT system are formulated and represented in Equations (11)–(16) [31].

The types of fuel consumed in the GT contain natural gas (NG) from the market and fire gas from industrial processes, and their total heat can be calculated using Equation (11), where variables FGTL_scen,h,g and FGTF_scen,h,g are the mass flowrate of NG and fire gas, respectively; parameters HQL and HQF are the lower calorific values of NG and fire gas, respectively. The power generated by the GT (PGT_scen,h,g) can be calculated using Equation (12), where variable QGT_scen,h,g is the total heat of fuel entering the GT and parameter EGT_g is the gas-to-power efficiency, which is taken to be 0.30 in this work.

Usually, there is an abundance of flue gas heat from the gas turbine that can be recovered and utilized by HRSGs for producing steam. The total heat of flue gas (QGTH_scen,h,g) entering the HRSG can be calculated using Equation (14), where parameter LGT_g is the heat loss coefficient of flue gas. The relationship between the heat recovery and steam generation in the HRSGs is described in Equation (15), where the heat efficiency of the HRSG (EHG) is 0.80, positive variable FHS_scen,h,g is the mass flow rate of steam generated in the HRSG. The operating boundary of the HRSG is constrained using Equation (16).

$$QG{T}_{scen,h,g}=FGT{L}_{scen,h,g}·HQL+FGT{F}_{scen,h,g}·HQF$$

(11)

$$PG{T}_{scen,h,g}·THS=QG{T}_{scen,h,g}·EG{T}_g$$

(12)

$$PG{T}_g^{\min }·Y_g\le PG{T}_{scen,h,g}\le PG{T}_g^{\max }·Y_g$$

(13)

$$QGT{H}_{scen,h,g}=(1-EG{T}_g-LG{T}_g)·QG{T}_{scen,h,g}$$

(14)

$$QGT{H}_{scen,h,g}·EHG={\displaystyle \sum _{i,\forall i\in SH{R}_{g,i}}FH{S}_{scen,h,g}·[L{W}_i·(EE{P}_i-E{P}_i)+L{H}_i+L{S}_i·(SS{P}_i-S{P}_i)]}$$

(15)

$$FH{S}_g^{\min }·Y_g\le FH{S}_{scen,h,g}\le FH{S}_g^{\max }·Y_g$$

(16)

4.1.4. Fuel Boilers

The boiler is another type of common equipment that consumes fuel to generate MPS in energy-intensive industries. The regression model proposed by Shang and Kokossis [32] is employed to describe boilers and is represented in Equation (17), and the mass flow rate of steam (FBS_scen,h,j) generated in the boiler is related to the total heat of fuels (QTB_scen,h,j) and the pressure level of steam. The heat resources of boilers are similar to those of the gas turbine and can be calculated using Equation (18). Equation (19) is formulated to constrain the operating boundaries of the boiler, where parameters $FB{S}_j^{\min }$ and $FB{S}_j^{\max }$ are the lower and upper boundary of boiler j, respectively; binary variable B_j denotes whether the boiler j is installed in the PHRES or not

$$\begin{array}{l}QT{B}_{scen,h,j}=[PBA·FB{L}_j^{\max }·B_j+(1+PBB)·FB{S}_{scen,h,j}]·\\ {\displaystyle \sum _{i,\forall i\in HB{S}_{j,i}}[L{H}_i+L{S}_i·(SS{P}_i-S{P}_i)+(1+BLD)·L{W}_i·(EE{P}_i-E{P}_i)]}\end{array}$$

(17)

$$QT{B}_{scen,h,j}=FB{L}_{scen,h,j}·HQL+FB{F}_{scen,h,j}·HQF$$

(18)

$$FB{S}_j^{\min }·B_j\le FB{S}_{scen,h,j}\le FB{S}_j^{\max }·B_j$$

(19)

4.1.5. Steam Turbines

The steam turbine installed in the PHRES can usually be classified into two types: One is the back-pressure steam turbine, and the other is the condensing steam turbine, both of which are employed to balance the steam of different levels at the total site and generate power. The mathematical model proposed by Mavromatis and Kokossis [33] is adopted to describe these two types of steam turbines and represented in Equation (20). The operating boundaries of the steam turbine are formulated and represented in Equation (21), where parameters $FS{T}_r^{\min }$ and $FS{T}_r^{\max }$ are the lower and upper boundary of steam turbine r, respectively; binary variable Z_r denotes whether the steam turbine r is installed in the PHRES or not.

$$PE{T}_{scen,h,r}·THS={\displaystyle \frac{6}{5}}·{\displaystyle \frac{1}{PT{B}_r}}·{\displaystyle \sum _{i,\forall i\in T{S}_{r,i}}\left[(DL{H}_i-\frac{PT{A}_r}{FS{T}_r^{\max }})·(FS{T}_{scen,h,r}-\frac{1}{6}·FS{T}_r^{\max }·Z_r)\right]}$$

(20)

$$FS{T}_r^{\min }·Z_r\le FS{T}_{scen,h,r}\le FS{T}_r^{\max }·Z_r$$

(21)

4.2. Modeling for the Energy Utilization Side

The energy requirements of process units determine the design boundaries of the IHRES. Accurately quantifying these demands is critical. As reviewed in publication [2], conventional methods use black-box models to fit energy consumption as linear functions of process throughputs. This rigid supply–utilization coupling artificially constrains the IHRES design space, yielding suboptimal solutions. In practice, energy interchangeabilities exist due to operational flexibilities. Thus, in that work, we introduced energy interchangeability and developed energy consumption relaxation models (ECRMs).

The core innovation on the demand side is to move away from treating process units as having fixed, inflexible energy demands. Instead, ECRMs are white-box models constructed using process-specific pinch analysis. They explicitly capture the two forms of energy interchangeability within certain units: (1) heat-based interchangeability between fuel consumption and steam usage, as the same temperature duty can be satisfied by different heat sources. (2) Shaft work-based interchangeability allows compressors or pumps to be driven by either electricity (via a motor) or steam (via a turbine). The key assumption is that this inherent operational flexibility exists and can be exploited without altering the core product yield or quality.

Thus, an ECRM does not prescribe a single utility demand but defines a feasible range [$E{R}_{p,e}^{\min }$, $E{R}_{p,e}^{\max }$] for each utility type e at unit p (Equation (22)), effectively relaxing the traditional rigid coupling and expanding the design space for the overall PHRES. The parameters $E{R}_{p,e}^{\min }$ and $E{R}_{p,e}^{\max }$ are the lower and upper boundaries of energy demand by process unit p, and set e denotes the energy demand types, concluding steam, fuel, and power. The lower and upper boundaries of energy demand can be obtained by solving the proposed mathematical framework twice. First, the proposed framework will be solved without considering energy interchangeability for minimizing the TAC to obtain the global solution results and the energy demands in terms of type and quantity. These levels will be marked as $E{R}_{p,e}^1$. Then, the proposed framework will be resolved to minimize utility requirements thoroughly, considering energy interchangeability and obtaining the energy demands in terms of type and quantity; these levels will be marked as $E{R}_{p,e}^2$. Last, the higher one between $E{R}_{p,e}^1$ and $E{R}_{p,e}^2$ will be the upper boundary, while the lower one will be the lower boundary.

Due to length constraints, we omit pinch analysis details (e.g., temperature intervals) and energy balance calculations. The detailed model and procedure for constructing the ECRMs can be found in the Supplementary Materials and publication [2].

$$E{R}_{p,e}^{\min }\le E{R}_{p,e}\le E{R}_{p,e}^{\max },\hspace{1em}e\in \left\{steam,fuel,power\right\}$$

(22)

4.3. Modeling for Energy Storage Systems

Considering the intermittency and uncertainty of renewable energy, energy storage systems are incorporated into the PHRES to enhance the reliability of the energy supply. Therefore, battery energy storage (BES) and thermal energy storage (TES) techniques are adopted to balance the energy supply and requirement at a total site and a temporal day scale.

4.3.1. Battery Energy Storage

The battery is used to store the surplus electricity for a time interval. Equations (23)–(27) model the state-of-charge balance and operating logic of the battery energy storage system. Equation (23) is the core energy balance, where the current charge state depends on the previous state (accounting for self-discharge), plus any charging or minus any discharging. The logical constraints in Equations (25)–(27), linearized via binary variables, ensure that the BES cannot be in both charging and discharging modes within the same time period h, which is a fundamental operational and physical safeguard.

The energy balances around the battery can be described using Equation (23) [34], where parameters ABAT, BBAT, and CBAT are the hourly self-discharge efficiency, charging efficiency, and discharging efficiency of the battery; positive variables EBT_scen,h and EBT_scen,h-1 are loads of battery at period h and h-1, respectively; and positive variables EBTC_scen,h and EBTD_scen,h are the charging load and discharging load of the battery at period h, respectively. Equation (24) is formulated to constrain the total charging load and the total discharging load, which should be equal during the whole period H [35].

Usually, the charging and discharging of an energy storage unit cannot occur in the same period. This logical constraint will lead to a bi-linear term; thus, a linearization method [36] is applied to eliminate the bi-linear term and is represented in Equations (25)–(27). Two binary variables BATX1 and BATX2 are introduced to denote whether electricity is charged and discharged, respectively. Parameter BM is big data.

$$EBA{T}_{scen,h}=EBA{T}_{scen,h-1}·(1-ABAT)+EBAT{C}_{scen,h}·BBAT-{\displaystyle \frac{EBAT{D}_{scen,h}}{CBAT}}, \forall h>1$$

(23)

$${\displaystyle \sum _{h=1}^{24}(EBAT{C}_{scen,h}·BBAT)}={\displaystyle \sum _{h=1}^{24}\frac{EBAT{D}_{scen,h}}{CBAT}}$$

(24)

$$EBAT{C}_{scen,h}\le BM·BATX1$$

(25)

$$EBAT{D}_{scen,h}\le BM·BATX2$$

(26)

$$BATX1+BATX2\le 1$$

(27)

4.3.2. Thermal Energy Storage

The thermal energy storage unit is employed to balance the surplus and shortage of steam at different periods. The quantity for hth thermal energy storage (QTES_scen,h) is related to the quantity for (h-1)th thermal energy storage (QTES_scen,h-1) and the operating mode of the thermal energy storage unit, i.e., heat storage or heat releasing. The energy balance can be described using Equation (28) [37], where parameter ATES is the hourly self-heat dissipation efficiency of the thermal energy storage unit; positive variables QTES_scen,h and QTES_scen,h-1 are loads of the thermal energy storage unit at period h and h-1, respectively; positive variables ETESC_scen,h and ETESD_scen,h are the heat storage load and heat releasing load of the thermal energy storage unit at period h, respectively. Equation (29) is formulated to constrain the total storage load and the total releasing load, which should be equal during the whole period H [35].

The medium for heat storage or its release in the thermal energy storage unit is steam. When steam is in surplus at a total site, it will be stored in the thermal energy storage unit, and the energy balance around the storage process is represented in Equation (30); when there is a shortage of steam at a total site, the stored heat energy will be released by generating steam, and the energy balance around the releasing processes can be described using Equation (31) [12]. Positive variables FTESC_scen,h,i and FTESD_scen,h,i are the flow rate of steam consumed and generated by the thermal energy storage unit, respectively; parameter ETES is the heat efficiency for consuming and generating steam.

Similarly to the battery, heat storage and its release in the thermal energy storage unit cannot occur simultaneously, and the logical constraints are represented in Equations (32)–(34), where two binary variables TESX1 and TESX2 are introduced to denote whether the thermal energy is charged and discharged, respectively.

$$QTE{S}_{scen,h}=QTE{S}_{scen,h-1}·(1-ATES)+QTES{C}_{scen,h}-QTES{D}_{scen,h}, \forall h>1$$

(28)

$${\displaystyle \sum _{h=1}^{24}QTES{C}_{scen,h}}={\displaystyle \sum _{h=1}^{24}QTES{D}_{scen,h}}$$

(29)

$$QTES{C}_{scen,h}·ETES={\displaystyle \sum _{i,\forall i\in TE{S}_i}\{FTES{C}_{scen,h,i}·[L{W}_i·(EE{P}_i-E{P}_i)+L{H}_i+L{S}_i·(SS{P}_i-S{P}_i)]}\}$$

(30)

$$QTES{D}_{scen,h}·ETES={\displaystyle \sum _{i,\forall i\in TE{S}_i}\{FTES{D}_{scen,h,i}·[L{W}_i·(EE{P}_i-E{P}_i)+L{H}_i+L{S}_i·(SS{P}_i-S{P}_i)]}\}$$

(31)

$$QTES{C}_{scen,h}\le BM·TESX1$$

(32)

$$QTES{D}_{scen,h}\le BM·TESX2$$

(33)

$$TESX1+TESX2\le 1$$

(34)

4.4. Balances and Constraints

The energy requirement of process units constrains the design boundaries of the PHRES, while the supply of by-products (i.e., fire gas) will influence the operating boundaries of the PHRES. Thus, the balances and constraints of energy flow and material flow over a total site are significant and need to be considered.

4.4.1. Balances in a Total Site

The balances of steam and power in a total site are constructed and represented in Equations (35) and (36). As shown in Equation (35), the balance of every pressure level of steam is formulated, and the left-hand side of the equation contains the steam generated in PT modules (FPT), HRSGs (FSH), boilers (FBS), steam turbines (FTS), process units (FPTS) and thermal energy storage units (FTESD); the right-hand side of the equation contains the steam consumed in steam turbines (FST), process units (SR), process units as a utility (FSTP), process units as a driving force (FSPE), auxiliary facilities (PSR) and thermal energy storage units (FTESC). The power balance is constrained in Equation (36), and the left side of the equation is the power generated in the PHRES, containing power from the grid (EPM), power generated by photovoltaic modules (PPV), wind turbines (TPWT), gas turbines (PGT), steam turbines (PET) and battery modules (EBATD); the right side of the equation is the power consumed at a total site, containing the power requirement of process units (PR), the power consumed by auxiliary facilities (PXR) and the power stored in the battery modules (EBATC). The balance of the fire gas in a total site is formulated in Equation (37), and the sum of fire gas consumed in the process units (FQF), gas turbines (FGTF), and boilers (FBF) should not exceed the supply quantity (TFG) from the process units.

$$\begin{array}{l}FP{T}_{scen,h,i}+{\displaystyle \sum _{g,\forall g\in SH{R}_{g,i}}FS{H}_{scen,h,g}}+{\displaystyle \sum _{j,\forall j\in HB{S}_{j,i}}FB{S}_{scen,h,j}}+{\displaystyle \sum _{r,\forall r\in T{O}_{r,i}}FT{S}_{scen,h,r}}+{\displaystyle \sum _{p}FPT{S}_{p,i}}+FTES{D}_{scen,h,i}\ge \\ {\displaystyle \sum _{r,\forall r\in T{S}_{r,i}}FS{T}_{scen,h,r}}+{\displaystyle \sum _{p}S{R}_{p,i}}+{\displaystyle \sum _{p}FST{P}_{p,i}}+{\displaystyle \sum _{p}FSP{E}_{p,i}}+FTES{C}_{scen,h,i}+PS{R}_i\end{array}$$

(35)

$$\begin{array}{l}EP{M}_{scen,h}+PP{V}_{scen,h}+TPW{T}_{scen,h}+{\displaystyle \sum _{g}PG{T}_{scen,h,g}}+{\displaystyle \sum _{r}PE{T}_{scen,h,r}}+EBAT{D}_{scen,h}\\ ={\displaystyle \sum _{p}P{R}_p}+PXR+EBAT{C}_{scen,h}\end{array}$$

(36)

$${\displaystyle \sum _{p}FQ{F}_p}+{\displaystyle \sum _{g}FGT{F}_{scen,h,g}}+{\displaystyle \sum _{j}FB{F}_{scen,h,j}}\le TFG$$

(37)

4.4.2. Capacity Boundary Constraints of Devices

The design optimization of the PHRES is a multi-scenario and multi-period problem; thus, the design capacity of every device in the system should be able to meet all requirements in different scenarios or periods. The maximum capacity of these devices can be obtained using Equation (38) [9], where the variable $CAP{A}_d^{\max }$ is the upper design boundary of devices installed in the PHRES, and the variable $CAP{A}_{d,scen,h}$ is the optimal design boundary of devices in every scenario scen or period h.

Equation (38) is a critical “design-for-all-scenarios” constraint. It ensures that the final installed capacity of any device (represented by the first-stage variable CAPA) is sized to be at least as large as the maximum capacity required by that device across all possible operational scenarios (scen) and time periods (h) considered in the second stage. This guarantees the feasibility of the chosen design under every represented uncertainty realization.

$$CAP{A}_d^{\max }\ge CAP{A}_{d,scen,h}, d\in \{PT,PV,WT,GT,HRSG,Boiler,ST,Battery,TES\}$$

(38)

5. Optimization Framework

5.1. Objective Function

For the design problem, economic performance is always the first choice to evaluate the design scheme; thus, the annual total cost (TAC) of the PHRES is employed in this work. As represented in Equation (39), the TAC consists of four parts: capital expense of devices (CAP), operation & maintenance (O&M) expense of devices (COM), operating expense (OP), and carbon taxes (CTAXT).

$$TAC=CAP+COM+{\displaystyle \sum _{scen=1}^{12}(PRO{B}_{scen}·O{P}_{scen})}+{\displaystyle \sum _{scen=1}^{12}(PRO{B}_{scen}·CTAX{T}_{scen})}$$

(39)

The total capital expense of devices (CAP) can be calculated using Equation (40), and the capital expense (CAPC_d) and cost recovery factor (CRF_d) of each device are considered, where the set of devices is indexed by d, containing PT modules, PV modules, WT modules, GTs, HRSGs, boilers, steam turbines, battery and TES; CRF can be calculated using Equation (41) [38], where parameters IR and LT are the interest rate and lifetime of the device, respectively; the capital expense of each device (CAPC_d) can be calculated using Equation (42), where parameter CAPD_d is the unitary capital expense of device d.

$$CAP={\displaystyle \sum _{d}CR{F}_d·CAP{C}_d}, d\in \{PT,PV,WT,GT,HRSG,Boiler,ST,Battery,TES\}$$

(40)

$$CR{F}_d={\displaystyle \frac{IR·{(1+IR)}^{L{T}_d}}{{(1+IR)}^{L{T}_d}-1}}, d\in \{PT,PV,WT,GT,HRSG,Boiler,ST,Battery,TES\}$$

(41)

$$CAP{C}_d=CAP{D}_d·CAP{A}_d^{\max }, d\in \{PT,PV,WT,GT,HRSG,Boiler,ST,Battery,TES\}$$

(42)

The O&M expense of devices (COM) can be calculated using Equation (43), where the positive variable (COMC_d) is the O&M of each device in the PHRES, and it can be calculated using Equation (44), where parameter COMD_d is the unitary O&M cost of each device.

$$COM={\displaystyle \sum _{d}COM{C}_d}, d\in \{PT,PV,WT,GT,HRSG,Boiler,ST,Battery,TES\}$$

(43)

$$COM{C}_d=COM{D}_d·CAP{A}_d^{\max }, d\in \{PT,PV,WT,GT,HRSG,Boiler,ST,Battery,TES\}$$

(44)

The operating expense (OP) of the PHRES can be calculated using Equation (45), where parameter OPT is the annual operating time, positive variables CFUEL and CEPM are the operating cost for purchasing fuel and electricity in the PHRES, and they can be obtained using Equations (46) and (47), respectively. In these equations, parameters PF and PE are the unitary costs for purchasing fuel and electricity, respectively. Note that the electricity price from the grid is set as an uncertainty parameter and fluctuates over 24 h.

$$O{P}_{scen}={\displaystyle \sum _{h=1}^{24}(CFUE{L}_{scen,h}+CEP{M}_{scen,h})·OPT}$$

(45)

$$CFUE{L}_{scen,h}=PF\times \left({\displaystyle {\sum }_{p}FQ{L}_p}+{\displaystyle {\sum }_{g}FGT{L}_{scen,h,p}}+{\displaystyle {\sum }_{j}FB{L}_{scen,h,j}}\right)$$

(46)

$$CEP{M}_{scen,h}=P{E}_h\times EP{M}_{scen,h}$$

(47)

The carbon taxes (CTAXT) of the PHRES can be calculated using Equations (48) and (49), where parameter CO2TAX is the unitary carbon tax and parameters CO2X1, CO2X2, and CO2X3 are the unitary carbon emission factors of fire gas, NG, and electricity from the grid, respectively. The internalization of carbon cost into the TAC is a standard scalarization technique in engineering economics, translating an environmental objective into a monetized constraint with a weight defined by policy or market (the carbon tax). This approach yields a single, economically rational solution under a specific carbon price scenario, providing direct practicality for decision-makers facing such regulations.

$$CTAX{T}_{scen}={\displaystyle \sum _{h=1}^{24}CTA{X}_{scen,h}}$$

(48)

$$CTA{X}_{scen,h}=CO2TAX·\left[\begin{array}{l}CO2X1·HQF·({\displaystyle \sum _{g}FGT{F}_{scen,h,g}}+{\displaystyle \sum _{j}FB{F}_{scen,h,j}})+\\ CO2X2·HQL·({\displaystyle \sum _{g}FGT{L}_{scen,h,g}}+{\displaystyle \sum _{j}FB{L}_{scen,h,j}})+\\ CO2X3·EP{M}_{scen,h}\end{array}\right]$$

(49)

5.2. Mathematical Framework

An MIP mathematical framework termed “ERS” is constructed based on the above models to conduct the design optimization of the PHRES, considering energy uncertainty, energy interchangeability, and energy storage to couple the energy supply side and energy utilization side.

(ERS) min TAC (Equation (39), the total annual cost of the PHRES)

s.t.

Equations (1)–(21) (modeling for energy supply side)

Equations (S1)–(S29) (modeling for energy utilization side)

Equations (23)–(34) (modeling for energy storage systems)

Equations (35)–(37) (balances and constraints in a total site)

Equations (40)–(49) (economic models)

6. Case Study

6.1. Case Background and Data

To enhance the energy supply reliability and reduce carbon emissions, a PHRES incorporating energy uncertainty, interchangeability, and storage is investigated. A refinery from our prior study [2] is adopted, where the PHRES consumes fuel (NG and fire gas) to generate steam and power meeting the energy requirements of process units. Process units consume energy to maintain normal operations while generating steam and fire gas. Steam generated by process units is integrated into steam networks, and fire gas is recycled as fuel for the PHRES and process units’ fired heater.

The refinery includes the following:

• Four crude distillation units (CDU1–4);
• Two fluid catalytic cracking units (FCCU1–2);
• One hydrocracking unit (HCU);
• Two diesel hydrotreating units (DHU1–2);
• One residual hydrotreating unit (RHU);
• One gasoline hydrotreating unit (GHU);
• One kerosene hydrotreating unit (KHU);
• One sewage stripping unit (SSU).

In this study, the pressure levels of steam are 3.5 MPa, 1.0 MPa, and 0.35 MPa for MPS, LPS, and LLPS. The by-product fire gas is used as fuel to be consumed in the energy system and fired heaters; their total quantity is 39 t/h. The annual operating time is 365 days, the purchase price of NG is 600 $/t, and the CO₂ tax is 35 $/t [20]. The CO₂ emission factors of fire gas, NG, and electricity from the grid are 0.075 tCO₂/GJ, 0.056 tCO₂/GJ, and 0.5810 tCO₂/MW; these values are adopted from the recommendation of the Intergovernmental Panel on Climate Change (IPCC) guidelines and the national standard of China. The technical specifications and economic parameters of devices are listed in

Table 2. Specification and cost parameters of devices [28,39,40,41].

Components	Lower Bound	Upper Bound	CAP ($/kW)	O&M ($/kW/yr)	Lifetime (yr)
PV	0 MW	3.0 MW	750	16	20
PT a,b	0 MW	280 t/h	78	19.8	20
WT	0 MW	2.0 MW	1200	55	20
GT	0 MW	30 MW	1210	20	10
HRSG	30 t/h	120 t/h	132	20	10
Boiler	180 t/h	280 t/h	75	26	10
ST c	130/100/50 t/h	150/130/80 t/h	800	60	10
Battery	0 MW	100 MW	414	6.2	5
TES	0 MW	100 MW	25	0	20

^a The measuring unit of cost parameters of photo-thermal is a square meter (m²). ^b The measurement standard for the photo-thermal capacity is the amount of steam produced. ^c There are three types of steam turbines corresponding to them, namely middle-pressure steam turbine, low-pressure steam turbine, and low-low-pressure steam turbine, and they are distinguished by “/”.

.

6.2. Scenarios Generation

The historical data (2016–2020) on solar irradiance and wind speed in South China are obtained from the National Solar Radiation Database [42]. The historical data are processed into 24 hourly intervals. Subsequently, the fast-forward selection algorithm [43] is applied to generate 12 representative scenarios from the dataset, with their probabilities shown in Figure 3. Given that the electricity price significantly impacts the operation of the battery energy storage system, a time-of-day tariff from the literature [44] is adopted for this study. To reflect regional conditions, adjustments are made to the reference tariff structure and are represented in Figure 4.

6.3. Case Description

Five cases are designed to evaluate the effects of energy interchangeability, energy uncertainty, and energy storage on the design optimization of the PHRES and to demonstrate the superiority of the proposed method:

Optimal Case: Energy interchangeability, renewable energy, and energy storage are simultaneously considered.

Base Case: Energy interchangeability, renewable energy, and energy storage are not considered.

Case 1: Only energy interchangeability is considered.

Case 2: Energy interchangeability and renewable energy are considered.

Case 3: Renewable energy and energy storage are considered.

When renewable energy is considered, the case will be optimized using the TSSP method, i.e., stochastic optimization. Conversely, cases excluding renewable energy employ deterministic optimization. The summary of the compared cases is listed in

Table 3. Summary of compared cases.

Case	With ECRMs	With RE	With ESS	Deterministic Optimization	Stochastic Optimization
Base Case	×	×	×	○	-
Optimal Case	√	√	√	-	○
Case 1	√	×	×	○	-
Case 2	√	√	×	-	○
Case 3	×	√	√	-	○

. All mathematical models and cases are coded and calculated in the General Algebraic Modeling System (GAMS) 24.7.1 [45] environment using a HUAWEI laptop with a 3.80 GHz Intel core Ultra7 155H CPU. The MIP solver CPLEX in GAMS 24.7.1 is utilized to solve the MIP models, and the relative optimality criterion is set as zero (OPTCR = 0%) to be the convergence condition.

It is crucial to distinguish between the five design Cases and the twelve uncertainty Scenarios. The Cases define the set of technologies considered in the optimization. The Scenarios, generated from historical weather data, represent equally probable realizations of renewable energy availability. Consequently, the Base Case and Case 1 are deterministic optimizations and are solved without considering multiple scenarios. In contrast, the Optimal Case, Case 2, and Case 3 are stochastic optimizations formulated within a TSSP framework. For these, a single design is optimized to perform well across all 12 scenarios simultaneously. The reported economic and environmental metrics (e.g., TAC) for these cases are therefore expected values, representing the probability-weighted average performance over the entire uncertain domain. This approach ensures the design’s robustness against renewable intermittency.

The economic performances of these five cases are compared in Figure 5. The TAC comprises four parts and is distinguished by different legends. CAP denotes the annualized capital cost of devices, COM denotes the operation and maintenance cost of devices, OP denotes the operating cost of the entire PHRES, and CTAXT denotes the total carbon taxes. As shown in Figure 5, the Optimal Case achieves minimal TAC using the proposed method in this work, though with marginally higher carbon emissions than the Base Case. The Case 3 yields the lowest carbon emission but incurs 7.67% higher than the Optimal Case due to the absence of the ECRMs, which constrain operational flexibility. This explicit comparison between Case 3 and the Optimal Case crystallizes the fundamental trade-off between economic and environmental objectives; achieving the lowest emissions incurs a substantial cost premium when system flexibility is constrained. Conversely, the Optimal Case demonstrates that the ECRMs serve as a critical enabler, allowing the system to navigate this trade-off space more efficiently and approach a superior point of balanced performance.

Moreover, because fire gas is regarded as cost-free fuel, it is fully consumed in all cases. The total consumption by process units is 11.40 t/h. The remainder is used by gas turbines and boilers, the quantity is 27.60 t/h, and the corresponding carbon emissions amount to 72.01 × 10⁴ t/year, accounting for over 90% of the total PHRES carbon emission in each case. As a result, the differences between the cases are not significant. As quantified in Figure 5, the Optimal Case achieves a 14.82% reduction in TAC compared to the Base Case, while Case 3, despite having the lowest emissions, incurs a 7.67% higher TAC than the Optimal Case. This directly quantifies the cost premium for deep decarbonization without full demand-side flexibility.

6.4. Comprehensive Performance Comparative Analysis

A cross-case synthesis of the results reveals the systematic impact of integrating energy interchangeability, renewable energy, and storage. As shown in

Table 4. Summary of energy generated and carbon emission in every case.

Case	Energy Generated (MW)	Energy Generated Ratio (%)			Carbon Emissions (t/h)
		NG	Fire Gas	Renewable Energy	NG	Fire Gas	Grid Electricity
Base Case	341.42	17.47	82.53	/	12.46	78.82	0
Optimal Case	325.27	5.42	90.22	4.35	3.69	82.81	5.42
Case 1	334.00	14.05	85.95	/	10.04	82.21	0
Case 2	325.90	5.72	89.91	4.38	3.90	82.21	5.19
Case 3	332.77	7.49	86.55	5.96	5.10	78.82	5.43

, the purely fossil-based Base Case established a benchmark, with its energy demand met entirely by natural gas and fire gas, leading to the highest dependence on purchased natural gas and its corresponding carbon emissions. Case 1, which introduced energy interchangeability (ECRMs) alone, demonstrated that operational flexibility could reduce natural gas consumption and thus lower both operating costs and emissions without capital investment in RE or storage.

The introduction of renewable energy fundamentally altered the system’s architecture and performance profile. Case 2 (with RE and ECRMs) achieved a significant penetration of solar energy (e.g., 4.38% energy generated ratio). However, the analysis of its cost distribution indicated higher volatility, reflecting the cost of managing RE intermittency without dedicated storage, which maintained a notable reliance on the grid and gas turbines. Case 3 (with RE and ESS) addressed this intermittency by incorporating a battery energy storage system, which stabilized operations and led to the highest RE permeability (e.g., 5.96% energy generated ratio) and the lowest expected carbon emissions among all cases. This environmental benefit, however, came with a higher TAC compared to Case 2, primarily due to storage capital costs, illustrating a clear cost-carbon trade-off.

Finally, the Optimal Case, which synergistically coupled all three elements, achieved the most economical solution. It maintained a high RE penetration comparable to Case 3, but through the co-optimization of demand-side flexibility with storage dispatch, it required less aggressive storage capacities and more efficient use of the gas turbine. This synergistic coordination resulted in the lowest expected TAC, while keeping emissions substantially lower than the Base Case. The progression from Case 2 to Case 3 to the Optimal Case, as visualized in the comparative CDFs of cost and carbon tax, clearly delineates the value of each additional layer of flexibility. RE reduces fuel consumption, ESS mitigates RE volatility, and ECRMs optimize the utilization of both, thereby providing a feasible path for industrial energy systems to navigate the trilemma of cost, reliability, and emission reduction.

6.5. Results Analysis and Discussion

6.5.1. Results Analysis and Discussion of Base Case

The Base Case designs a fossil energy-based energy system excluding renewable energy, energy interchangeability, and energy storage. The total variables in the Base Case are 5137, whereas the discrete ones are 106. Prior to solving the MIP model, a relaxed MIP problem is solved using the solver CONOPT to accelerate computation. As RE is excluded, the Base Case is a deterministic optimization problem, and the optimal solution results can be obtained in 0.20 s. The optimal TAC of this case is $80.24 M/year, where the total annualized capital cost of devices is $12.82 M/year, the O&M cost of devices is $10.28 M/year, the total carbon tax is $27.99 M/year, and the operating cost of the entire system is $29.15 M/year. The proportions and compositions of each item are shown in Figure 6. As shown in the Sunburst chart, the contributions to the total capital cost come from the capital cost of GT, boiler, steam turbine, and HRSG, in that order. The contributions to the total O&M cost come from the O&M cost of the boiler, steam turbine, HRSG, and GT, in that order. According to the solution results, the power generated in the energy system can exactly meet the requirement from process units; therefore, the operating cost consists solely of the purchase cost of NG. Simultaneously, the carbon emissions result from the combustion of NG and fire gas, and the carbon tax cost is therefore attributable to them.

The configuration and key parameters of the energy system are represented in Figure 7. The system comprises two GT systems, one boiler, two MPS turbines, two LPS turbines, and one LLPS turbine. The total power generation is 73.71 MW, whereas the power generated by GTs and steam turbines is 35.67 MW and 38.04 MW, respectively. The total MPS generated by HRSG and boilers is 319.06 t/h. All the steam and power generated by the energy system can meet the energy requirements of the process units.

6.5.2. Results Analysis and Discussion of Optimal Case

In the Optimal Case, renewable energy, energy interchangeability, and energy storage are simultaneously incorporated into the energy system; then, the proposed method in this work is employed to solve the Optimal Case for designing a PHRES. The total variables in the Optimal Case are 30,414, whereas the discrete ones are 2700. Because the RE is considered, this case is thus a stochastic optimization problem, and the optimal solution results can be obtained in 2150 s. Due to the introduction of renewable energy, the problem becomes a stochastic optimization issue; thus, the TAC of the Optimal Case is an expected value. The optimal expected TAC of this case is $68.35 M/year, where the total annualized capital cost of devices is $12.77 M/year; the O&M cost of devices is $11.68 M/year; the total carbon tax is $15.90 M/year; the operating cost of the entire system is $28.00 M/year. As shown in Figure 8, which compares the results with the Base Case and details the proportions and compositions of each item, the TAC was reduced by 14.82%, with only a marginal increase in carbon emissions.

The cost decrease is mainly attributed to lower operating costs, resulting from the reduced demand for NG. In the Base Case, the purchased mass flow rate of NG is 5.54 t/h. This value is significantly reduced in the Optimal Case. Owing to the nature of the TSSP method, the purchased NG flow rate in the Optimal Case represents the expected value. The maximum purchased NG flow rate is 2.38 t/h, and the average purchased NG flow rate across all scenarios over the 24-h period is 1.69 t/h. This reduction contributes to an operating cost decrease of at least $16.61 M/year. Although this strategy results in increased electricity purchases (with a maximum grid purchase of 22.17 MW), the associated cost is at most $0.19 M/year. Moreover, compared to Case 1, using the method proposed in our previous publication [2], the TAC of the Optimal Case is also reduced by 7.69%; the reason for this reduction is also primarily due to reduced NG demand.

To further present the TAC tendency in the uncertain space, the cumulative distribution function (CDF) is analyzed and shown in Figure 9. A CDF is constructed by sorting the TAC of all scenarios in ascending order, where the vertical axis shows the probability that the TAC is not greater than the target limitation indicated on the horizontal axis. The CDFs provide the complete probabilistic synthesis of outcomes across all 12 scenarios for a given stochastic case. This representation is fundamentally more informative than listing individual scenario results, as it directly visualizes the risk profile, i.e., the full range, variance, and likelihood (probability) of meeting or exceeding any given cost or emission target under uncertainty. For instance, the CDF for the Optimal Case shows that there is over a 40% probability that the realized annual cost will be lower than its expected value, quantitatively demonstrating the design’s risk-mitigating potential. Therefore, Figure 9 effectively supplies the ‘all scenario modelling and optimization results’ in their most distilled and decision-relevant form.

By adding the minimum expected TAC of the Optimal Case to the CDF, it is found that in the uncertain space, the probability of a practical TAC being lower than the expected TAC is over 40%. Furthermore, the highest TAC of the Optimal Case is also obviously lower than the TAC of the Base Case. Meanwhile, to demonstrate the carbon mitigation potential of the proposed method, the CDF of the total carbon tax (CTAXT) is also generated and shown in Figure 9b. It can be seen that, in the uncertain space, the probability of a practical CTAXT being lower than both the expected CTAXT of the Optimal Case and the CTAXT of the Base Case is also over 40%; compared to the Case 1, this probability can reach 80%.

The configuration and main parameters of the PHRES in the Optimal Case are represented in Figure 10. Compared to the Base Case, only one GT system needs to be installed, and the LLPS-steam turbine is omitted. The number of GT systems, boilers, MPS turbines, and LPS turbines is 1, 1, 2, and 2, respectively. The main reason for omitting the LLPS turbine is the lower energy efficiency of the condensing steam turbine [2]. Furthermore, due to the implementation of a battery energy storage system for electricity regulation, the power generation load of the GT system is significantly reduced; therefore, only one GT system is required. Solar energy is utilized through PV and PT modules; 12 PV modules are installed with a maximum power output of 25.12 MW. PT modules are employed for MPS generation, with an installed area of 49,744 m² and a maximum MPS mass flow rate of 26.58 t/h. Wind energy is not utilized as the wind speed is insufficient for reaching the cost-effectiveness threshold.

The battery energy storage system is employed to maintain the electricity balance across the total site over the entire time period H. The maximum charging and discharging capacities are 6.69 MW and 5.69 MW, respectively. The detailed loads of battery over the period H for each scenario are provided in Figure 11. As can be seen, battery charging occurs primarily between 4 a.m. to 6 a.m. and 2 p.m. to 7 p.m. daily. To illustrate the typical operating logic of the battery energy storage system under uncertainty, we select Scenario 4 from the set of 12 scenarios for a detailed time-of-day analysis. This scenario exhibits a clear diurnal pattern that effectively demonstrates the system’s response to variable PV generation and time-of-day electricity prices. The hourly battery load, PV output, grid electricity purchase, and the electricity price for Scenario 4 are shown in Figure 12. It is evident that the battery effectively regulates electricity by charging during periods of low electricity costs and high PV output and discharging during periods of high electricity prices and low PV output, thereby achieving peak shaving and valley filling.

6.5.3. Analysis and Discussion of Energy Interchanges in the Optimal Case

The results of energy interchanges in the Optimal Case are also obtained. As aforementioned, two kinds of energy interchangeability are considered in this work. One is the heat energy-based energy interchange, and the other is the shaft work-based energy interchange. These two kinds of energy interchangeability had also been investigated and analyzed in our previous publication [2], where the heat energy-based energy interchanges occurred in CDU1, DHU1, and KHU, while the shaft work-based energy interchanges emerged in RHU, GHU, and KHU.

In this work, the heat energy-based energy interchanges occur in DHU1 and KHU. The GCC (Grand Composite Curve) for these two units in the Base Case and Optimal Case are drawn and shown in Figure 13, where (a) and (b) are the GCCs of DHU1 and KHU, respectively. MPS is used in DHU1 for energy interchanges. The mass flow rate of MPS is 5.46 t/h, while HPS and MPS are simultaneously used in KHU; their mass flow rates are 1.82 t/h and 2.88 t/h, respectively. The shaft work-based energy interchanges occur in HCU, GHU, and KHU. The pressure levels of steam used in these units to drive the hydrogen compressors are all LLPS; the mass flow rates of steam are 25.59 t/h, 6.60 t/h, and 4.58 t/h, respectively.

7. Conclusions

With the rapid development of renewable energy, designing reliable energy systems that integrate renewable and fossil energy poses significant challenges and has been intensively investigated. In this study, a design optimization method for PHRESs from an energy supply–utilization coupling perspective is proposed, incorporating energy interchangeability, energy uncertainty, and energy storage. Heat energy-based and shaft work-based energy interchangeability are considered on the energy utilization side; the uncertainty of renewable energy and the time-of-day grid are considered on the energy supply side. These two sides are then coupled by energy mediums and energy storage systems, i.e., battery and thermal energy storage systems. A TSSMIP model is formulated for the design optimization of PHRESs using TAC as the objective and applied to a large-scale refinery. The model can be solved using a global deterministic optimization algorithm. Compared to the Base Case and Case 1, the Optimal Case utilizing our methodology demonstrates 14.82% and 7.69% reductions in TAC while achieving probabilistic carbon tax savings. Stochastic assessment reveals over 40% and 80% probabilities of total carbon tax expenditure being lower than the Base Case and the Case 1, indicating significant carbon mitigation potential through systematic integration.

While specific optimization results depend on the input parameters and scenario sets, the values of the synergistic strategy, which simultaneously considers uncertainty, flexibility, and storage, and its methodological framework are universal, providing a solid foundation for subsequent research. The proposed methodological framework offers strong generality and extensibility. Future work could explore the following directions:

(1)

A deeper investigation into the mechanistic impact of system parameters like carbon tax, cost-effectiveness threshold of wind speed, energy storage capacity, charge–discharge efficiencies of energy storage, and so on is warranted.

(2)

The optimal design of a hybrid renewable energy system that integrates multiple types of renewable energy and various energy storage technologies.

(3)

Building upon the validated deterministic MIP model core, employ methods such as the ε-constraint technique to perform multi-objective optimization, mapping the Pareto frontiers among total cost, carbon emissions, and system reliability.

(4)

Apply the framework to other energy-intensive sectors (e.g., chemicals, steelmaking) and explore its integration with real-time operational strategies to verify its broader applicability.

(5)

Integrate life cycle assessment (LCA) or a multi-criteria sustainability framework to incorporate a more comprehensive set of environmental and social indicators alongside economic objectives, supporting holistic decision-making for industrial energy transition.