# Behavioral analytics for optimized self-scheduling in sustainable local multi-carrier energy systems: A prospect theory approach

Sobhan Dorahaki, S M Muyeen, Nima Amjady, Syed Shuibul Qarnain, Mohamed Benbouzid

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## To cite this version:

Sobhan Dorahaki, S M Muyeen, Nima Amjady, Syed Shuibul Qarnain, Mohamed Benbouzid. Behavioral analytics for optimized self-scheduling in sustainable local multi-carrier energy systems: A prospect theory approach. Sustainable Energy, Grids and Networks, 2025, 42, 10.1016/j.segan.2025.101679. hal-05373671

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https://hal.univ-brest.fr/hal-05373671v1

Submitted on 19 Nov 2025

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# Behavioral analytics for optimized self-scheduling in sustainable local multi-carrier energy systems: A prospect theory approach

Sobhan Dorahaki<sup>a</sup>, S.M. Muyeen<sup>a,*</sup>, Nima Amjady<sup>b,†</sup>, Syed Shuibul Qarnain<sup>c</sup>, Mohamed Benbouzid<sup>d</sup>

## ARTICLE INFO

### Keywords:

Behavioral economics

Local multi-carrier energy system (LMCES)

Optimization

Prospect theory

Self-scheduling

## ABSTRACT

The transition towards sustainable energy systems demands innovative solutions to overcome the challenges of integrating diverse energy carriers, fluctuating market dynamics, and operator decision-making complexities. The active involvement of local multi-carrier energy systems (LMCES) as virtual power plants in upstream energy markets is particularly hindered by the limitations of conventional optimization methods, which fail to capture the nuanced behavioral aspects of decision-making. This paper presents a novel prescriptive behavioral analytics framework for LMCES self-scheduling, integrating insights from prospect theory to address the operator's behavioral tendencies, including loss aversion, subjective risk attitudes, and mental reference points. By embedding these behavioral considerations into a mixed integer linear programming (MILP) model, the proposed approach accounts for real-world decision-making complexities often overlooked in conventional economic theories based on rationality. Comparative analyses demonstrate that the proposed framework not only enhances the modeling of LMCES operators' decision-making processes but also improves energy scheduling efficiency and supports sustainable energy transitions. The findings provide actionable insights for optimizing LMCES operations, advancing their role in achieving energy sustainability goals.

## Nomenclature

### Indices

$s, N_s$ Set and index of scenarios.

$t, N_t$ Set and index of time.

$l, N_l$ Set and index of linear segments.

### Parameters

$\pi_{s,t}^{E,S}$ Selling electricity price to the upstream grid ($/kWh).

$\pi_{s,t}^{H,S}$ Selling heat price to the upstream grid ($/kWh).

$\pi^{G,B}$ Buying gas price from the upstream grid ($/kWh).

$\eta_l^{CH/DCH,E}$ Charging and discharging electrical energy storage efficiency.

$\eta_l^{CH/DCH,H}$ Charging and discharging thermal energy storage efficiency.

$\delta^{CHP/MT \& BO}$ CHP, microturbine and boiler conversion rate.

$\rho_s$ Probability of each scenario.

$\xi$ Decision weight function coefficient.

$\varphi_l^{Pos, min/max}$ Minimum and maximum of positive auxiliary variables.

$\varphi_l^{Neg, min/max}$ Minimum and maximum of negative auxiliary variables.

$\alpha, \beta, \lambda$ Value function coefficients.

$G_{max}^B$ Maximum allowable gas procurement from the upstream grid (kW).

$H_{max}^{BO}$ Maximum heat output capacity of the boiler unit (kW).

$H_{A,B,C,D}^{CHP}$ Thermal coefficients of the CHP unit (kW).

$M_{1-4}$ Big constant numbers.

$P_{max}^{CH/DCH,E}$ Maximum charge and discharge capacity of the electrical energy storage system (kW).

$H_{max}^{CH/DCH,H}$ Maximum charge and discharge capacity of the thermal energy storage system (kW).

$P_{max}^{MT}$ Maximum output of microturbine unit (kW).

$P_{max}^{E,S}$ Maximum selling electricity to the grid (kW).

$H_{max}^{H,S}$ Maximum selling heat to the grid (kW).

$P_t^{ref}$ Profit reference point of the LMCES operator ($).

<footnote><sup>a</sup> Electrical Engineering Department, Qatar University, Doha, Qatar</footnote>

<footnote><sup>b</sup> Centre for New Energy Transition Research (CfNETR), Federation University Australia, Mt Helen, VIC 3353, Australia</footnote>

<footnote><sup>c</sup> Qatar General Electricity and Water Corporation-KAHRAMAA, Doha, Qatar</footnote>

<footnote><sup>d</sup> University of Brest, UMR CNRS 6027 IRDL, Brest 29238, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: sobhan.dorahaki@qu.edu.qa, sobhandorahaki@gmail.com (S. Dorahaki), sm.muyeen@qu.edu.qa (S.M. Muyeen).</footnote>
$P_{A,B,C,D}^{CHP}$ Power coefficients of the CHP unit (kW).

$SoC_{min/max}^{E}$ Lower and upper bounds for the SoC of electrical energy storage (kWh).

$SoC_{min/max}^{H}$ Lower and upper bounds for the SoC of thermal energy storage (kWh).

$Slop_{i}^{Pos/Neg}$ Slope of the value function in the positive (gain) and negative (loss) parts.

### Variables

$u_{s,t}^{Pos/Neg}$ Binary variables of the positive and negative parts of the value function.

$\varphi_{l,s,t}^{Pos/Neg}$ Positive and negative auxiliary variables.

$\epsilon_{s,t}^{Pos}, \epsilon_{s,t}^{Neg}$ The auxiliary variables used for the linearization of the positive/negative segments of the value function.

$G_{s,t}^{B}$ Buying gas from the grid (kW).

$G_{s,t}^{CHP/MT/BO}$ Gas consumption of CHP, microturbine, and boiler units (kW).

$H_{s,t}^{BO/CHP}$ Heat energy output from the boiler and CHP unit (kW).

$I_{s,t}^{CH,E/H}$ Charging binary variables for electrical and thermal energy storage units (kW).

$I_{s,t}^{DCH,E/H}$ Discharging binary variables for electrical and thermal energy storage units (kW).

$I_{s,t}^{CHP}$ Binary indicator for CHP unit operation.

$P_{s,t}$ LMCES operator profit ($).

$p_{s,t}^{CHP}$ CHP unit generated power output (kW).

$P_{s,t}^{E,S}$ Selling electricity to the grid (kW).

$H_{s,t}^{H,S}$ Selling heat to the grid (kW).

$P_{s,t}^{CH/DCH,E}$ Charging and discharging powers of electrical storage (kW).

$H_{s,t}^{CH/DCH,H}$ Charging and discharging powers of thermal storage (kW).

$P_{s,t}^{MT}$ Microturbine unit generated power output (kW).

$SoC_{s,t}^{E/H}$ Electrical and thermal SoC (kWh).

$V_{s,t}$ Value function.

$W_{s,t}$ Decision weight function.

# 1. Introduction

## 1.1. Motivation and background

The local multi-carrier energy system (LMCES) has demonstrated significant benefits in terms of energy cost efficiency and environmental impact [1]. The LMCES self-scheduling involves optimizing the operation of various energy generation, storage, and conversion assets to maximize the operational profit [2]. However, the success of integrating LMCES into the electrical and thermal markets as a virtual power plant (VPP) hinges significantly on the attitudes and behaviors of its operator [3]. Understanding the decision-making dynamics of the LMCES operator is paramount for effective energy scheduling within LMCES [4]. In fact, human decision-making is complex and influenced by cognitive biases, risk preferences, and subjective evaluations of gains and losses [5]. Conventional economic theories, including classical and neoclassical theories, assume that economic agents act optimally to maximize utility or profit based on complete information and well-defined preferences [6]. These theories underpin many conventional optimization models in energy systems, where decision-makers are considered fully rational entities that evaluate all possible outcomes logically and without bias. However, real-world decision-making often deviates from these assumptions due to cognitive biases, risk perceptions, and subjective valuations, as highlighted in behavioral economics. By contrast, behavioral economics recognizes that individuals do not always behave in strictly rational ways [7]. Instead, they are influenced by various psychological, emotional, and cultural factors. This field integrates

insights from psychology and economics to explain why people may overestimate low-probability risks or underestimate high-probability ones, why loss aversion affects decisions, and how heuristics and biases lead to departures from purely rational behaviors [8]. As a result, behavioral economics provides a richer framework for capturing how real decision-makers operate under uncertainty and risk—factors highly relevant to energy system modeling. Moreover, traditional uncertainty management techniques based on conventional economic theories, such as stochastic programming, robust optimization, and distributionally robust optimization (DRO), rely on objective probability distributions, worst-case scenarios, or worst-case probability distributions to model uncertainties in energy systems. These methods assume that decision-makers assess probabilities rationally and make optimal decisions accordingly. However, empirical studies have shown that individuals do not perceive probabilities in a purely linear manner. Instead, they tend to overestimate low-probability events and underestimate high-probability events, leading to cognitive biases and deviations from rational decision-making. Thus, there is a pressing need for prescriptive analytics frameworks that can inform decision-making by accommodating these behavioral aspects in the LMCES operation.

To address this challenge, this paper introduces a prescriptive behavioral analytics approach for LMCES self-scheduling by integrating principles derived from behavioral economics, specifically prospect theory. By incorporating prospect theory into the optimization framework, the model goes beyond traditional economic paradigms to provide a more realistic depiction of the decision-maker's behaviors, guiding more effective decision-making processes within LMCES operations.

## 1.2. Literature review

Traditional energy systems are being transformed by the emergence of LMCES, where energy can be locally generated, stored, and consumed [9]. This shift towards decarbonized, digitalized, and decentralized energy systems necessitates new approaches to energy management and scheduling [10]. LMCES offers the potential for greater efficiency, resilience, and sustainability; however, it also presents new challenges in terms of optimization and control [11]. Several energy management strategies have been discussed in existing literature concerning the optimal self-scheduling of LMCES as a VPP based on the classic economic theory. In [12], the authors introduced a deterministic model aimed at economically operating LMCES. This model incorporated various components including combined thermal and power (CHP) units, energy storage systems (ESSs), boilers, and renewable energy sources (RESs) to serve as local generation, storage, and conversion equipment. The work [13] proposed a stochastic method for optimizing the LMCES scheduling that relied on the conditional value-at-risk methodology to address the uncertainties inherent in the system. The study [14] provided an LMCES framework utilizing information gap decision theory (IGDT) to optimize operations within community hydrogen parking lots. In [15], a flexibility-oriented model for a sustainable LMCES is proposed using a hybrid multi-objective probabilistic-IGDT optimization approach, aiming to enhance system adaptability under uncertainty. In [16], a two-stage stochastic optimization framework is developed to optimize the dispatch scheduling process within an LMCES, explicitly accounting for the challenges posed by an unbalanced active low-voltage distribution network. In [17], a market-based integrated thermal and electrical energy distribution model in LMCES is developed to capture energy flow dynamics and strategic behaviors of the system operator. The study [18] presented a transactive energy framework for a fuel cell-based LMCES, incorporating Conditional Value-at-Risk (CVaR) to manage uncertainty and optimize energy transactions. In [19], a decentralized management strategy for unbalanced multi-agent distribution systems is proposed, leveraging a customized alternating direction method of multipliers with transactive energy signals for autonomous operation while employing CVaR-based risk management to address uncertainty and
mitigate asymmetrical conditions at the connection point with the upstream network. In [20], a co-scheduling model for VPPs is introduced, concurrently scheduling energy alongside voltage and frequency regulation, using a data-driven distributionally robust optimization framework to manage prosumers' behavioral uncertainties, while an adaptive alternating direction method of multipliers mechanism ensures privacy and scalability for multi-VPP collaboration. In [21], a distributed optimization framework is employed to represent a multi-carrier energy system, utilizing the alternating direction method of multipliers (ADMM) for efficient problem-solving. In [22], an optimization model is proposed to address the operation and coordination of LMCES in smart microgrids using a stochastic approach. [23], highlights advanced optimization techniques for managing grid-connected renewable energy hubs, focusing on energy hubs that integrate compressed air storage, hydrogen, and thermal storage systems. However, all of these mentioned works model the problem based on the rational classic economic theory.

Classical economics forms the foundational basis of economic theory, assuming all economic agents to be rational decision-makers, as discussed in [24]. Economists have scrutinized this assumption, revealing that classic theories fail to account for certain economic agent behaviors. In this regard, the neoclassic economy emerged where the human decision-makers are modeled by the utility function [25]. The utility function captures various human behaviors and explains decision-making in economic contexts, making it a widely applied approach in energy studies. For instance, [26] modeled prosumers behavior in demand response program participation using the utility function to represent the decision-making dynamics. The authors in [27] developed a hybrid local energy market trading framework to optimize energy transactions of prosumers by modeling market participant behaviors through a utility function, incorporating risk attributes and network constraints. The work [28] used a utility function-based priority selection mechanism to model the demand reduction bidding of prosumers in a local energy market. Another work [29] proposed a peer-to-peer transaction platform to promote renewable energy consumption and carbon emission reduction using a utility function-based user model to guide electricity consumption behavior while optimizing generation costs, user utility, and network fees through a distributed optimal scheduling model. The authors in [30] provided an asymmetric Nash bargaining-based strategy, utilizing a utility function to optimize peer-to-peer energy sharing, benefit allocation, and distributed coordination, enhancing prosumers' participation and efficiency. Nevertheless, behavioral actions such as loss aversion, mental reference points, subjective risk aversion or seeking behavior, and human responses to uncertainty are inadequately explained by the utility function theory rooted in neoclassical economics [31].

This gap has led to the proposal of prospect theory as a behavioral economic framework to address these aspects. The prospect theory, which was developed by Kahneman (Nobel Prize laureate) and Tversky, provides a framework for understanding how people make decisions under uncertainty [32]. By integrating these behavioral insights into scheduling algorithms, energy systems can be optimized in a way that accounts for the real-world behaviors of system participants.

Prospect theory has been increasingly applied in the energy domain in recent years. In the energy context, prospect theory has been used to study a range of topics, including energy efficiency investment [7], energy sharing problem [33], mobile energy storage configuration [8], and renewable energy adoption [34]. [35] provides an understanding of energy retrofit decisions among homeowners in the Netherlands, revealing how behavioral economics can optimize decisions under uncertainty in the energy domain. [36] evaluates the risk management of photovoltaic (PV) power plants by introducing a novel fuzzy multi-criteria decision-making (MCDM) method grounded in prospect theory. [37] develops an incentive-based demand response program for VPPs, incorporating consumer behavior and renewable energy fluctuations through a Prospect Theory-based model to optimize energy scheduling in uncertain environments.

Table 1 provides a comparison between traditional and behavioral approaches in LMCES self-scheduling problem. Based on the literature review and Table 1, existing research on energy management and self-scheduling for LMCES, like [3], [4] and [38], predominantly relies on classic economic theory, neglecting the behavioral complexities inherent in decision-making. While deterministic and stochastic models have been proposed, they often overlook factors such as loss aversion, mental reference point, and subjective risk preferences. Therefore, the models proposed in the literature may not provide a realistic environment for LMCES operator in the self-scheduling problem.

### 1.3. Contributions

To fill such a crucial gap, in this paper, a behavioral self-scheduling model is proposed by incorporating the prospect theory approach. The proposed behavioral LMCES self-scheduling model works as a VPP to maximize profit in the electrical and thermal energy markets. Within this framework, the impacts of several human behavioral attitudes, such as loss aversion, mental reference points, and subjective risk-averse or risk-seeking behaviors, on the LMCES self-scheduling are evaluated. Moreover, the issue of the upstream price uncertainty is addressed by modeling it through several scenarios, allowing for an evaluation of LMCES behavior in response to this uncertainty. Importantly, the perspective of the LMCES operator as a human decision-maker facing uncertainty is modeled by utilizing the decision weight function inherent in the prospect theory, providing actionable insights that guide the self-scheduling decisions. The main contributions of the paper are as follows:

* *The attitudes of the LMCES operator in the self-scheduling problem are modeled using the prospect theory approach:* The paper integrates principles from behavioral economics, particularly from the prospect theory, into the LMCES self-scheduling problem. By doing so, it acknowledges and incorporates human behavioral attitudes, such as mental reference point, loss aversion, and subjective risk preferences, offering a more realistic representation of decision-making dynamics in energy scheduling.

* *The impact of price uncertainty on the LMCES operator's decision-making is analyzed from both the behavioral and rational perspectives:* Building on this novel approach in the self-scheduling field, the impacts of the human attitudes of the LMCES operator in facing uncertainty (overestimation and underestimation) are evaluated.

* *Enhanced Realism in Multi-Energy System Optimization:* By embedding human behavioral nuances such as subjective risk attitudes and reference-dependent preferences, the model bridges the gap between theoretical optimization and practical application, providing actionable strategies for real-world LMCES operators to navigate complex and uncertain energy markets effectively.

This study excludes renewable energy sources to maintain a manageable number of uncertainty scenarios, ensuring clear observation of behavioral effects in decision-making. Adding renewables would complicate uncertainty modeling and distort probability perception that may impact the presentation of the core ideas. However, extending the proposed behavioral self-scheduling model to incorporate renewable energy sources can be considered in the future works. Furthermore, the VPP is modeled as a single-node system, focusing on LMCES self-scheduling rather than grid-level power flow, and the heat network is simplified using energy balance constraints to avoid the computational complexity of detailed thermodynamic modeling, both aligning with common practices in VPP self-scheduling studies [4,38–41].

## 2. The proposed methodology

The conceptual structure and mathematical formulation for the
<table><caption>**Table 1**
Comparison of traditional and behavioral approaches in LMCES self-scheduling problem [1–4,9,11–15,17,18,38].</caption><thead><tr><th>Ref</th><th>Self-Scheduling</th><th>LMCES<br>Structure</th><th>Behavioral<br>Analytics</th><th>Uncertainty Modeling<br>Approach</th><th>Optimization<br>Type</th></tr></thead><tbody><tr><td>[1]</td><td>✗</td><td>✓</td><td>✗</td><td>Scenario-Based</td><td>MILP</td></tr><tr><td>[2]</td><td>✗</td><td>✓</td><td>✗</td><td>✗</td><td>MILP</td></tr><tr><td>[3]</td><td>✓</td><td>✓</td><td>✗</td><td>✗</td><td>MINLP</td></tr><tr><td>[4]</td><td>✓</td><td>✓</td><td>✗</td><td>IGDT</td><td>MILP</td></tr><tr><td>[9]</td><td>✗</td><td>✓</td><td>✗</td><td>Scenario-Based</td><td>MILP</td></tr><tr><td>[11]</td><td>✗</td><td>✓</td><td>✗</td><td>IGDT</td><td>MILP</td></tr><tr><td>[12]</td><td>✗</td><td>✓</td><td>✗</td><td>✗</td><td>MINLP</td></tr><tr><td>[13]</td><td>✗</td><td>✓</td><td>✗</td><td>Scenario-Based</td><td>MILP</td></tr><tr><td>[14]</td><td>✗</td><td>✓</td><td>✗</td><td>IGDT</td><td>MILP</td></tr><tr><td>[15]</td><td>✗</td><td>✓</td><td>✗</td><td>IGDT</td><td>MILP</td></tr><tr><td>[14]</td><td>✗</td><td>✓</td><td>✗</td><td>Scenario-Based</td><td>MINLP</td></tr><tr><td>[17]</td><td>✗</td><td>✓</td><td>✗</td><td>✗</td><td>MILP</td></tr><tr><td>[18]</td><td>✗</td><td>✓</td><td>✗</td><td>Scenario-Based</td><td>MILP</td></tr><tr><td>[38]</td><td>✓</td><td>✓</td><td>✗</td><td>Stochastic/IGDT</td><td>MINLP</td></tr><tr><td>This Paper</td><td>✓</td><td>✓</td><td>✓</td><td>Behavioural Scenario-Based</td><td>MILP</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 1.** The proposed self-scheduling model for the LMCES in the electrical and thermal markets.</figcaption></figure>
introduced behavioral self-scheduling model are provided in this section. The first subsection of this section outlines the conceptual framework of the proposed model. The second subsection covers the conventional self-scheduling approach using the expected theory, while the third subsection introduces the proposed formulation based on prospect theory.

## 2.1. The conceptual framework for the proposed LMCES self-scheduling model

The proposed framework for LMCES self-scheduling is depicted in Fig. 1. In this setup, the LMCES operates as a VPP, actively participating in trading activities within both upstream electrical and thermal markets. The VPP operator, rather than individual prosumers, manages generation, storage, and market participation under uncertainty to maximize profit in upstream energy markets [42]. By integrating prospect theory which captures subjective risk perceptions and probability distortions, this approach offers a more realistic representation of how VPP operators develop trading strategies and make decisions in uncertain environments. Moreover, unlike traditional unit commitment models, the self-scheduling framework does not explicitly consider energy demand constraints. Instead, the LMCES operator strategically schedules its energy resources to maximize economic benefits while ensuring the LMCES system feasibility. The energy balance constraints inherently maintain system operational consistency, ensuring that energy supply and storage decisions align with technical and market-based limitations. This aligns with existing literature on self-scheduling frameworks in multi-energy systems, where the focus is on optimizing energy transactions and scheduling decisions rather than meeting a predefined demand [4], [39]. Additionally, the gas consumption of the LMCES energy resources is met by the upstream gas network. The proposed LMCES includes electrical and thermal energy storage, which can enhance the LMCES's profit in upstream energy trading [43]. Moreover, the proposed LMCES features CHP, boiler, and microturbine units, which are active in converting gas energy into electrical and thermal energy. Additionally, the LMCES operator encounters fluctuating prices in the upstream electrical and thermal energy markets. To address this, a scenario-based approach is employed to model the uncertainties of electricity and heat prices in the upstream grids.

In LMCES self-scheduling, the operator's decisions are modeled using behavioral analytics, specifically prospect theory, to incorporate human biases and real-world behaviors, providing more realistic decision-making than traditional models that assume rationality. Each monetary outcome is quantified using the value function, while the probability of each uncertain scenario is adjusted through the decision weight function. Comprehensive details of the proposed model are provided in the subsequent subsections.

## 2.2. LMCES self-scheduling model based on the expected theory

The formulation of the objective function for the LMCES self-scheduling problem, grounded in the principles of expected theory, is expressed in Eq. (1):

$$ \text{Max} \left\{ \sum_{s=1}^{N_s} \sum_{t=1}^{N_t} \rho_s P_{s,t} \right\} \qquad (1) $$

The objective function of the conventional self-scheduling problem, seeks to maximize the expected profit of the LMCES operator. Here, $\rho_s$ represents the probability of each scenario, while $P_{s,t}$ denotes the LMCES operator's profit, which is calculated using Eq. (2) as follows:

$$ P_{s,t} = P_{s,t}^{E,S} \pi_{s,t}^{E,S} + H_{s,t}^{H,S} \pi_{s,t}^{H,S} - G_{s,t}^{B} \pi_{s,t}^{G,B} \quad \forall s, t \qquad (2) $$

The electrical and thermal energy storage systems are modeled by Eqs. (3)–(7) and Eqs. (8)–(12), respectively, as follows [44]:

$$ SoC_{s,t}^{E} = SoC_{s,t-1}^{E} + P_{s,t}^{CH,E} \eta^{CH,E} - P_{s,t}^{DCH,E} / \eta^{DCH,E} \quad \forall s, t \qquad (3) $$

$$ SoC_{min}^{E} \leq SoC_{s,t}^{E} \leq SoC_{max}^{E} \quad \forall s, t \qquad (4) $$

$$ 0 \leq P_{s,t}^{CH,E} \leq I_{s,t}^{CH,E} P_{max}^{CH,E} \quad \forall s, t \qquad (5) $$

$$ 0 \leq P_{s,t}^{DCH,E} \leq I_{s,t}^{DCH,E} P_{max}^{DCH,E} \quad \forall s, t \qquad (6) $$

$$ I_{s,t}^{CH,E} + I_{s,t}^{DCH,E} \leq 1 \quad \forall s, t \qquad (7) $$

$$ SoC_{s,t}^{H} = SoC_{s,t-1}^{H} + H_{s,t}^{CH,H} \eta^{CH,H} - H_{s,t}^{DCH,H} / \eta^{DCH,H} \quad \forall s, t \qquad (8) $$

$$ SoC_{min}^{H} \leq SoC_{s,t}^{H} \leq SoC_{max}^{H} \quad \forall s, t \qquad (9) $$

$$ 0 \leq H_{s,t}^{CH,H} \leq I_{s,t}^{CH,H} H_{max}^{CH,H} \quad \forall s, t \qquad (10) $$

$$ 0 \leq H_{s,t}^{DCH,H} \leq I_{s,t}^{DCH,H} H_{max}^{DCH,H} \quad \forall s, t \qquad (11) $$

$$ I_{s,t}^{CH,H} + I_{s,t}^{DCH,H} \leq 1 \quad \forall s, t \qquad (12) $$

In the above formulation, Eq. (3) and Eq. (8) represent the energy balance equation for the State of Charge (SoC) of the electrical and thermal energy storage systems, respectively. Also, Eq. (4) and Eq. (9) ensure that the electrical and thermal SoCs remain within their pre-defined minimum and maximum limits, respectively. Constraints Eq. (5) and Eq. (10) limit the charging power of the electrical and thermal energy storage systems, respectively. They specify that the charging electricity and heat must be non-negative and cannot exceed the maximum charging power limit. Similarly, Eq. (6) and Eq. (11) limit the discharging power of the electrical and thermal energy storage systems, respectively. Constraints Eq. (7) and Eq. (12) ensure that the electrical and thermal energy storage systems cannot simultaneously charge and discharge.

The CHP unit is modeled by (13)–(18) [45].

$$ P_{s,t}^{CHP} - P_{A}^{CHP} - \frac{P_{A}^{CHP} - P_{B}^{CHP}}{H_{A}^{CHP} - H_{B}^{CHP}} (H_{s,t}^{CHP} - H_{A}^{CHP}) \leq 0 \quad \forall s, t \qquad (13) $$

$$ P_{s,t}^{CHP} - P_{B}^{CHP} - \frac{P_{B}^{CHP} - P_{C}^{CHP}}{H_{B}^{CHP} - H_{C}^{CHP}} (H_{s,t}^{CHP} - H_{B}^{CHP}) \geq - (1 - I_{s,t}^{CHP}) M_{1} \quad \forall s, t \qquad (14) $$

$$ P_{s,t}^{CHP} - P_{C}^{CHP} - \frac{P_{C}^{CHP} - P_{D}^{CHP}}{H_{C}^{CHP} - H_{D}^{CHP}} (H_{s,t}^{CHP} - H_{C}^{CHP}) \geq - (1 - I_{s,t}^{CHP}) M_{1} \quad \forall s, t \qquad (15) $$

$$ 0 \leq P_{s,t}^{CHP} \leq P_{A}^{CHP} \quad \forall s, t \qquad (16) $$

$$ 0 \leq H_{s,t}^{CHP} \leq H_{B}^{CHP} \quad \forall s, t \qquad (17) $$

$$ P_{s,t}^{CHP} = G_{s,t}^{CHP} \delta^{CHP} \quad \forall s, t \qquad (18) $$

Eqs. (13)–(15) ensure that the operation of the CHP unit adheres to its technical limits. Eq. (16) defines the allowable range for the power output of the CHP unit, ensuring that it remains between zero and the maximum power output. Eq. (17) sets the limits for the heat output, ensuring that it stays within the permissible range of zero to the maximum heat output. Eq. (18) establishes the relationship between the CHP unit's power output and the gas consumption.

Constraints Eq. (19) and Eq. (20) pertain to the boiler system performance, ensuring its operational constraints are met.

$$ 0 \leq H_{s,t}^{BO} \leq H_{max}^{BO} \quad \forall s, t \qquad (19) $$
$$H_{s,t}^{BO} = G_{s,t}^{BO} \delta^{BO} \quad \forall s, t \qquad (20)$$

Additionally, Eq. (21) and Eq. (22) model the performance of the microturbine unit.

$$0 \le P_{s,t}^{MT} \le P_{max}^{MT} \quad \forall s, t \qquad (21)$$

$$P_{s,t}^{MT} = G_{s,t}^{MT} \delta^{MT} \quad \forall s, t \qquad (22)$$

The electricity, heat, and gas balance constraints are represented in Eqs. (23)–(25), respectively.

$$P_{s,t}^{CHP} + P_{s,t}^{MT} + P_{s,t}^{DCH.E} - P_{s,t}^{CH.E} = P_{s,t}^{ES} \quad \forall s, t \qquad (23)$$

$$H_{s,t}^{CHP} + H_{s,t}^{BO} + H_{s,t}^{DCH.H} - H_{s,t}^{CH.H} = H_{s,t}^{HS} \quad \forall s, t \qquad (24)$$

$$G_{s,t}^{MT} + G_{s,t}^{CHP} + G_{s,t}^{BO} = G_{s,t}^{B} \quad \forall s, t \qquad (25)$$

The electricity, heat, and gas trading with the upstream grid are limited by Eqs. (26)–(28).

$$0 \le P_{s,t}^{ES} \le P_{max}^{ES} \quad \forall s, t \qquad (26)$$

$$0 \le H_{s,t}^{HS} \le H_{max}^{HS} \quad \forall s, t \qquad (27)$$

$$0 \le G_{s,t}^{B} \le G_{max}^{B} \quad \forall s, t \qquad (28)$$

## 2.3. LMCES self-scheduling model based on the prospect theory

Recent research in behavioral economics reveals a significant departure from the assumptions of rational decision-making that are foundational in traditional economic theories [6]. Classical economic models, such as expected utility theory, assume that individuals consistently make optimal decisions by evaluating all available information logically and without bias. However, empirical studies suggest that real-world decision-making often deviates from these rational assumptions, as humans tend to rely on heuristics, emotions, and cognitive biases when making choices, particularly under uncertainty and risk [46]. These deviations manifest in various ways, including loss aversion, probability distortion, and reference-dependent evaluations, which significantly influence economic and energy-related decision-making.

A prominent framework that addresses these deviations is prospect theory, which provides a behavioral perspective on how individuals make decisions under uncertainty. The prospect theory challenges the rationality assumption by demonstrating that individuals subjectively evaluate potential gains and losses relative to a reference point rather than making absolute comparisons. It is primarily composed of two key components: the *Value Function* and the *Decision Weight Function*, both of which influence how decision-makers, including LMCES operators, respond to uncertain market conditions and risk-related trade-offs.

* **Value Function:** This function describes how individuals perceive and assign value to gains and losses, emphasizing its asymmetric shape, where losses feel more impactful than comparable gains. It explains why decision-makers may prioritize loss avoidance over potential gains, leading to risk-averse behavior in certain scenarios.

* **Decision Weight Function:** This function models how individuals subjectively distort probabilities, often overestimating rare events and underestimating high-probability occurrences. This distortion affects decision-making by influencing how uncertainty is perceived, which may lead LMCES operators to make conservative or overly cautious self-scheduling choices even when rational models suggest otherwise.

In the following sections, the value function, decision weight function, and the final LMCES self-scheduling formulation based on prospect theory are presented.

### 2.3.1. Value function

The human brain perceives gains and losses asymmetrically, affecting decision-making in uncertain environments. In the LMCES self-scheduling framework, this behavior influences how operators respond to profit changes, leading to risk-averse tendencies in the gain domain and heightened sensitivity to losses. As an illustrative example of risk-averse behavior in the gain area, an LMCES operator perceives an increase in operational profit from $0 to $200 as more significant than an increase from $200 to $400. Conversely, in the loss area, the operator experiences a stronger negative impact if the profit shifts from $0 to -$200, compared to a further decline from -$200 to -$400. This asymmetric perception leads the operator to be more cautious in maximizing gains, while strongly avoiding additional losses in scheduling decisions. This asymmetry is further reflected in loss aversion, where individuals tend to weigh losses more heavily than equivalent gains. As an illustrative example of loss aversion, consider an LMCES operator evaluating two scenarios. In the first scenario, the operator gains $300, while in the second scenario, the operator loses $300. Although the absolute value of the gain and loss is the same, the operator perceives the pain of losing $300 as significantly stronger than the pleasure of gaining $300. This asymmetry causes the operator to prioritize strategies that minimize potential losses rather than those that maximize potential gains, influencing self-scheduling decisions and market participation under uncertainty.

The abovementioned behaviors are systematically explained in prospect theory, where the value function is steeper in the loss domain than in the gain domain, reflecting how decision-makers react more strongly to losses than to equivalent gains. In the context of the self-scheduling problem, the value function of the LMCES operator is defined as follows:

$$V_{s,t} = \begin{cases} (P_{s,t} - P_t^{ref})^\alpha & \text{if } (P_{s,t} - P_t^{ref}) \ge 0 \\ -\lambda( (P_{s,t} - P_t^{ref}) )^\beta & \text{if } (P_{s,t} - P_t^{ref}) < 0 \end{cases} \quad \forall s, t \qquad (29)$$

where $P_{s,t}$ represents the monetary profit of the sth scenario in the th time slot and $P_t^{ref}$ denotes the mental monetary reference point. The mental monetary reference point is related to the past experiences and may also be influenced by other LMCES operators based on the peer effect theory [47]. Moreover, $V_{s,t}$ represents the value or subjective profit that the LMCES operator brain assigns to $P_{s,t}$. Additionally, the constant coefficients $\alpha, \beta$, and $\lambda$ of the value function satisfy the conditions $\alpha \le 1, \beta < 1, \lambda \ge 1$. The constant parameters in the value function of prospect theory are determined empirically through experiments where participants make choices between different probabilistic outcomes. Researchers analyze these choices to estimate parameters that best fit observed behaviors, typically using methods like choice experiments and questionnaires [48]. As illustrated in Fig. 2, the prospect theory

<figure><img src="image_6.png" /><figcaption>Fig. 2. The value function of the prospect theory.</figcaption></figure>
value function described in Eq. (29) exhibits a concave shape in the gain domain and a convex shape in the loss domain [32]. This reflects human behavioral tendencies, where individuals display risk aversion in the context of gains and risk-seeking behavior when faced with potential losses. This figure clearly indicates that unlike other risk modeling approaches, such as IGDT and CVaR, which model risk rationally by focusing on minimizing losses or targeting worst-case scenarios, the proposed approach, based on the prospect theory, uniquely accounts for the psychological biases of the LMCES operator in the loss and gain areas. This approach offers a more nuanced and accurate representation of the human behavior in uncertain environments, particularly when personal attitudes toward risk play a significant role in decision-making. Moreover, in Fig. 2, the positive subjective profit, "+V," for a monetary gain "+X" is less than the negative subjective profit, "-(V + V)," for a monetary loss "-X" (comparing the absolute values). This illustrates the principle of loss aversion in the value function, which is controlled by the coefficient λ, greater than 1, highlighting the greater psychological impact of losses compared to gains. Also, the mental reference point (denoted as $P_t^{ref}$) in Fig. 2 is the baseline from which the LMCES operator evaluates outcomes. Outcomes above this point are perceived as gains, while those below are perceived as losses.

### 2.3.1.1. Linearization formulations of the value function.
The value function plays a critical role in modeling the decision-making behavior of the LMCES operator under uncertainty. However, due to its non-linear nature, directly incorporating it into the LMCES self-scheduling optimization model significantly increases computational complexity, making the problem more challenging to solve. Mixed-integer non-linear programming (MINLP) models, which arise from such formulations, require substantial computational resources and may result in impractical solution times for real-world applications. To address this issue, linearization techniques are applied to transform the non-linear value function into a piecewise linear function, allowing the problem to be reformulated as a MILP model. This transformation significantly enhances the model's solvability while preserving the essential behavioral properties of prospect theory. Initially, the piecewise non-linear function Eq. (29) is transformed into a single non-linear function through the application of Eqs. (30)–(32):

$$-M_2(1 - u_{s,t}^{\text{Pos}}) \le (P_{s,t} - P_t^{\text{ref}}) \le u_{s,t}^{\text{Pos}} M_2 \quad \forall s, t \quad (30)$$

$$-M_2 u_{s,t}^{\text{Neg}} \le (P_{s,t} - P_t^{\text{ref}}) \le M_2 (1 - u_{s,t}^{\text{Neg}}) \quad \forall s, t \quad (31)$$

$$u_{s,t}^{\text{Pos}} + u_{s,t}^{\text{Neg}} \le 1 \quad \forall s, t \quad (32)$$

where $u_{s,t}^{\text{Pos}}$ and $u_{s,t}^{\text{Neg}}$ are binary variables and $M_2$ is the big constant number. By multiplying these binary variables by the respected positive and negative parts of Eq. (29), $V_{s,t}$ is reformulated as the following single function:

$$V_{s,t} = u_{s,t}^{\text{Pos}} ((P_{s,t} - P_t^{\text{ref}})^a) + u_{s,t}^{\text{Neg}} (-\lambda ((P_{s,t} - P_t^{\text{ref}}))^\beta) \quad \forall s, t \quad (33)$$

$(P_{s,t} - P_t^{\text{ref}})^a$ and $(- (P_{s,t} - P_t^{\text{ref}}))^\beta$ are another aspect of non-linearity. The piecewise linear function method is used to linearize them as follows [33]:

$$-u_{s,t}^{\text{Neg}} M_3 \le (P_{s,t} - P_t^{\text{ref}}) - \sum_{l=1}^{N_l} q_l^{\text{Pos}} \le M_3 (1 - u_{s,t}^{\text{Pos}}) \quad \forall s, t \quad (34)$$

$$-u_{s,t}^{\text{Pos}} M_3 \le (P_{s,t} - P_t^{\text{ref}}) - \sum_{l=1}^{N_l} q_l^{\text{Neg}} \le M_3 (1 - u_{s,t}^{\text{Neg}}) \quad \forall s, t \quad (35)$$

$$0 \le \sum_{l=1}^{N_l} Slop_l^{\text{Pos}} q_{l,s,t}^{\text{Pos}} \le M_3 u_{s,t}^{\text{Pos}} \quad \forall s, t \quad (36)$$

$$0 \le \sum_{l=1}^{N_l} Slop_l^{\text{Neg}} q_{l,s,t}^{\text{Neg}} \le M_3 u_{s,t}^{\text{Neg}} \quad \forall s, t \quad (37)$$

$$q_l^{\text{Pos\_min}} \le q_{l,s,t}^{\text{Pos}} \le q_l^{\text{Pos\_max}} \quad (38)$$

$$q_l^{\text{Neg\_min}} \le q_{l,s,t}^{\text{Neg}} \le q_l^{\text{Neg\_max}} \quad (39)$$

Based on (34)–(39):

$$\text{if } u_{s,t}^{\text{Pos}} = 1 \rightarrow (P_{s,t} - P_t^{\text{ref}}) = \sum_{l=1}^{N_l} q_{l,s,t}^{\text{Pos}} \Rightarrow (P_{s,t} - P_t^{\text{ref}})^a = \sum_{l=1}^{N_l} Slop_l^{\text{Pos}} q_{l,s,t}^{\text{Pos}} \quad (40)$$

$$\text{if } u_{s,t}^{\text{Neg}} = 1 \rightarrow (P_{s,t} - P_t^{\text{ref}}) = \sum_{l=1}^{N_l} q_{l,s,t}^{\text{Neg}} \Rightarrow (- (P_{s,t} - P_t^{\text{ref}}))^\beta = \sum_{l=1}^{N_l} Slop_l^{\text{Neg}} q_{l,s,t}^{\text{Neg}} \quad (41)$$

The modified version of Eq. (33) is obtained, by substituting $(P_{s,t} - P_t^{\text{ref}})^a$ and $(- (P_{s,t} - P_t^{\text{ref}}))^\beta$ by $\sum_{l=1}^{N_l} Slop_l^{\text{Pos}} q_{l,s,t}^{\text{Pos}}$ and $\sum_{l=1}^{N_l} Slop_l^{\text{Neg}} q_{l,s,t}^{\text{Neg}}$, respectively, as follows:

$$V_{s,t} = u_{s,t}^{\text{Pos}} \left( \sum_{l=1}^{N_l} Slop_l^{\text{Pos}} q_{l,s,t}^{\text{Pos}} \right) + u_{s,t}^{\text{Neg}} \left( (-\lambda) \sum_{l=1}^{N_l} Slop_l^{\text{Neg}} q_{l,s,t}^{\text{Neg}} \right) \quad \forall s, t \quad (42)$$

Next, $u_{s,t}^{\text{Pos}} \sum_{l=1}^{N_l} Slop_l^{\text{Pos}} q_{l,s,t}^{\text{Pos}}$ and $u_{s,t}^{\text{Neg}} \sum_{l=1}^{N_l} Slop_l^{\text{Neg}} q_{l,s,t}^{\text{Neg}}$ are linearized using Eqs. (43)–(44) and Eqs. (45)–(46), respectively.

$$\sum_{l=1}^{N_l} Slop_l^{\text{Pos}} q_{l,s,t}^{\text{Pos}} - (1 - u_{s,t}^{\text{Pos}}) M_4 \le \varepsilon_{s,t}^{\text{Pos}} \le u_{s,t}^{\text{Pos}} M_4 \quad \forall s, t \quad (43)$$

$$0 \le \varepsilon_{s,t}^{\text{Pos}} \le \sum_{l=1}^{N_l} Slop_l^{\text{Pos}} q_{l,s,t}^{\text{Pos}} \quad \forall s, t \quad (44)$$

$$\sum_{l=1}^{N_l} Slop_l^{\text{Neg}} q_{l,s,t}^{\text{Neg}} - (1 - u_{s,t}^{\text{Neg}}) M_4 \le \varepsilon_{s,t}^{\text{Neg}} \le u_{s,t}^{\text{Neg}} M_4 \quad \forall s, t \quad (45)$$

$$0 \le \varepsilon_{s,t}^{\text{Neg}} \le \sum_{l=1}^{N_l} Slop_l^{\text{Neg}} q_{l,s,t}^{\text{Neg}} \quad \forall s, t \quad (46)$$

By substituting $u_{s,t}^{\text{Pos}} \sum_{l=1}^{N_l} Slop_l^{\text{Pos}} q_{l,s,t}^{\text{Pos}}$ and $u_{s,t}^{\text{Neg}} \sum_{l=1}^{N_l} Slop_l^{\text{Neg}} q_{l,s,t}^{\text{Neg}}$ in (42) by $\varepsilon_{s,t}^{\text{Pos}}$ and $\varepsilon_{s,t}^{\text{Neg}}$ and using the constraints Eqs. (43)–(46) a linear form of $V_{s,t}$ is obtained as follows:

$$V_{s,t} = \varepsilon_{s,t}^{\text{Pos}} - \lambda \varepsilon_{s,t}^{\text{Neg}} \quad \forall s, t \quad (47)$$

### 2.3.2. Decision weight function

The human brain often struggles with accurately assessing and dealing with probability. In other words, human brains tend to subjectively assign weights to the probabilities of uncertain outcomes [49]. As an illustrative example of human decision-making, consider an LMCES operator who must determine whether to schedule additional energy generation under uncertain market prices. The operator faces two possible price scenarios: a high-price scenario, where the electricity selling price is $0.25/kWh with an 80% probability, and a low-price scenario, where the electricity selling price is $0.05/kWh with a 20% probability. The generation cost (e.g., fuel cost) for producing electricity is $0.18/kWh.

* Under expected utility theory, the operator evaluates decisions using objective probabilities and calculates an expected profit of $0.03/kWh after subtracting the generation cost and applying the
probabilities as: ($0.25/kWh- $0.18/kWh) × 80% + ($0.05/kWh- $0.18/kWh) × 20% = $0.03/kWh. Since the expected profit is positive, the operator rationally chooses to schedule additional generation, assuming probabilities are assessed objectively.

* Under prospect theory, the operator subjectively perceives probabilities, underestimating high-probability events and overestimating low-probability events. For instance, the 80% probability may be perceived as 65% and the 20% probability may be perceived as 35%. Applying these decision weights, the subjectively perceived profit becomes zero as: ($0.25/kWh- $0.18/kWh) × 65% + ($0.05/kWh- $0.18/kWh) × 35% = $0.00/kWh. This means the operator perceives no financial gain from scheduling additional generation. As a result, the operator may avoid scheduling additional generation, even though the rational model suggests it is beneficial.

This subjective weighting is systematically captured in the prospect theory framework, where the decision weight function reflects the psychological impact of these probabilities on decision-making. The decision weight function of the prospect theory, based on the Prelec approach, is given as follows [50]:

$$W_s = \exp(-(-\ln\rho_s)^2) \quad 0 \le \xi \le 1 \qquad (48)$$

where $\xi$ captures the degree of non-linearity in the weighting of probabilities. This parameter is determined empirically through experimental and observational studies [48]. The decision weight function for different values of $\xi$ is shown in Fig. 3. In Fig. 3 the probability of the probabilistic event is shown on the horizontal axis, and the decision weight, which represents the human brain's perception of that uncertainty, is displayed on the vertical axis. The figure illustrates that LMCES operators may not perceive probabilities linearly. Instead, they tend to overestimate low probabilities and underestimate high probabilities. This behavior is captured by the decision weight function, which curves away from the diagonal line representing linear weighting. Different curves in the figure represent different values of the parameter $\xi$, which controls the degree of non-linearity in the probability weighting. A lower value of $\xi$ results in a more pronounced distortion, where small probabilities are heavily overweighted, and large probabilities are underweighted. Conversely, a higher $\xi$ value results in a less distorted weighting curve, closer to the linear, rational expectation.

Moreover, it should be mentioned that, based on the Prelec approach and the main principles of prospect theory, the sum of decision weights (or behavioral probabilities) is not always equal to 1. This phenomenon is referred to as "Subcertainty" in the prospect theory approach [49]. By incorporating the decision weight function shown in Fig. 3, the proposed self-scheduling model can more realistically simulate the behavior of the LMCES operator under uncertainty. This enhances the model's ability to predict real-world decision-making outcomes, particularly in scenarios where probability judgments play a critical role.

<figure><img src="image_7.png" /><figcaption>Fig. 3. The decision weight function of the prospect theory.</figcaption></figure>

### 2.3.3. Self-scheduling formulation based on the prospect theory

The primary goal of the objective function in the proposed approach is to maximize the prospected subjective profit for the LMCES operator by incorporating both technical constraints (as detailed in (2)-(28)) and behavioral constraints (outlined in (30)-(32), (34)-(39), and (43)-(48)). Therefore, the proposed LMCES self-scheduling model, based on the prospect theory, can be summarized as follows:

$$\text{Max} \left\{ \sum_{s=1}^{N_s} \sum_{t=1}^{N_t} W_s V_{s,t} \right\} \qquad (49)$$

$$\text{s.t Constraints} \ (2) - (28) \qquad (50)$$

$$\text{Constraints} \ (30)-(32), \ (34)-(39) \ \text{and} \ (43)-(48) \qquad (51)$$

This formulation marks significant advancements over conventional self-scheduling LMCES models by enhancing both the objective function and constraints. The proposed objective function, detailed in Eq. (49), shifts the focus to maximizing prospected subjective profit, unlike traditional models that aim to maximize expected profit as shown in Eq. (1). This distinction is crucial because while conventional models rely on rational decision-making and statistical averages, the proposed model integrates prospect theory, accounting for psychological biases, such as loss aversion and diminishing sensitivity to gains and losses. Furthermore, by incorporating both technical constraints Eqs. (2)-(28) and behavioral constraints Eqs. (30)-(32), Eqs. (34)-(39), and (43)-(48), the model offers a more comprehensive framework that addresses not only the operational challenges but also the psychological factors influencing decisions. Fig. 4 provides an overview of the proposed behavioral self-scheduling model based on the prospect theory. As shown, the value function (Eq. (29)) models the LMCES operator's perceived gains and losses relative to a mental reference point, reflecting behaviors such as loss aversion and risk aversion/seeking. Meanwhile, the decision weight function (Eq. (48)) captures the operator's tendency to overestimate low

<figure><img src="image_8.png" /><figcaption>Fig. 4. The prospect theory mathematical modeling process.</figcaption></figure>
probabilities and underestimate high probabilities. However, the objective function—obtained by multiplying these two functions—is nonlinear and must be linearized through Eqs. (30)–(47). Finally, by incorporating Eq. (49)–(51), the behavioral self-scheduling problem can be solved.

## 3. Numerical study

### 3.1. Input data

The electricity and heat prices used for the case study are shown in Fig. 5 [11]. As shown in the previous section, a scenario approach (in which the scenarios are indexed by *s*) has been used in the proposed framework to model the uncertainties of selling electricity price ($\pi_{s,t}^{E,S}$) and selling heat price ($\pi_{s,t}^{H,S}$). However, in the implementation of the case study, to evaluate the behavior of the LMCES operator in the face of price uncertainties with a manageable number of scenarios, only two scenarios are assumed here. The first scenario, with a higher electricity and heat selling price, has a probability of 0.8, while the second scenario, with a lower electricity and heat selling price, has a probability of 0.2. The choice of only two scenarios was driven by the need to preserve and illustrate the behavioral significance of probability distortions. Expanding the number of scenarios would reduce individual scenario probabilities, pushing them into the overestimation region of the decision weight function and distorting the assessment of the uncertainty bias and risk preferences. A higher number of scenarios might improve the model’s statistical depth in a conventional probabilistic framework but would blur the differentiation between overestimation of small probabilities and underestimation of large probabilities, which is the critical point in this study. By maintaining one high-probability and one low-probability scenario, the model ensures meaningful differentiation between probability overestimation and underestimation, which is crucial for accurately analyzing behavioral decision-making under uncertainty. The high-price scenario reflects conditions where the market is favorable, while the low-price scenario represents less favorable conditions. These two scenarios are sufficient to capture the range of

<table><caption>**Table 2**
The input parameters of the proposed model.</caption><thead><tr><th>Parameters</th><th>Value</th><th>Parameters</th><th>Value</th></tr></thead><tbody><tr><td>π<sup>G.B</sup></td><td>0.05 $/kWh</td><td>H<sup>CHP</sup><sub>A</sub></td><td>0 kW</td></tr><tr><td>η<sup>CH.E</sup></td><td>0.95</td><td>H<sup>CHP</sup><sub>B</sub></td><td>210 kW</td></tr><tr><td>η<sup>DCH.E</sup></td><td>0.95</td><td>H<sup>CHP</sup><sub>C</sub></td><td>120 kW</td></tr><tr><td>SoC<sup>E</sup><sub>min</sub></td><td>200 kWh</td><td>H<sup>CHP</sup><sub>D</sub></td><td>0 kW</td></tr><tr><td>SoC<sup>E</sup><sub>max</sub></td><td>1000 kWh</td><td>δ<sup>CHP</sup></td><td>0.6</td></tr><tr><td>p<sup>CH.E</sup><sub>max</sub></td><td>300 kW</td><td>H<sup>BO</sup><sub>max</sub></td><td>200 kW</td></tr><tr><td>p<sup>DCH.E</sup><sub>max</sub></td><td>300 kW</td><td>δ<sup>BO</sup></td><td>0.75</td></tr><tr><td>η<sup>CH.H</sup></td><td>0.95</td><td>P<sup>MT</sup><sub>max</sub></td><td>300 kW</td></tr><tr><td>η<sup>DCH.H</sup></td><td>0.95</td><td>δ<sup>MT</sup></td><td>0.68</td></tr><tr><td>SoC<sup>H</sup><sub>min</sub></td><td>200 kWh</td><td>P<sup>E.S</sup><sub>max</sub></td><td>1000 kW</td></tr><tr><td>SoC<sup>H</sup><sub>max</sub></td><td>1000 kWh</td><td>H<sup>H.S</sup><sub>max</sub></td><td>1000 kW</td></tr><tr><td>H<sup>CH.H</sup><sub>max</sub></td><td>300 kW</td><td>G<sup>B</sup><sub>max</sub></td><td>2000 kW</td></tr><tr><td>H<sup>DCH.H</sup><sub>max</sub></td><td>300 kW</td><td>λ</td><td>1.2</td></tr><tr><td>p<sup>CHP</sup><sub>A</sub></td><td>463 kW</td><td>ζ</td><td>0.8</td></tr><tr><td>p<sup>CHP</sup><sub>B</sub></td><td>295 kW</td><td>P<sup>CHP</sup><sub>D</sub></td><td>156 kW</td></tr><tr><td>p<sup>CHP</sup><sub>C</sub></td><td>145 kW</td><td></td><td></td></tr></tbody></table>

possible behaviors under different market conditions without overwhelming the analysis with too many variables.

The input parameters for the proposed LMCES are presented in Table 2 [44]. The proposed LMCES has a 1 MW capacity, allowing the VPP to meet participation thresholds in certain wholesale markets, where similar-scale units are permitted to engage in energy trading. However, for smaller-scale VPPs that do not meet wholesale market entry requirements, local markets, such as peer-to-peer (P2P) trading and community energy markets, may provide a more viable alternative. These markets enable direct energy transactions between participants, offering flexibility and improved market access for distributed energy resources while enhancing local energy resilience and efficiency [51]–[54]. The proposed model is solved using a computer equipped with Intel Core i7 CPU and 16 GB of RAM. The CPLEX solver in the GAMS environment is used to solve the problem [55].

### 3.2. Numerical results

In the first part of this subsection, the results of the conventional self-scheduling model, based on the expected profit of the classic economic, are presented and discussed. Afterward, the proposed self-scheduling model, based on the prospect theory, is tested in the second part of this subsection.

#### 3.2.1. Results of the conventional self-scheduling model

The conventional self-scheduling model, outlined in (1)–(28), aims to maximize the expected profit of the LMCES, adhering to the foundational principles of classical economics.

In this regard, Fig. 6 shows the optimal hourly monetary profit ($P_{s,t}$)

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Selling price for scenarios #1 and #2. a: Electricity and b: Heat.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 6.** The optimal hourly LMCES monetary profit in scenarios #1 and #2 in the expected theory approach.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 7. The optimal selling of electricity and heat to the upstream grid in the conventional approach (expected profit approach).</figcaption></figure>

for each scenario. The results clearly indicate that the first scenario, which assumes higher electricity and heat selling prices, yields a higher hourly profit compared to the second scenario. The highest profit occurs in hour 24, primarily due to the high heat prices at that time and the strategic utilization of energy storage and generation resources. Fig. 7 shows the optimal selling of electricity and heat to the upstream grid. The results indicate that the maximum selling of electricity to the upstream grid occurs during hours 8, 9, 20, 22, and 24, when the electricity price is high (as shown in Fig. 5. a). Conversely, during the valley price hours, such as hours 13 and 15, the electric sold to the grid drops to zero. Moreover, the results for selling heat to the grid show that the peak selling of heat occurs during hours 5, 13, 17, and 24, when the heat selling price is high. Conversely, the selling of heat to the upstream grid drops to zero during the valley hours of 7, 9, 11, and 16. In this regard, the obtained results can be described by the rational economic theory, particularly considering the fluctuations in electricity and heat prices. For instance, higher selling activities during peak price hours align with the economic principle of maximizing profit by capitalizing on favorable market conditions.

The SoC of the electrical and thermal energy storage systems is shown in Fig. 8. The electricity storage system charges during hours 4, 6, 12, 13, 15, 18, and 23 when the electricity price has low values. Conversely, it discharges during hours 3, 8, 9, 20, 22, and 24 when the electricity price is high. Similarly, the heat storage operates based on fluctuations in heat prices, charging during low-heat-price periods and discharging during high-heat-price periods.

The output energy of the boiler and microturbine units as well as the electrical power output of the CHP unit is shown in Fig. 9. The boiler unit is off during hours 7–9, 11, and 16, when the heat price is low, based on the gas price and the efficiency of the boiler unit. Indeed, this operational decision is driven by the economics of energy conversion: with a gas price ($\pi^{G.B}$) of 0.05 ($/kWh) and a boiler efficiency ($\delta^{BO}$) of 75 %, the cost of generating thermal energy during these hours exceeds the potential revenue from selling it at the prevailing low heat prices.

Moreover, the microturbine unit does not generate any power during

<figure><img src="image_12.png" /><figcaption>Fig. 8. The optimal SoC of electrical and thermal energy storage systems in the conventional approach (expected profit approach).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 9. The output energy of microturbine, boiler, and CHP units.</figcaption></figure>

hours 4, 6, 12, 13, 15–18, and 23, due to low electricity price at these hours as well as the gas price ($\pi^{G.B}$) of 0.05 ($/kWh) and the microturbine unit efficiency ($\delta^{MT}$) of 68 %. At these times, the revenue from electricity generation would not sufficiently cover the production costs, making it uneconomical to operate the microturbine. Consequently, the unit remains offline during these hours to avoid incurring losses. Additionally, the CHP unit, which plays an active role in generating both electricity and heat, operates during all the hours of the operation period.

### 3.2.2. Results of the proposed self-scheduling model based on the prospect theory

In this section, the results of the proposed LMCES self-scheduling model, based on the prospect theory approach, are presented. This model considers human biases and investigates their impacts on the LMCES profit. Table 3 compares the conventional and proposed self-scheduling models. The results indicate that considering risk-averse and risk-seeking behaviors in gains and losses, loss aversion, and mental reference points reduces the expected profit in the self-scheduling problem by 4.55 %. This reduction is captured through the prospect theory-based model, where the value function in (29) reflects risk behaviors, the loss aversion parameter $\lambda$ adjusts for the psychological impact of losses, and the mental reference points influence the evaluation of outcomes as shown in (29). These behavioral factors lead to more conservative decision-making, resulting in a lower overall expected profit. However, the significantly higher prospected subjective profit reported in Table 3 under the prospect theory compared to the expected theory highlights how the proposed model better aligns with the decision-maker’s psychological preferences. This suggests that while the expected profit is slightly lower in the proposed model, the decision-maker perceives its outcomes much more favorably, reflecting a more realistic and satisfactory decision-making process from the perspective of the LMCES operator.

The hourly monetary profit of the LMCES ($P_{s,t}$) under the proposed approach is illustrated in Fig. 10. Comparing this with the hourly monetary profit under the conventional approach in Fig. 6 reveals the impact of loss aversion on decision-making. Based on the principles of the prospect theory, the LMCES operator tries to avoid losses (negative profits) in Fig. 10 compared with Fig. 6. Furthermore, the findings indicate a risk-averse behavior in the profit maximization, leading to a slight decrease in the overall profit. This cautious approach reflects the LMCES operator’s tendency to prioritize risk avoidance over potential gains, particularly evident in the proposed approach.

<table><caption>**Table 3**
The expected profit and prospected subjective profit in conventional LMCES self-scheduling and proposed LMCES self-scheduling.</caption><thead><tr><td></td><td>Expected Profit ($)</td><td>Prospected Subjective Profit ($)</td></tr></thead><tbody><tr><td>Conventional</td><td>334.09</td><td>25.42</td></tr><tr><td>Proposed</td><td>318.86</td><td>92.47</td></tr></tbody></table>
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** The optimal hourly LMCES profit in scenarios #1 and #2 in the proposed approach.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** The optimal selling electricity and heat to the upstream grid in the proposed approach (Prospect theory approach).</figcaption></figure>

The optimal hourly selling of electricity and heat to the upstream grid in the proposed approach is illustrated in Fig. 11. The results reveal that the LMCES operator's decisions are significantly influenced by loss aversion, as demonstrated in Fig. 10. This conservative approach translates to increased selling during certain hours, prioritizing the avoidance of potential losses over fully capitalizing on high-gain opportunities. For instance, during hour 6, as depicted in Fig. 6, the monetary profit of the LMCES is negative in the expected theory approach, which contradicts the loss aversion principle in the prospect theory. In such instances, the typical strategy would involve charging the electrical energy storage during these periods (valley price hours), as shown in Fig. 8, to maximize future monetary profits. Consequently, the monetary profit of the LMCES dips into the loss area in the charging hours. However, according to the principles of the prospect theory, the pain of experiencing a negative profit is greater than the pleasure derived from gains. Therefore, the LMCES operator prefers to avoid charging energy during hour 6 (as shown in Fig. 12) and opts to sell energy to the grid instead. Additionally, the risk-averse strategy of the

<figure><img src="image_16.png" /><figcaption>**Fig. 12.** The optimal SoC of electrical and thermal energy storage systems in the proposed approach (Prospect theory approach).</figcaption></figure>

LMCES operator, as modeled by the concave curve in the gain area of the prospect theory value function (depicted in Fig. 2), results in a reduction in the profit during positive profit hours, particularly noticeable during hours such as 8–10.

The real and behavioral probabilities of the prospect theory approach with different values of the decision weight function parameter ($\xi$) are shown in Fig. 13. The results indicate that when $\xi$ is 1, the decision weight aligns with the real probability (0.8 probability for scenario 1 and 0.2 probability for scenario 2). However, as $\xi$ decreases from 1 to 0.1, the bias of the LMCES operator in dealing with uncertainty increases. Specifically, decreasing $\xi$ results in the lower probability scenario (scenario #2) being given a higher decision weight (overestimation), while the higher probability scenario (scenario #1) is given a lower decision weight (underestimation).

This bias of the LMCES operator has a significant impact on the prospected subjective profit, modeled by the objective function in (49), as shown in Fig. 14. Based on the obtained results, the prospected subjective profit of the LMCES operator is lower in lower $\xi$ values because, in this situation, the low selling price scenario #2 with the lower probability has a higher decision weight value (and vice versa for the high selling price scenario #1).

The impact of the loss aversion parameter ($\lambda$) on the prospected subjective profit is shown in Fig. 15. The results indicate that the prospected subjective profit decreases as $\lambda$ increases. This means that as the LMCES operator becomes more sensitive to losses compared to gains, their tendency to avoid potential losses intensifies, leading to more conservative decision-making and, consequently, a lower overall prospected value.

The pruning analysis in Table 4 evaluates the impact of neglecting participation in either the electrical or thermal market on the LMCES operator's profitability and decision-making process. This evaluation is essential in the prospect theory-based self-scheduling model because it reveals how market participation influences the operator's decision-making. The analysis helps quantify how the absence of market options affects profitability and risk perception, demonstrating the significance of behavioral considerations in the self-scheduling optimization.

The pruning analysis consists of three cases. Case #1 serves as the baseline scenario where the LMCES participates in both the electrical and thermal markets. This scenario allows the operator to make trading decisions considering both markets, resulting in the highest prospected subjective profit in Table 4. Case #2 removes electricity market participation, meaning the LMCES can only sell heat to the upstream grid. The results indicate a 60.23 % reduction in the prospected subjective profit, decreasing from $92.47 to $36.77. This significant drop suggests that electricity trading plays a dominant role in optimizing financial outcomes, and its absence introduces a higher perceived risk. Case #3 removes participation in the thermal market, allowing the LMCES to trade only electricity. The results show a 50.79 % decrease in the prospected subjective profit, reducing it to $45.50. While the impact is still

<figure><img src="image_17.png" /><figcaption>**Fig. 13.** The real and behavioral probabilities based on the decision weight function.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Fig. 14.** The decision weight function parameter (ξ) impacts on the prospected subjective profit.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Fig. 15.** The loss aversion parameter (λ) impacts on the prospected subjective profit.</figcaption></figure>

significant, the decline is less severe than in Case #2, indicating that heat trading is less critical than electricity trading for this LMCES.

To further investigate the impact of market participation on the LMCES self-scheduling, Figs. 16–18 present a sensitivity analysis of the influence of gas, electricity, and heat prices on the prospected subjective profit. As illustrated in Fig. 16, a 5% increase in electricity price—while keeping gas price at their initial value—results in a 6.83% rise in the subjective profit. Conversely, a 5% increase in gas price, with the electricity price held constant at its initial value, leads to a 7.49% reduction in the subjective profit. Notably, when both electricity and gas prices increase by the same percentage, the overall prospected subjective profit declines, highlighting the dominant cost burden of rising gas price. Fig. 17 further explores the effect of heat price variations. When heat price rise by 5% while maintaining gas price at their initial level, the prospected subjective profit increases by 4.27%. However, a 5% increase in gas price—while keeping heat price constant—causes a 7.49% decline in the subjective profit. These findings suggest that although higher heat price contribute to increased profitability, their impact is less pronounced than that of electricity price fluctuations. In contrast, rising gas price significantly elevate operational costs, leading to a notable reduction in overall profit. Finally, Fig. 18 reveals that when

<figure><img src="image_20.png" /><figcaption>**Fig. 16.** Sensitivity analysis of the impact of gas and electricity prices on the prospected subjective profit.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. 17.** Sensitivity analysis of the impact of heat and gas prices on the prospected subjective profit.</figcaption></figure>

both electricity and heat prices increase by 5%, the prospected subjective profit rises by 10.85%. This result underscores the potential profitability of favorable electricity and heat prices conditions while reaffirming the adverse impact of gas price hikes on overall economic performance.

The computational time for the conventional self-scheduling problem is 0.4 s, while for the proposed self-scheduling problem, based on the prospect theory, is 8 s. To assess the tightness and compactness of the proposed MILP model, a computational analysis is conducted. Fig. 19 illustrates the convergence behavior by tracking the progression of the

<caption>**Table 4**</caption>

Pruning analysis of electrical and thermal energy market participation in LMCES self-scheduling.

<table><thead><tr><th></th><th>Electrical Market</th><th>Thermal Market</th><th>Prospected Subjective Profit ($)</th><th>Change Percentage (%)</th></tr></thead><tbody><tr><td>Case #1</td><td>✓</td><td>✓</td><td>92.47</td><td>---</td></tr><tr><td>Case #2</td><td>✗</td><td>✓</td><td>36.77</td><td>-60.23</td></tr><tr><td>Case #3</td><td>✓</td><td>✗</td><td>45.503</td><td>50.79</td></tr></tbody></table>
<figure><img src="image_22.png" /><figcaption>**Fig. 18.** Sensitivity analysis of the impact of heat and electricity prices on the prospected subjective profit.</figcaption></figure>

"Best Integer" and "Best Bound" values as the optimality gap decreases. The "Best Integer remains" approximately stable, while the "Best Bound" steadily improves, demonstrating the solver's efficiency in narrowing the search space and achieving near-optimal solutions with a reasonable computational effort. Table 5 presents additional computational results, including the number of explored nodes and total iterations, further validating the model's computational efficiency and suitability. From Table 5, it is seen that the optimality gap is zero for which the tightness of the solution is indicated in Fig. 19.

## 4. Conclusion

This study provides a new and more realistic LMCES self-scheduling model by incorporating principles from behavioral economics, specifically from the prospect theory. The key points and important findings of the paper can be summarized below:

* The classical economy approach to self-scheduling focuses on maximizing expected profit, with optimal profits achieved during peak prices periods. However, this approach does not account for human behavioral biases, potentially leading to unrealistic decision-making models.

* The prospect theory approach reveals significant differences with respect to the classical economy approach due to considering subjective loss aversion and risk-averse behaviors. LMCES operator demonstrates conservative selling strategies to avoid potential losses, even at the cost of not fully capitalizing on high-gain opportunities. By modeling this attitude of the LMCES operator, the proposed approach leads to slightly lower expected profits but aligns more closely with real-world decision-making processes. Furthermore, differences between the subjective loss aversion and risk-averse behaviors, modelled in this work, and probabilistic/stochastic risk, modelled in the previous works, have been described in the paper.

* The behavioral attitudes of the LMCES operator result in more conservative management of the electrical and thermal energy storage systems, leading to less fluctuation in their usage.

* The decision weight function parameter ($\xi$) significantly affects the prospected subjective profit, with lower $\xi$ values leading to overestimation of low probability scenarios and underestimation of high probability scenarios, thereby reducing the prospected value.

* Increased loss aversion ($\lambda$) in the LMCES operator leads to more conservative decision-making and a lower overall prospected subjective profit.

A limitation of this study is that it does not explicitly consider other

<figure><img src="image_23.png" /><figcaption>**Fig. 19.** Convergence analysis of the proposed model.</figcaption></figure>

<table><caption>**Table 5**
Computational results of the proposed optimization model.</caption><thead><tr><td>Metric</td><td>Value</td></tr></thead><tbody><tr><td>Final Best Integer</td><td>92.4756</td></tr><tr><td>Final Best Bound</td><td>92.4756</td></tr><tr><td>Initial Optimality Gap (%)</td><td>25.04</td></tr><tr><td>Final Optimality Gap (%)</td><td>0</td></tr><tr><td>Total Nodes Explored</td><td>6180</td></tr></tbody></table>

psychological biases, such as overconfidence and regret aversion. While this simplification facilitates model tractability, it does not violate the generality of real-world decision-making, as the selected behavioral factors capture key aspects of human decision-making under uncertainty. However, incorporating additional behavioral factors that evolve over time could further enhance the model's applicability to real-world energy markets. Future research can focus on incorporating other human behavioral factors, such as overconfidence and regret aversion, into the optimization framework to provide a more comprehensive model of decision-making under uncertainty.

## CRediT authorship contribution statement

**dorahaki sobhan:** Writing – original draft, Visualization, Software, Investigation, Formal analysis, Conceptualization. **Muyeen S.M.:** Visualization, Supervision, Project administration, Methodology, Funding acquisition, Formal analysis, Conceptualization. **Amjady Nima:** Writing – review & editing, Supervision, Resources, Project administration, Funding acquisition, Formal analysis. **Qarnain Syed Shuibul:** Writing – original draft, Visualization, Software, Methodology, Investigation, Formal analysis, Conceptualization. **Benbouzid Mohamed:** Writing – review & editing, Visualization, Supervision, Methodology, Investigation, Formal analysis, Conceptualization.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgment

This publication was made possible by the 1st Cycle of ARG Grant no. ARG01-0504-230073, from the Qatar Research, Development and Innovation (QRDI) Council, Qatar. The findings herein reflect the work, and are solely the responsibility, of the authors. The authors also gratefully acknowledge support from Qatar University. Open Access funding provided by the Qatar National Library.
# Data availability

Data will be made available on request.

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