Energy expansion planning with a human evolutionary model

Abstract

This study presents a novel method for planning the expansion of transmission lines and energy storage systems while considering the interconnectedness of electricity and gas networks. We developed a two-level stochastic planning model that addresses both the expansion of transmission and battery systems in the electrical grid and the behavior of the gas network. This research explores the challenges and effects of integrating high levels of renewable energy sources while ensuring security within both networks. Our model uses a stochastic mixed-integer non-linear programming approach. To solve this complex model, we applied the Human Evolutionary Model (HEM). We tested our approach on two case studies: a simple 6-node network and the more complex IEEE RTS 24-bus network for the electricity grid, combined with 5-node and 10-node gas networks, respectively. The results demonstrate the effectiveness of our model, particularly in scenarios where connections in the power and gas networks are disrupted, preventing load shedding even when integrated network connections are cut.

Introduction

While significant progress has been made in planning the expansion of energy storage and transmission systems, existing models often fall short in several key areas. First, many models primarily focus on either the electricity or the gas network, without adequately considering the intricate interactions between the two. This omission can lead to suboptimal planning outcomes, especially given the increasing interdependence of these networks. Second, the integration of security constraints that account for potential disruptions in both networks is frequently inadequate, leaving systems vulnerable to failures. Lastly, many existing models do not effectively address the uncertainties introduced by high levels of renewable energy penetration, leading to less reliable planning under stochastic conditions.

Our study aims to bridge these gaps by developing a comprehensive bi-level stochastic planning model that concurrently addresses transmission and battery expansion in the electric network and the dynamics of the gas network. By employing a stochastic mixed-integer non-linear programming approach and leveraging the Human Evolutionary Model (HEM) for solving complex optimization challenges, our research provides a more robust framework for strategic planning. This model not only improves the coordination between electricity and gas infrastructures but also enhances resilience against potential disruptions and accommodates the uncertainties associated with renewable energy sources. Through detailed case studies, we demonstrate the effectiveness of our approach in preventing load shedding and maintaining network stability even under severe conditions.

In recent years, the adoption of commercial-scale energy systems has significantly increased due to their flexibility, cost-effectiveness, and ability to quickly respond to energy demands. This shift aims to harness renewable resources, particularly wind and solar energy, more efficiently. However, alongside these renewable energy sources, power systems still rely on traditional technologies like gas-fired units, which saw an increase from 37% in 2019 to 39% in 2020. This reliance has intensified the interconnection between natural gas and electricity networks, impacting the supply–demand balance and introducing operational challenges. Natural disasters further risk disrupting electricity and gas network pipelines, underscoring the need for strategic planning that accounts for potential line interruptions (U.S. 2020).

The rise in energy storage systems, integration of gas-fired units, and expansion of renewable energy sources highlight the necessity for coordinated development between electricity and gas companies, with a focus on security constraints. This study targets strategic planning for energy storage investments, aiming to synchronize the expansion of power and natural gas infrastructures while addressing security constraints (Waseem and Manshadi 2020).

Literature review

Transmission and battery expansion models

Several studies have explored various aspects of transmission and battery expansion. Reference (Luburić et al. 2020) presents a bi-level investment model for expanding battery storage, thyristor-controlled series compensators, and transmission lines using Benders decomposition. Reference (Qiu et al. Jan. 2017) introduces a multi-stage stochastic model for concurrent transmission and battery planning, integrating renewable energy dynamics and load demand. Reference (Yao et al. 2020) outlines an integrated strategy to bolster power system resilience, incorporating transmission expansion and strategic battery placement for extreme events. Reference (Moradi-Sepahvand et al. March 2022) develops a multi-stage model for simultaneous planning of transmission lines, batteries, and wind farm expansions to enhance system resilience. Reference (Kazemi and Ansari 2022) presents a method for concurrent transmission and battery planning to improve network reliability with security constraints. Reference (Aguado et al. 2017) advocates a multi-objective approach to increase network flexibility and profitability through line and battery expansion. Reference (Moradi-Sepahvand and Amraee 2021) proposes an integrated multi-period model covering battery, transmission, and power plant expansion, utilizing Benders decomposition. Reference (Alobaidi et al. 2021) offers a stochastic bi-level model using Benders decomposition for integrated transmission and battery expansion, especially under high renewable energy penetration scenarios. Reference (Gan et al. 2019) presents a bi-level model for transmission expansion planning, managing network units and energy storage systems with Benders decomposition, focusing on security constraints.

Integration of electricity and gas networks

Research has also focused on the integration of electricity and gas networks. Reference (Gbadamosi and Nwulu 2020) examines the impact of renewable energy and demand response on generation and transmission expansion. Reference (Barati et al. 2015) introduces a framework for integrated planning of generation, transmission, and natural gas networks using genetic algorithms. Reference (Shao et al. 2017) proposes a resilient bi-level model for expanding gas and transmission networks to enhance electricity network adaptability. Reference (Zhang et al. Nov. 2018) outlines a mixed-integer linear programming framework for natural gas and electricity system expansion, optimizing cost efficiency and network security. Reference (Bakhshi Yamchi et al. 2021) unveils a stochastic model for concurrent transmission and gas network expansion, accounting for contingencies like outages. Reference (Zahedi Rad et al. 2019) presents a model integrating gas network with transmission and generation planning to enhance social welfare. Reference (Jokar and Bahmani-Firouzi 2022) proposes an optimal operation model for transmission networks and distribution substations using a bi-level model and the Reformulation and Decomposition Method. Reference (Unsihuay-Vila et al. 2010) presents a long-term, multiarea, and multistage expansion planning model for integrated electricity and gas grids. Reference (Fan et al. 2020) offers a planning model for multi-energy systems, coordinating the expansion of generators, transmission lines, gas boilers, CHP units, and gas pipelines. Reference (Bent et al. Nov. 2018) proposes an integrated electricity and gas expansion planning model, focusing on cost minimization and addressing gas price fluctuations due to congestion. Reference (Lin et al. 2022) explores integrating hydrogen production from renewable energy curtailments into transmission expansion planning using a mixed-integer nonlinear programming problem. Reference (Zhao and Conejo and R. Sioshansi 2018) uses a two-stage stochastic optimization model for coordinated expansion planning of natural gas and electric power systems, highlighting the importance of uncertainty. Reference (Odetayo et al. Nov. 2018) addresses the dependence on natural gas for electricity by proposing an integrated planning approach using a chance-constrained program to minimize investment costs for new gas-fired generators, pipelines, compressors, and storage facilities, focusing on managing short-term uncertainties. Reference (Qiu et al. July 2015) highlights the importance of simultaneous expansion planning for gas power plants, transmission lines, and pipelines, using a mixed-integer nonlinear programming model to optimize co-expansion and maximize social welfare.

Addressing uncertainties with high renewable penetration

High levels of renewable energy introduce significant uncertainties that many existing models inadequately address. Reference (Safari et al. 2021) introduces a framework for power grid frequency analysis through stochastic expansion planning of transmission and gas networks, considering load growth and security constraints. Reference (Aldarajee et al. 2020) introduces a three-level nonlinear programming approach for investment in integrated electricity and natural gas networks, focusing on extreme events. Reference (Silva et al. Aug. 2000) uses an advanced genetic algorithm for transmission network expansion planning. Reference (Zhang et al. Sept. 2015) presents an optimal expansion planning model for an energy hub integrating electricity, natural gas, and heating, focusing on investment strategies to optimize performance, reliability, energy efficiency, and emissions reduction.

Our study aims to fill these gaps by developing a bi-level stochastic planning model that integrates transmission and battery expansion in the electric network with the dynamics of the gas network. This model employs a stochastic mixed-integer non-linear programming approach, addressing the complexities introduced by high renewable energy penetration and security constraints in both networks.

To solve this complex model, we utilize the Human Evolutionary Model (HEM), a method particularly suited for addressing the stochastic bi-level optimization challenge. We validate our approach through two case studies: a simplified 6-node network and the more complex IEEE RTS 24-bus network for the electric side, paired respectively with 5-node and 10-node systems for the gas network. Our results demonstrate the model’s effectiveness, especially in scenarios involving disruptions in power and gas network lines, and highlight its capability to prevent load shedding even under severe conditions of severed integrated network connections.

Table 1 provides a comparative analysis showing how our model uniquely integrates stochastic approaches, accommodates high renewable energy penetration, and accounts for line outages in both electricity and gas networks (Zeng and An 2014; Zeng and Zhao 2013; Karimianfard et al. Dec. 2022). Given the complexity of our bi-level model with binary variables at both levels, conventional solution methods like strong duality or KKT conditions are impractical. Instead, we adopt the Human Evolutionary Model (HEM) from Ma and Wang (2020) to effectively address this stochastic bi-level optimization challenge.

Table 1 Comparison between this paper and similar papers

In (Borozan et al. 2022), the strategic planning of network expansion with the concepts of smart charging of electric vehicles is proposed as investment options. In the study (Vinasco et al. Nov. 2011), an intelligent strategy is proposed to solve the multi-stage transmission development planning problem. In reference (Konstantelos et al. March 2017), the authors presented a model for the strategic valuation of smart grid technology options in distribution networks. In the study (Ergun et al. 2016), a hierarchical model is proposed to optimize the step-by-step investment plan for expanding the transmission system on a large scale and in multiple regions. In (Giannelos et al. 2021) long-term development planning of transmission network in India under multi-dimensional uncertainty is proposed.

Research motivation

This research optimizes energy systems, focusing on gas-grid integration, electrical network security, renewable energy, and multi-level analysis. Motivations include enhancing energy planning with gas infrastructure, ensuring grid stability, increasing renewable energy use, comprehensive energy system analysis, managing stochastic elements, expanding battery storage, optimizing transmission, and ensuring gas network security. This research is pivotal for sustainable energy systems, addressing challenges in integrating renewables and emphasizing innovative solutions for network optimization. It stresses the importance of flexibility, especially with intermittent renewables, advocating for integrated electricity and gas networks to enhance flexibility. Strategic transmission expansion and energy storage integration are crucial for balancing supply–demand variations. Cost-effectiveness is highlighted, emphasizing long-term savings from optimized infrastructure. Resilience is enhanced by considering security constraints and outage scenarios. The bi-level stochastic planning model manages outages without load shedding, offering benefits in flexibility, cost savings, and resilience for modern energy transitions and network design.

Research GAP

This study offers a holistic approach to energy system optimization, covering gas-grid and electrical network security, renewable energy integration, multi-level decision-making, stochastic optimization, and battery and transmission expansion. It surpasses previous studies by employing Mixed-Integer Non-Linear Programming (MINLP), adopting a multi-objective approach, incorporating stochastic optimization, addressing both battery and transmission expansion, and including gas network security. Its innovative use of Human Evolutionary techniques, multi-objective considerations, and handling of uncertainties provide a more comprehensive solution than previous works. The 28 references reviewed have limitations: some miss aspects like gas-grid and electrical network security, overlook renewable energy integration, or lack comprehensive addressing of electrical network security and other critical factors. Overall, these references have gaps in addressing the multifaceted challenges of energy system optimization, which the current study aims to fill with its comprehensive and innovative approach.

Contribution

This paper introduces key innovations:

• 1. Integrated network planning: it advances by concurrently addressing electricity and gas network expansion, offering a comprehensive analysis for renewable energy assimilation and network resilience.

• 2. Stochastic modeling: it incorporates stochastic modeling for realistic network assessment, enhancing applicability.

• 3. Mixed-Integer Non-Linear Programming (MINLP): employing MINLP ensures robust optimization grounded in mathematics, vital for intricate planning.

• 4. Novel solution algorithm: a new solution algorithm inspired by the Human Evolutionary model efficiently addresses the complex bi-level stochastic optimization, emphasizing practical solutions for network expansion.

Paper organization

The paper details the modeling and innovative algorithm in the next section, while the fourth section presents integrated systems and simulation results. Concluding remarks and future recommendations follow in the final section.

Proposed model and method

This section outlines a stochastic bi-level problem for expanding transmission and battery storage in integrated gas and electricity systems, offering key benefits:

• 1. Enhanced Flexibility: integrating networks enables better adaptation to changing energy demands, crucial for managing renewable energy variability.

• 2. Optimized Renewables Integration: plans expansions to accommodate intermittent renewables, maintaining system stability.

• 3. Cost Savings: identifies synergies to reduce infrastructure costs by optimizing transmission, storage, and gas networks.

• 4. Improved Resilience: proposes robust expansion plans to enhance system resilience by identifying potential failure points.

• 5. Load Shedding Mitigation: prevents load shedding during disruptions, ensuring stable energy supply.

• 6. Real-World Applicability: tested on experimental networks, demonstrating potential for real-world use.

In summary, integrating electricity and gas networks enhances flexibility, cost efficiency, resilience, and reliability, vital for sustainable energy systems amidst evolving challenges.

Upper-level problem

This paper focuses on an upper-level problem involving stochastic security-constrained planning for transmission and battery expansion. The objective function for this upper-level problem is detailed in Eq. (1).

$$\begin{gathered} {\text{min}}\sum\limits_{{l \in L}} {\hat{\alpha }_{l} } x_{l} + \sum\limits_{{b \in B}} {\hat{\beta }_{b} } x_{b} + 365 \times 24\sum\limits_{{s \in S}} {\sigma _{s} } \hfill \\ \left( {\begin{array}{*{20}c} {\sum\limits_{{b \in B,t \in T}} {g_{{b,t,s}} } c_{b}^{g} + u_{{b,t,s}} c_{b}^{n} + s_{{b,t,s}} c_{b}^{{st}} + v_{{b,t,s}} c_{b}^{{sd}} } \\ { + r_{{b,t,s}} c_{b}^{r} + \left( {p_{{b,t,s}}^{c} + p_{{b,t,s}}^{d} } \right)c_{b}^{{bt}} + \left( {\tilde{p}_{{b,t,s}}^{{re}} - p_{{b,t,s}}^{{re}} } \right)c_{b}^{{re}} } \\ \end{array} } \right) \hfill \\ \end{gathered}$$

(1)

In Eq. (1), the terms $L$, $B$, $S$, and T represent the sets of power grid lines, power grid buses, scenarios, and hours in a sample day of the target year, indexed by $l$, $b$, $s$, and $t$ respectively. The upper-level problem's objective function aims to minimize several costs, including the investment in lines and batteries, operational expenses, costs associated with no-load, startup, and shutdown of units, load curtailment costs, and costs related to battery charging/discharging and renewable energy power curtailment. Key variables include ${\widehat{\alpha }}_{l}$ (annual investment cost for new lines), ${x}_{l}$ (binary variable for new line installation status), ${\widehat{\beta }}_{b}$ (annual battery investment cost), ${x}_{b}$ (installed battery capacity), and ${\sigma }_{s}$ (scenario probability). Other variables cover unit production ${g}_{b,t,s}$ and production cost (${c}_{b}^{g}$), unit commitment status (${u}_{b,t,s}$) and no-load cost (${c}_{b}^{n}$), startup status (${s}_{b,t,s}$) and cost (${c}_{b}^{st}$), shutdown status (${v}_{b,t,s}$) and cost (${c}_{b}^{sd}$), load curtailment and its cost (${r}_{b,t,s}$ and ${c}_{b}^{r}$), battery charging/discharging (${p}_{b,t,s}^{c}$, ${p}_{b,t,s}^{d}$) and corresponding costs (${c}_{b}^{bt}$), actual and operated amounts of renewable energy production (${\widetilde{p}}_{b,t,s}^{re}$, ${p}_{b,t,s}^{re}$), and the cost of renewable energy power outage (${c}_{b}^{re}$).

Equations (2) to (21) detail the constraints of the upper-level problem. Specifically, Eq. (2) outlines the operational limits for generation units. Within this context, ${g}_{b}^{min}$ and ${g}_{b}^{max}$ represent the minimum and maximum production limits for each unit at bus $b$, respectively. These constraints ensure that the output of each generation unit stays within specified boundaries.

$${g}_{b}^{min}{u}_{b,t,s}\le {g}_{b,t,s}\le {g}_{b}^{max}{u}_{b,t,s}\quad \forall b\in B,t\in T,s\in S$$

(2)

Equation (3) presents the power balance in the electrical grid. This equation includes terms such as ${\widetilde{g}}_{b,t,s}$, representing the output of gas-fired units, ${f}_{ij,t,s}$ which denotes the power flow between bus $i$ and bus $j$, and ${d}_{b,t,s}$, signifying the electrical load at bus $b$ at time $t$ in scenario $s$. This balance equation ensures that the total generation, including gas-fired units and the flow of electricity between buses, matches the network's load at each bus, time, and scenario.

$${g}_{b,t,s}+{\widetilde{g}}_{b,t,s}+{r}_{b,t,s}+{p}_{b,t,s}^{d}+{p}_{b,t,s}^{re}+\sum_{ij\in b}{f}_{ij,t,s}-\sum_{ji\in b}{f}_{ji,t,s}-{p}_{b,t,s}^{c}-{d}_{b,t,s}=0 \quad\forall b\in B,t\in T,s\in S$$

(3)

Relation (4) outlines constraints for renewable energy use in each scenario, ensuring wind and solar operations meet specific criteria for effective integration into the energy system.

$$0\le {p}_{b,t,s}^{re}\le {\widetilde{p}}_{b,t,s}^{re} \quad\forall b\in B,t\in T,s\in S$$

(4)

Equation (5) sets constraints on load curtailment for each bus at time t in every scenario, establishing limits for reducing electricity supply to ensure controlled load management.

$$0\le {r}_{b,t,s}\le {d}_{b,t,s}\quad \forall b\in B,t\in T,s\in S$$

(5)

Constraints (6) and (7) set the upper limits for battery charging and discharging, preventing overcharging and excessive depletion to maintain battery efficiency and longevity.

$$0\le {p}_{b,t,s}^{c}\le {x}_{b}\left({1-z}_{b,t,s}\right)\quad \forall b\in B,t\in T,s\in S$$

(6)

$$0\le {p}_{b,t,s}^{d}\le {x}_{b}{z}_{b,t,s}\quad \forall b\in B,t\in T,s\in S$$

(7)

Constraints (8) and (9) guarantee that the selected battery capacity at each bus adheres to predefined discrete limits. These constraints involve a set, $O$, which represents the range of possible battery capacities, each indexed by o. The auxiliary binary variable ${y}_{b,o}$ is used in these constraints to ensure that the battery capacity (${x}_{b}$) at a specific bus ($b$) aligns with one of the specified capacities in the set ${A}_{o}$. This approach ensures that battery capacities chosen for installation are consistent with the predetermined options available within the set ${A}_{o}$.

$$\sum_{o\in O}{y}_{b,o}\le 1\quad \forall b\in B$$

(8)

$${x}_{b}=\sum_{o\in O}{y}_{b,o}{A}_{o}\quad \forall b\in B$$

(9)

Relationships (10) and (11) in the study outline the available energy in the battery and its constraints. In these equations, ${e}_{b,t,s}$ represents the energy status of the battery, indicating the amount of energy stored in the battery at a specific bus (b), at a particular time (t), and under a given scenario (s). These relationships help manage and limit the energy levels within the battery, ensuring they stay within operational and design specifications.

$${e}_{b,t+1,s}={e}_{b,t,s}+{p}_{b,t+1,s}^{c}-{p}_{b,t+1,s}^{d}\quad \forall b\in B,t\in T,s\in S$$

(10)

$$0\le {e}_{b,t,s}\le {x}_{b}\quad \forall b\in B,t\in T,s\in S$$

(11)

Equations (12) and (13) ensure the battery can only charge in the first hour and starts with no initial capacity, crucial for accurate modeling and planning.

$${e}_{b,t,s}={p}_{b,t,s}^{c}-{p}_{b,t,s}^{d}\quad \forall b\in B,t=1,s\in S$$

(12)

$${p}_{b,t,s}^{d}=0\quad \forall b\in B,t=1,s\in S$$

(13)

Equation (14) defines the interaction between commitment binary variables, indicating unit on/off status, crucial for managing the operational status of units within the energy system.

$${u}_{b,t+1,s}-{u}_{b,t,s}={s}_{b,t,s}-{v}_{b,t,s}\quad \forall b\in B,t\in T,s\in S$$

(14)

Equation (15) in the study illustrates the constraint on the budget for investment purposes. The variable $\psi$ is used to represent the maximum available budget for these investments. This equation ensures that the total cost of investments made in the energy system does not exceed the predetermined budget limit, $\psi$, thereby maintaining financial feasibility and discipline in the planning and expansion process.

$$\sum_{l\in L}{\alpha }_{l}{x}_{l}+\sum_{b\in B}{\beta }_{b}{x}_{b}\le \psi$$

(15)

Equations (16) and (17) in the paper define the power flow and set limits on the power line. Specifically, ${\gamma }_{l}$ represents the line susceptance, which is a measure of how easily a power line can conduct an alternating current. The voltage angle at a specific point in the network, at a given time and scenario, is denoted by ${\theta }_{i,t,s}$. The maximum allowable power flow through a line is indicated by ${f}_{l,t,s}^{max}$. Finally, ${\varrho }_{l,t,s}$ is a binary variable representing the status of a line outage, indicating whether the line is operational or not. These equations are crucial for modeling the physical constraints and operational status of the power transmission lines in the energy network.

$${f}_{l,t,s}={x}_{l}{\gamma }_{l}\left({\theta }_{i,t,s}-{\theta }_{j,t,s}\right)\quad \forall l\in L,i,j\in B,t\in T,s\in S$$

(16)

$$-{f}_{l,t,s}^{max}{\varrho }_{l,t,s}\le {f}_{l,t,s}\le {f}_{l,t,s}^{max}{\varrho }_{l,t,s}\quad \forall b\in B,t=1,s\in S$$

(17)

Equation (18) in the paper outlines the limitations on power line outages. In this context, $\Gamma$ represents the maximum number of power line outages that can be tolerated in the network. The set of all lines within the network is denoted as $\widetilde{L}$. This equation is critical for ensuring the reliability and stability of the power network by setting a cap on the number of allowable outages, thus helping to maintain continuous and uninterrupted power supply within the system.

$$\sum_{l\in \widetilde{L}}\left(1-{\varrho }_{l,t,s}\right)=\Gamma\quad \forall t\in T,s\in S$$

(18)

Relations (19) and (20) set constraints on voltage angles to maintain grid stability and ensure efficient, reliable power transmission.

$$-\pi \le {\theta }_{b,t,s}\le \pi\quad \forall b\in B,t\in T,s\in S$$

(19)

$${\theta }_{b,t,s}=0\quad \forall b=ref,t\in T,s\in S$$

(20)

Equation (21) in the study illustrates the process of converting gas into electricity using gas-fired units. In this equation, $\rho$ represents the factor that converts gas to electric power, while ${w}_{n,t,s}$ denotes the gas demand of these units. The set of gas nodes in the system is represented by $N$, with each node individually indexed by $n$. This equation is essential for modeling and understanding the interplay between gas consumption and electricity generation in gas-fired power units.

$${\widetilde{g}}_{b,t,s}=\rho {w}_{n,t,s}\quad \forall b\in B,n\in N,t\in T,s\in S$$

(21)

Lower-level problem

In the discussed paper, the modeling of the gas network is addressed as a lower-level problem within a bi-level optimization framework. This lower-level issue is formulated as a mixed integer linear optimization model. Equation (22) outlines the objective function for this lower-level problem, which aims at increasing social welfare. Key variables in this function include ${w}_{n,t,s}$, representing the gas demand at gas-fired units, and ${c}_{n}^{w}$, denoting the bid price of these units. Additionally, ${\omega }_{n,t,s}$ signifies the gas supplied at each node in the gas network, while ${c}_{n}^{\omega }$ corresponds to the marginal cost of supplying gas at these network nodes.

$$\text{max}\sum_{n\in N,t\in T,s\in S}{w}_{n,t,s}{c}_{n}^{w}-\sum_{n\in N,t\in T,s\in S}{\omega }_{n,t,s}{c}_{n}^{\omega }$$

(22)

Equation (23, 24, 2526) in the paper detail the constraints associated with the lower-level problem, focusing on the gas network. Specifically, Eq. (23) establishes the balance of natural gas at each node in the gas network. This equation incorporates $\zeta_{{}}$, which represents the flow of gas through each pipeline, and $d_{n,t,s}^{g}$, denoting the consistent demand for gas. These components work together to ensure that the flow and demand for natural gas are appropriately balanced at each node within the network.

$${{\omega }_{n,t,s}-w}_{n,t,s}+\sum_{nm\in N}{\zeta }_{nm,t,s}-\sum_{mn\in N}{\zeta }_{mn,t,s}-{d}_{n,t,s}^{g}=0\quad \forall n\in N,t\in T,s\in S$$

(23)

Equation (24) in the paper describes the restrictions on gas flow within each pipeline of the gas network. This equation introduces ${\zeta }_{k}^{max}$, which represents the maximum allowable gas flow in each pipeline. The gas network's pipelines are collectively denoted by the set $K$, with individual pipelines indexed by $k$. Additionally, the equation includes$Z_{k,t,s}$, a binary variable that signifies the operational status of each pipeline in the gas network. This ensures that the flow in each pipeline does not exceed its designated maximum capacity, maintaining the integrity and efficiency of the gas distribution system.

$${-\zeta }_{k}^{max}{\mathcal{z}}_{k,t,s}\le {\zeta }_{k,t,s}\le {\zeta }_{k}^{max}{\mathcal{z}}_{k,t,s}\quad \forall k\in K,t\in T,s\in S$$

(24)

Equation (25) in the research paper sets the constraints for gas supply from gas wells. It specifies the minimum (${\omega }_{n}^{min}$) and maximum (${\omega }_{n}^{max}$) limits of gas that can be utilized from each gas well. This equation ensures that the usage of gas from these wells stays within a predefined range, balancing the need for sufficient gas supply with the constraints of gas well capacities.

$${\omega }_{n}^{min}\le {\omega }_{n,t,s}\le {\omega }_{n}^{max}\quad \forall n\in N,t\in T,s\in S$$

(25)

In the research paper, Eq. (26) defines the consumption limits for gas in gas-fired units. This constraint ensures that the gas usage in these units remains within a specified range, with a minimum limit denoted by ${w}_{n}^{min}$ and a maximum limit indicated by ${w}_{n}^{max}$. This boundary is crucial for maintaining an efficient and balanced operation of gas-fired units within the gas network.

$${w}_{n}^{min}\le {w}_{n,t,s}\le {w}_{n}^{max}\quad \forall n\in N,t\in T,s\in S$$

(26)

Equation (27) ensures gas network reliability by addressing pipeline outage security measures, integral to the bi-level optimization model.

$$\sum\limits_{{k \in K}} {{\text{z}}_{{k,t,s}} } = K - 1\quad\forall t \in T,s \in S$$

(27)

The paper states that both the upper and lower level models are mixed integer non-linear programming problems. Due to the binary variable in the lower model, it can't be simplified using KKT conditions. Thus, a solution algorithm inspired by the Human Evolutionary model is proposed to address the complex bi-level model.

Proposed solution

The paper’s stochastic bi-level optimization model includes Eqs. (28) to (31): (28) upper-level objective, (29) upper-level constraints, (30) lower-level objective, and (31) lower-level constraints (see Fig. 1).

Fig. 1

Framework of the proposed approach

$$\text{min}\sum_{l\in L}{\widehat{\alpha }}_{l}{x}_{l}+\sum_{b\in B}{\widehat{\beta }}_{b}{x}_{b}+365\times 24\sum_{s\in S}{\sigma }_{s}$$

(28)

$$\left(\begin{array}{c}\sum_{b\in B,t\in T}{g}_{b,t,s}{c}_{b}^{g}+{u}_{b,t,s}{c}_{b}^{n}+{s}_{b,t,s}{c}_{b}^{st}+{v}_{b,t,s}{c}_{b}^{sd}\\ +{r}_{b,t,s}{c}_{b}^{r}+\left({p}_{b,t,s}^{c}+{p}_{b,t,s}^{d}\right){c}_{b}^{bt}+\left({\widetilde{p}}_{b,t,s}^{re}-{p}_{b,t,s}^{re}\right){c}_{b}^{re}\end{array}\right)$$

$$s.t. \left(2\right)-(21)$$

(29)

$${\mathbf{w}}_{\mathbf{n},\mathbf{t},\mathbf{s}}\in \text{argmax}\sum_{n\in N,t\in T,s\in S}{w}_{n,t,s}{c}_{n}^{w}-\sum_{n\in N,t\in T,s\in S}{\omega }_{n,t,s}{c}_{n}^{\omega }$$

(30)

$$s.t. \left(23\right)-(27)$$

(31)

The bi-level optimization model (Eqs. 28–31) includes binary configurations at both levels. A specific approach from wan] is used to address this binary nature.

The Human Evolutionary Model (HEM), introduced by (Montiel et al. 2007). Employs Mediative Fuzzy Logic (MFL) for adaptation, ensuring excellent global search performance in complex environments. HEM comprises genetic representation (decision variables), genetic effects (individual attributes), and objective value (fitness function). It emulates human expert parameter selection using MFL to reconcile conflicting information and make effective decisions in the evolutionary process.

MFL, based on intuitive fuzzy logic systems, resolves conflicts in information. In the HEM, the Artificial Intelligent Intuitive System (AIIS) uses MFL to regulate population size, enhancing HEM’s convergence speed. AIIS adjusts the number of individuals created or removed based on ‘variance’ and ‘cycling’ derived from fitness values, tracked by ‘numCreate’ and ‘numDelete’. Reference (Ma and Wang 2020) offers detailed insights into AIIS and its role in HEM.

An algorithm integrates the HEM with bilevel programming problems (BLPP), where each individual’s genetic representation combines solutions from upper and lower levels (see Fig. 2). HEM guides evolutionary direction for both levels, while the lower level is solved using a genetic algorithm (GA). This approach redefines individual generation for solving BLPP, detailed in the following steps:

Fig. 2

Process of adaptive intelligent/intuitive system (AIIS)

$\text{x}={\left({x}_{1},{x}_{2},\cdots ,{x}_{{n}_{x}}\right)}^{T}$: upper level decision variable,

$\text{y}={\left({y}_{1},{y}_{2},\cdots ,{y}_{{n}_{y}}\right)}^{T}$: lower level decision variable,

F: upper level objective function,

$f$: lower level objective function,

N: quantity of new individuals.

The detailed process for generating new individuals in the context of the Human Evolutionary Model (HEM) for solving the bi-level programming problem (BLPP) is as follows:

Step 1 involves randomly generating a set of upper decision variables, represented by X = (X₁, XX_N) where the size of the set is $N$. At this stage, it is not possible to define the individual genetic representation.

Step 2 involves treating each upper-level variable as a known quantity and solving the corresponding lower-level problem using a genetic algorithm. This process yields a set of relevant solutions, denoted as Y = (y₁, y₂, _⋯, y_N) These optimal solutions from the lower level are then conveyed to the upper level for further processing.

Step 3: In this step, the genetic representation for each individual in the population is formed by combining the upper and lower decision variables. This is denoted as ${\mathbf{g}\mathbf{r}}_{i}=\left({\mathbf{x}}_{i}^{T},{\mathbf{y}}_{i}^{T}\right)=\left({x}_{i1},{x}_{i2},\cdots ,{x}_{i{n}_{x}},{y}_{i1},{y}_{i2},\cdots ,{y}_{i{n}_{y}}\right)$ for each individual $i$, where $i$ ranges from 1 to $N$, and in X and in Y represent the number of decision variables at the upper and lower levels, respectively. At this stage, a new population of size $N$ is established, where the genetic representation of each individual is expressed as $\mathbf{G}\mathbf{R}=\left({\mathbf{g}\mathbf{r}}_{1};{\mathbf{g}\mathbf{r}}_{2};\cdots ;{\mathbf{g}\mathbf{r}}_{N}\right)$. Similarly, the genetic effects for each individual are denoted as $\mathbf{G}\mathbf{E}=\left({\mathbf{g}\mathbf{e}}_{1};{\mathbf{g}\mathbf{e}}_{2};\cdots ;{\mathbf{g}\mathbf{e}}_{N}\right)$. An important part of this step is to adjust the population to ensure that each individual meets the constraints of the bi-level programming problem.

Step 4: In this step of the Human Evolutionary Model (HEM), the genetic effects for each individual in the population are randomly generated. This is represented as ${\text{ge}}_{i}=\left(g{e}_{i1},g{e}_{i2},g{e}_{i3},g{e}_{i4}\right)$ for each individual $i$. The elements of the genetic effect vector represent different attributes:

$g{r}_{i1}$: Represents the individual's gender. This is a characteristic used in the model to simulate diversity in the population, akin to biological populations.

$g{r}_{i2}$: Indicates the actual age of the individual. This factor could be used to simulate the evolution of the solutions over time and how they adapt or change as they 'age'.

$g{r}_{i3}$: Specifies the maximum age for the individual. This element might be used to determine the lifespan of a solution in the population, after which it could be replaced or removed.

4. $g{r}_{i4}$: Represents the pheromone levels of the individual. This is an interesting aspect that might be analogous to the concept of pheromones in biological systems, potentially used to influence the selection or reproduction process in the algorithm.

In the context of optimization, these genetic effects could be used to introduce various strategies for maintaining diversity, selection pressure, and other evolutionary dynamics within the algorithm, thus influencing how solutions are generated, evolved, and selected for subsequent generations.

Step 5: This step involves the evaluation of the fitness of each individual in the population within the Human Evolutionary Model (HEM) framework. The process is as follows:

a. Calculate the Objective Value: For each individual in the population, the upper-level objective function is calculated. This is denoted as $ov(i)=F\left(g{r}_{i}\right)$, where $ov(i)$ is the objective value or fitness of the i-th individual, and $F\left(g{r}_{i}\right)$ represents the calculation of the upper-level objective function using the genetic representation $g{r}_{i}$ of the individual.

b. Fitness Assessment: The fitness in this context refers to how well an individual (a solution) meets the objectives of the optimization problem. In optimization terms, this could be about minimizing costs, maximizing efficiency, or achieving the best balance of multiple factors, depending on the specifics of the upper-level objective function.

c. Formation of the Population Matrix: once the objective values for all individuals are calculated, the complete population can be represented in a matrix form, shown as $\mathbf{P}=(\mathbf{G}\mathbf{R},\mathbf{G}\mathbf{E},\mathbf{O}\mathbf{V})$. In this matrix:

GR represents the Genetic Representations of all individuals.

GE denotes the Genetic Effects for each individual.

$\mathbf{O}\mathbf{V}$ contains the Objective Values (fitness) of each individual.

$\mathbf{O}\mathbf{V}$ contains the Objective Values (fitness) of each individual.

d. Use in Evolutionary Process: The calculated fitness values play a crucial role in the evolutionary process. They are used to guide the selection of individuals for reproduction (to generate new solutions) and for survival (to decide which solutions are kept for the next generation). This process mimics natural selection, where individuals better adapted to their environment (in this case, those with better objective values) are more likely to contribute to future generations.

By completing Step 5, the algorithm effectively assesses the current state of the population in terms of how well each solution performs according to the upper-level objective function, setting the stage for the next steps in the evolutionary process.

The steps of the proposed algorithm for constructing new individuals (NewP) in the context of the HEM applied to a bi-level programming problem are outlined below. These steps detail the process from initialization to the formation of a new generation of solutions:

• Step 1: Parameter Initialization. Set the population size to $N$, establish the upper bound ($ub$) and lower bound ($lb$), and define the maximum number of generations as ${M}_{-}Gen$. Initially, set the generation count, $t$, to 0.

• Step 2: Generate an initial population, noted as ${P}^{t}$, with size $N$ using the NewP function.

• Step 3: In the t-th generation, the optimal individual, denoted as ${p}^{\text{Best}}$, is stored in a Tabu List (TL), designated as $t{t}^{t}$. This list precludes repetitive storage of the same individual. Additionally, each entry in $TL={\left\{t{t}^{t}\right\}}_{t=1}^{L}$ is assigned a radius $\rho$, symbolizing a subspace within the search domain.

• Step 4: Refresh the population denoted by ${\mathbf{P}}^{t}$. Employ both the genetic operator and the predation operator across the entire population to manage genetic influences and representations effectively.

• Step 5: Perform individual selection based on the pyramidal selection rule, choosing individuals that meet the specified criteria to form a new population.

• Step 6: Calculate and record the variance of the top-performing individual from recent generations as 'Variance'. Based on this value, determine the 'Cycling' variable. Subsequently, ascertain the number of individuals to be removed ('Delete') and generated ('Create'), utilizing the AIIS approach, guided by the 'Variance' and 'Cycling' variables.

• Step 7: Proceed to remove individuals displaying lower fitness, as dictated by the 'Delete' variable. Concurrently, generate new individuals in alignment with the 'Create' variable, employing the Tabu Search (TS) method.

• Step 8: Assess if the terminal condition has been met. If so, preserve the optimal individual, whose genetic representation constitutes the optimal solution, and the corresponding objective value represents the upper optimum. If the condition is not satisfied, revert to Step 3 for further iteration.

Building on the steps outlined previously, the workflow of the proposed algorithm is visually represented in Fig. 3.

Fig. 3

Flowchart of the proposed algorithm

In the following section, we introduce the proposed test networks and thoroughly analyze the simulation results (see Fig. 3).

Numerical results

To validate our proposed algorithm and model, we tested it on two IEEE power networks (6-bus and 24-bus) and two gas networks (4 and 10 nodes). We conducted evaluations using the Julia programming language, considering a 10-year planning horizon with a 7% interest rate. Results are presented for each system separately, providing insights into the model's applicability and effectiveness.

First integrated system

The first integrated system combines a 6-bus power network with a 4-node gas network. This configuration was selected to represent a small-scale but realistic scenario, often seen in regional grids and gas distribution systems. The chosen setup allows us to analyze the interaction between electricity and gas networks under various operational conditions and expansion strategies. By focusing on this smaller system, we can thoroughly investigate the model's behavior and effectiveness in managing integrated network operations and expansions (see Fig. 4).

Fig. 4

The first integrated system (Farokhzad Rostami et al. 2024)

The 6-bus power network includes key elements such as photovoltaic (PV) plants, wind plants, gas-fired units, and potential battery storage locations. The network is depicted with potential expansion lines shown as dotted lines.

• PV Plants: located at buses 2 and 3.

• Wind Plants: located at buses 5 and 6.

• Gas Units: located at buses 3 and 4, connected to the gas network nodes 1 and 3.

• Battery Storage: each bus in the power grid can host a battery.

The 4-node gas network supports the gas-fired units in the power network and represents a small-scale distribution network.

Nodes 1 and 3: connected to gas units in the power network.

The budget for the system development is set at $20 million. This budget constrains the extent of expansions and upgrades that can be implemented within the planning horizon. Our goal is to optimize the allocation of these funds to achieve the most effective integration and enhancement of the electricity and gas networks (Silva et al. Aug. 2000).

To demonstrate the model's effectiveness, we analyzed different scenarios, considering varying numbers of line outages in both networks:

• Case 1: no line outages in either network. This scenario serves as the baseline, showing the system's performance under ideal conditions with no disruptions.

• Case 2: one line outage in each network. This scenario introduces a single line outage in both the power and gas networks, testing the system's resilience and response to minor disruptions.

• Case 3: two line outages in each network. This scenario further challenges the system by introducing two line outages in both networks, evaluating the model’s ability to manage more severe disruptions and maintain reliable operations.

Table 2 summarizes the simulation results for the initial system, showing how the objective function value increases as the number of outages rises in both networks. Objective function values for the first three cases are $322 million, $365 million, and $518 million, respectively, revealing the impact of line outages on system performance. Decision variables vary based on outage scenarios: for instance, with one line and one pipeline outage (case 2), the plan shifts to include different lines and an upgraded battery unit. Budget adjustments also occur, with the budget increasing from $18 million to $20 million in case 1. Problem-solving time rises significantly with outages, from 12 to 21 s. These results underscore the model's performance and the influence of outages on decisions and outcomes.

Table 2 The results of the simulation in the first integrated system

Table 3 compares the impact of integrating gas-fired units into the gas network. Results show a significant reduction in the objective function compared to Table 2, emphasizing the gas network's role in their operational strategies. This difference underscores the importance of gas network modeling in electricity network expansion planning, leading to distinct solutions. Objective function values for the first through third cases are $164 million, $169 million, and $327 million, respectively, indicating the escalation due to line outages and their effect on decision-making processes.

Table 3 The results of the simulation without connecting the gas-fired units to the gas network in the first integrated system

Table 2 presents the results of the simulation for the first integrated system, highlighting the impact of different outage scenarios on the objective cost, investments, and system performance. It demonstrates how increasing complexity and number of outages (lines and pipelines) necessitate higher investments and budgets, while the system successfully avoids load shedding. These results support the findings that strategic investments in transmission lines and batteries can mitigate the effects of disruptions, maintaining system reliability.

Table 3 illustrates the simulation results when gas-fired units are not connected to the gas network, showing a different pattern in objective costs, investments, and system performance compared to the integrated system. The results indicate that, without gas network connections, the system incurs lower costs and budgets while maintaining zero load shedding. This highlights the cost efficiency and reliability of the system under these conditions, reinforcing the conclusion that strategic planning can optimize performance and cost-effectiveness in varied scenarios.

The results highlight the importance of strategic investments in infrastructure to minimize costs and ensure reliability. Even with increased disruptions (outages), the right mix of investments in transmission lines and batteries can mitigate the impacts on system costs. Outages in key infrastructure components like transmission lines and pipelines significantly increase system costs. Ensuring redundancy and resilience in these systems is critical for maintaining economic efficiency and system reliability. Investing in battery storage at strategic bus locations plays a vital role in managing disruptions and maintaining system stability without resorting to load shedding. Policy-makers should consider the findings when designing regulations and incentives for infrastructure investments. Supporting robust grid infrastructure and strategic energy storage investments can lead to long-term savings and enhanced grid stability. The contrasting results between integrated systems and those without gas-fired units connected to the gas network suggest areas for further research, such as exploring hybrid energy systems and the optimal integration of different energy sources. Overall, these findings provide valuable insights for optimizing energy systems, enhancing resilience, and guiding policy and future research.

Second integrated system

The second integrated system involves the IEEE RTS 24-bus power grid and a 10-node gas grid, analyzed using scenarios similar to the first system. Gas-fired units at buses 3, 10, and 24 are linked to gas network nodes 2, 4, and 8. Photovoltaic installations are at buses 2, 3, 8, 9, 14, 15, 20, and 21, and wind power plants at buses 5, 6, 11, 12, 17, 18, 23, and 24. Potential battery investments are considered for buses 1, 6, 7, 12, 13, 18, 19, and 24.

Table 4 summarizes the results for the second integrated system. Case 3, with two line outages in both networks, incurs the highest cost. The budgets allocated are $20 million for Case 1, $25 million for Case 2, and $50 million for Case 3. Interestingly, there's no load shedding in any case. The objective function values for Cases 1 to 3 are $1149 million, $1154 million, and $1167 million, respectively. In Case 1, lines 43, 47, 48, 49, 55, and 59 are installed, with a 50 MW battery planned for bus 20. Case 3 involves additional battery capacity due to network adjustments, resulting in a higher budget. The study demonstrates the model's effectiveness across various network sizes and underscores the impact of network disruptions on outcomes.

Table 4 The results of the simulation in the second integrated system

Third integrated system

In our analysis of the third integrated system—IEEE 118-bus power grid paired with a 20-node gas grid—different scenarios, labeled 'cases', were examined to understand its dynamics. Ten gas-fired units are strategically located at various buses in the electricity network and seamlessly connected to nodes in the gas network. PV resources and wind power plants further bolster the system's resilience and eco-friendliness. Key findings are summarized in Table 5, with the highest cost occurring in case 3, marked by simultaneous outages of two critical lines in both networks.

Table 5 The results of the simulation in the third integrated system

Table 5 assesses three scenarios in the third integrated system. In Case 1, with no outages, the cost is $16,804 million. Case 2, with one outage per network, sees costs rise to $16,887 million. Case 3, with two outages per network, incurs the highest cost of $16,969 million. These escalating costs demonstrate the financial impact of infrastructure vulnerabilities. Detailed outage locations and required investments are outlined, along with battery investments for stability. Despite the disruptions, all scenarios report zero load shedding, reflecting effective mitigation strategies. However, Case 3 requires significantly longer CPU time, highlighting computational complexities. Overall, the table illustrates the system's response to outage scenarios, emphasizing resilience and adaptability through strategic investments and load management.

Conclusion

This study introduces an innovative bi-level stochastic planning model for integrating electricity and gas networks, specifically designed to address the research questions and hypotheses concerning network expansion and resilience. Using the Human Evolutionary Model (HEM), the study effectively demonstrates the model’s capability to manage high levels of renewable energy integration and prevent load shedding in scenarios of network disruptions.

Summary of key findings

• Integrated system costs: the objective cost increases with the complexity and number of outages, highlighting the importance of strategic investments in transmission lines and battery capacities.

• Reliability: despite higher investments, the system can avoid load shedding, ensuring reliable operation.

• Cost efficiency: scenarios without connecting gas-fired units to the gas network show lower objective costs and budgets while maintaining zero load shedding.

• Model validation: the findings validate our hypotheses by demonstrating the model's effectiveness in maintaining network security and performance under challenging conditions.

• Strategic expansion: the results meet the research objectives of ensuring robust and reliable network operation through strategic transmission and storage system expansion.

Future research directions

• Sensitivity analysis: conducting comprehensive sensitivity analyses to understand the impact of various parameters on the model's performance. This includes varying the levels of renewable energy penetration, investment costs, outage probabilities, and load growth rates to assess how changes in these factors influence the overall system reliability and cost-effectiveness.

• Demand response integration: investigating the integration of demand response strategies into the planning model. This includes exploring how demand-side management can be used to enhance system flexibility and resilience, potentially reducing the need for costly investments in infrastructure and improving overall system efficiency.

• Policy implications: analyzing the policy implications of the model's findings to inform regulatory and policy frameworks. This involves evaluating how different policy measures, such as incentives for renewable energy integration and infrastructure investments, can support the model's recommendations and promote more resilient and sustainable energy networks.

• Environmental impact assessment: assessing the environmental impacts of the proposed planning strategies. This includes evaluating the reduction in greenhouse gas emissions and other environmental benefits resulting from optimized network expansion and the integration of renewable energy sources, contributing to broader sustainability goals.

• Resilience and risk management: developing methodologies to enhance the resilience of energy networks against extreme events and uncertainties. This involves incorporating risk management strategies and contingency planning into the model to ensure that the networks can withstand and recover from severe disruptions and maintain continuous operation.

These avenues can advance integrated energy system optimization, addressing critical challenges for sustainable energy systems.