In modern energy systems, the integration of renewable energy sources has become paramount for sustainability. However, the inherent uncertainty of wind and solar power often leads to significant curtailment issues, compromising system efficiency. To address this, Power-to-Gas (P2G) technology has emerged as a promising solution, converting surplus electricity into natural gas and enhancing renewable energy utilization. Despite its benefits, P2G suffers from low energy efficiency due to the loss of reaction heat during methanation. Concurrently, the widespread adoption of energy storage batteries in Integrated Energy Systems (IES) improves stability, but improper usage accelerates lifespan degradation, raising economic and operational concerns. Traditional optimization approaches often neglect these aspects, such as the impact of discharge depth on battery life and the associated degradation costs. Moreover, the complex, multi-variable, and strongly coupled nature of IES poses challenges for solution algorithms, necessitating improved strategies. In this article, I explore an IES framework that incorporates P2G heating and explicitly accounts for energy storage battery lifespan degradation. A multi-objective optimization scheduling model is developed, minimizing operational costs, environmental costs, and energy losses. To solve this model, I propose an Adaptive Evolutionary Multi-Objective Particle Swarm Optimization (AEMOPSO) algorithm, which enhances convergence and avoids local optima. Through comparative analysis, I demonstrate the effectiveness of the proposed approach in reducing total costs, improving energy efficiency, and balancing battery health with economic benefits.
The IES under study comprises various energy generation and conversion devices. Electrical power is primarily supplied by a 1000 kW wind turbine, an 850 kW Combined Heat and Power (CHP) system, a 100 kW diesel generator, and a 250 kWh energy storage battery, with additional power drawn from the main grid. Thermal energy is generated by a 700 kW gas boiler, a 200 kW electric boiler, and the CHP system. Excess electricity is utilized by the P2G unit to produce natural gas, while its reaction heat is recovered for heating. Natural gas is sourced from external purchases and P2G production. This integrated setup allows for synergistic energy flows, but its optimization requires detailed modeling of each component.
Modeling of Integrated Energy System Components
The CHP system simultaneously generates electricity and heat. Its electrical output $P_{\text{CHP},e}$ and thermal output $P_{\text{CHP},h}$ are governed by the following equations:
$$P_{\text{CHP},e} = \eta_{\text{CHP},e} W_g v_{\text{CHP},g} / K_{\text{he}}$$
$$P_{\text{CHP},h} = \eta_{\text{CHP},h} (1 – \eta_{\text{CHP},e}) W_g v_{\text{CHP},g} / K_{\text{he}}$$
Here, $\eta_{\text{CHP},e}$ and $\eta_{\text{CHP},h}$ denote the electrical efficiency and heat recovery efficiency, respectively. $W_g$ is the calorific value of natural gas, $v_{\text{CHP},g}$ is the natural gas flow rate consumed by the CHP, and $K_{\text{he}}$ is the heat-to-electricity conversion coefficient. This model captures the cogeneration特性, enabling efficient energy use.
The electric boiler converts electrical power into heat with efficiency $\eta_{\text{EB}}$. Its thermal output $P_{\text{EB},h}$ is given by:
$$P_{\text{EB},h} = \eta_{\text{EB}} P_{\text{EB},\text{in}}$$
where $P_{\text{EB},\text{in}}$ is the electrical power input. This device provides flexibility in meeting thermal demands using electricity, especially during periods of renewable abundance.
The P2G unit is crucial for renewable energy absorption. It consumes electrical power $P_{\text{P2G}}$ to produce natural gas and reaction heat. The gas production flow rate $v_{\text{P2G},\text{CH}_4}$ and thermal output $P_{\text{P2G},h}$ are expressed as:
$$v_{\text{P2G},\text{CH}_4} = \eta_{\text{CH}_4} P_{\text{P2G}}$$
$$P_{\text{P2G},h} = \mu_{\text{P2G},h} P_{\text{P2G}}$$
Here, $\eta_{\text{CH}_4}$ is the conversion coefficient for methane production, and $\mu_{\text{P2G},h}$ is the heating efficiency. By recovering reaction heat, the overall energy efficiency of P2G improves, addressing one of its key limitations.
The energy storage battery plays a pivotal role in balancing supply and demand. Its state of charge (SOC) at time step $i$ with interval $\Delta t$ is updated as:
$$
\text{SOC}^i =
\begin{cases}
\text{SOC}^{i-1} – P_{\text{BT}}^i \Delta t / \eta_{\text{BT}}, & P_{\text{BT}}^i > 0 \\
\text{SOC}^{i-1} – \eta_{\text{BT}} P_{\text{BT}}^i \Delta t, & P_{\text{BT}}^i \leq 0
\end{cases}
$$
where $P_{\text{BT}}^i$ is the charging (negative) or discharging (positive) power, and $\eta_{\text{BT}}$ is the charge-discharge efficiency. The energy storage battery’s lifespan is highly sensitive to usage patterns, particularly discharge depth. Each discharge cycle incurs degradation, which accumulates over time. To quantify this, the lifespan degradation cost $C_{\text{BT},\text{dis}}$ for each discharge is derived from the initial investment cost $C_{\text{BT},\text{cost}}$, rated total discharge capacity $Q_r$, and actual discharge amount $Q_{\text{act}}$:
$$C_{\text{BT},\text{dis}} = C_{\text{BT},\text{cost}} \frac{Q_{\text{act}}}{Q_r} \left(1 – \frac{S_{\text{SOC}}}{D_r}\right)^{\mu_1} e^{\mu_2 \left(1 – \frac{S_{\text{SOC}}}{D_r} – 1\right)}$$
$$Q_{\text{act}} = \max(P_{\text{BT}}, 0) \Delta t$$
In these equations, $S_{\text{SOC}}$ is the SOC percentage, $D_r$ is the rated discharge depth, and $\mu_1$, $\mu_2$ are current fitting coefficients. This model emphasizes that deeper discharges accelerate battery aging, thus incorporating lifespan considerations directly into economic dispatch. The energy storage battery’s health is critical for long-term system reliability and cost-effectiveness.
The gas boiler and heat storage tank are modeled similarly to the CHP and energy storage battery, respectively. The gas boiler’s thermal output $P_{\text{GB},h}$ depends on gas consumption and efficiency, while the heat storage tank’s state $H_{\text{HST}}^i$ follows energy balance principles. These components add flexibility to thermal management, allowing for decoupled heat production and consumption.
Multi-Objective Optimization Scheduling Model
The optimization aims to balance economic, environmental, and efficiency objectives. The decision variables include power outputs of all devices, grid interactions, and charge-discharge schedules for the energy storage battery.
Objective Functions
The first objective, operational cost $F_1$, encompasses energy purchase costs $C_{\text{buy}}$, maintenance costs $C_m$, energy storage battery lifespan degradation costs $\sum_{i=1}^I C_{\text{BT},\text{dis}}^i$, and revenue from electricity sales $C_{\text{sell}}$:
$$F_1 = C_{\text{buy}} + C_m + \sum_{i=1}^I C_{\text{BT},\text{dis}}^i – C_{\text{sell}}$$
where:
$$C_{\text{buy}} = \sum_{i=1}^I (C_{\text{buy},e} P_{\text{grid},\text{buy}}^i + C_g v_g^i + C_d P_{\text{DE}}^i) \Delta t$$
$$C_m = \sum_{i=1}^I \sum_{n=1}^N C_{\text{dev},n} P_{\text{dev},n}^i \Delta t$$
$$C_{\text{sell}} = \sum_{i=1}^I C_{\text{sell},e} P_{\text{grid},\text{sell}}^i \Delta t$$
Here, $C_{\text{buy},e}$ and $C_{\text{sell},e}$ are electricity purchase and selling prices, $P_{\text{grid},\text{buy}}^i$ and $P_{\text{grid},\text{sell}}^i$ are power bought and sold from the grid, $C_g$ is natural gas price, $C_d$ is diesel price, $v_g^i$ is purchased gas flow, $P_{\text{DE}}^i$ is diesel generator output, $C_{\text{dev},n}$ is maintenance cost coefficient for device $n$, $P_{\text{dev},n}^i$ is power of device $n$, $I$ is number of time steps, and $N$ is number of device types. The inclusion of energy storage battery degradation cost ensures that battery health is factored into economic decisions.
The second objective, environmental cost $F_2$, accounts for pollution treatment costs from CO$_2$, nitrogen oxides, sulfur oxides, and particulate matter emitted by CHP, gas boiler, and diesel generator, offset by CO$_2$ absorption from P2G:
$$F_2 = \sum_{i=1}^I \sum_{j=1}^J C_{\text{pol},j} \sum_{n=1}^N \varepsilon_{nj} P_{\text{dev},n}^i \Delta t – \alpha_{\text{CO}_2} P_{\text{P2G}}^i \Delta t$$
$C_{\text{pol},j}$ is unit treatment cost for pollutant $j$, $\varepsilon_{nj}$ is emission coefficient of device $n$ for pollutant $j$, and $\alpha_{\text{CO}_2}$ is a coefficient reflecting reduced treatment cost due to P2G’s CO$_2$ absorption.
The third objective, energy loss $F_3$, includes device energy losses $L_{\text{dev}}$ and wind curtailment $L_{\text{WT}}$:
$$F_3 = L_{\text{dev}} + L_{\text{WT}}$$
$$L_{\text{dev}} = \sum_{i=1}^I \sum_{n=1}^N \gamma_{\text{loss},n} P_{\text{dev},n}^i \Delta t$$
$$L_{\text{WT}} = \sum_{i=1}^I (P_{\text{WT}}^i – P_{\text{WT},\text{act}}^i) \Delta t$$
$\gamma_{\text{loss},n}$ is energy loss rate of device $n$, $P_{\text{WT}}^i$ is available wind power, and $P_{\text{WT},\text{act}}^i$ is utilized wind power. Minimizing $F_3$ promotes efficient energy use and renewable integration.
Constraints
The optimization is subject to various operational constraints:
1. Device and grid power limits: For each device, power outputs must remain within bounds, and ramping rates are constrained:
$$P_{\text{dev},\min} \leq P_{\text{dev}} \leq P_{\text{dev},\max}$$
$$P_{\text{dev},\text{low}} \leq P_{\text{dev}}^i – P_{\text{dev}}^{i-1} \leq P_{\text{dev},\text{up}}$$
2. Energy balance constraints: Electrical, thermal, and gas power must balance at each time step:
$$\sum_{n=1}^N P_{\text{dev},e,n} + P_{\text{grid},\text{buy}} + P_{\text{BT}} = P_{\text{load},e} + P_{\text{P2G}} + P_{\text{EB},\text{in}} + P_{\text{grid},\text{sell}}$$
$$P_{\text{GB},h} + P_{\text{CHP},h} + P_{\text{EB},h} + P_{\text{P2G},h} + P_{\text{HST}} = P_{\text{load},h}$$
$$v_g + v_{\text{P2G},\text{CH}_4} = v_{\text{CHP},g} + v_{\text{GB},g} + v_{\text{load},g}$$
These ensure supply-demand matching across all energy carriers.
3. Energy storage battery constraints: The SOC must stay within safe limits (e.g., 20% to 100%), and charge-discharge power is limited by battery ratings. This directly impacts the energy storage battery’s lifespan and performance.
Adaptive Evolutionary Multi-Objective Particle Swarm Optimization Algorithm
Solving the multi-objective optimization model requires an efficient algorithm capable of handling non-linearity and multiple competing objectives. Traditional Multi-Objective Particle Swarm Optimization (MOPSO) often suffers from slow convergence and premature stagnation. To overcome this, I propose the AEMOPSO algorithm, which incorporates adaptive mechanisms for inertia weight and mutation.
In particle swarm optimization, each particle represents a potential solution, updating its position based on personal and global best experiences. The velocity update equation is:
$$\mathbf{v}_a^{k+1} = \omega_a^k \mathbf{v}_a^k + c_1 r_1 (\mathbf{p}_{\text{best},a} – \mathbf{x}_a^k) + c_2 r_2 (\mathbf{g}_{\text{best}} – \mathbf{x}_a^k)$$
where $\mathbf{v}_a^k$ and $\mathbf{x}_a^k$ are velocity and position of particle $a$ at iteration $k$, $\omega_a^k$ is inertia weight, $c_1$ and $c_2$ are acceleration coefficients, $r_1$ and $r_2$ are random numbers, $\mathbf{p}_{\text{best},a}$ is personal best position, and $\mathbf{g}_{\text{best}}$ is global best position. In AEMOPSO, $\omega_a^k$ is adjusted adaptively based on the particle’s social status coefficient $S_a^k$, which measures distance to the global best:
$$S_a^k = \frac{1}{D} \sum_{d=1}^D \frac{|g_d^k – x_{ad}^k|}{x_{ad,\max} – x_{ad,\min}}$$
Here, $D$ is problem dimension, $g_d^k$ is $d$-th dimension of global best, $x_{ad}^k$ is particle’s position, and $(x_{ad,\min}, x_{ad,\max})$ is variable range. The inertia weight is then updated as:
$$\omega_a^k = \omega_{\max} – (\omega_{\max} – \omega_{\min}) (S_a^k – 1)^2$$
with $\omega_{\min}$ and $\omega_{\max}$ as bounds. This adaption allows particles farther from the optimum to explore more broadly, while those closer refine search.
To enhance diversity and avoid local optima, adaptive mutation is introduced. The mutation rate $p_m^k$ at iteration $k$ is:
$$p_m^k = 1 – \left( \frac{k – 1}{k_{\max} – 1} \right)^{\frac{1}{mu}}$$
where $k_{\max}$ is maximum iterations, and $mu$ is a mutation coefficient. For each particle, if a random number $r < p_m^k$, $M$ variables are randomly selected for mutation. A variable $x_{ad}$ is mutated to $x_{ad}’$ within a range $[x_{ad} – \Delta x, x_{ad} + \Delta x]$, where $\Delta x = p_m^k (x_{ad,\max} – x_{ad,\min})$. The new position is accepted only if it is feasible and dominates the original. This mechanism maintains population diversity and improves convergence to the Pareto front.
I tested AEMOPSO on benchmark functions like Rastrigin and Ackley (20-dimensional, population 500, 100 iterations). Compared to NSGA-II and standard MOPSO, AEMOPSO demonstrated superior convergence and ability to escape local optima, validating its efficacy for complex IES optimization.
Case Study and Results Analysis
The proposed model and algorithm are applied to a park-level IES with typical daily profiles for wind power and electrical, thermal, and gas loads, as shown in the following table summarizing key parameters:
| Component | Parameter | Value |
|---|---|---|
| Wind Turbine | Rated Power | 1000 kW |
| CHP | Electrical Efficiency ($\eta_{\text{CHP},e}$) | 0.35 |
| CHP | Heat Recovery Efficiency ($\eta_{\text{CHP},h}$) | 0.45 |
| P2G | Gas Conversion Coefficient ($\eta_{\text{CH}_4}$) | 0.6 m³/kWh |
| P2G | Heating Efficiency ($\mu_{\text{P2G},h}$) | 0.12 |
| Energy Storage Battery | Capacity | 250 kWh |
| Energy Storage Battery | Charge-Discharge Efficiency ($\eta_{\text{BT}}$) | 0.95 |
| Energy Storage Battery | Rated Discharge Depth ($D_r$) | 80% |
| Energy Storage Battery | Initial Cost ($C_{\text{BT},\text{cost}}$) | $50,000 |
| Gas Boiler | Efficiency | 0.85 |
| Electric Boiler | Efficiency ($\eta_{\text{EB}}$) | 0.95 |
Four scenarios are defined to evaluate the impact of P2G heating and energy storage battery lifespan consideration:
- Scenario 1: Baseline without P2G.
- Scenario 2: With P2G but no reaction heat recovery.
- Scenario 3: With P2G and reaction heat used for heating.
- Scenario 4: Scenario 3 plus explicit inclusion of energy storage battery lifespan degradation cost.
Each scenario is optimized using AEMOPSO over 24 time steps (hourly intervals). The algorithm parameters are set as: population size 100, iterations 200, $\omega_{\min}=0.4$, $\omega_{\max}=0.9$, $c_1=c_2=2.0$, $mu=5$, $M=3$. From the Pareto-optimal solution set, a compromise solution is selected based on normalized objective weights (e.g., equal weighting) for comparison.
The results for costs and energy losses across scenarios are summarized below:
| Metric | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 |
|---|---|---|---|---|
| Operational Cost ($F_1$) in $ | 9,068.6 | 8,858.4 | 8,710.5 | 8,689.8 |
| Environmental Cost ($F_2$) in $ | 816.5 | 769.7 | 746.8 | 745.6 |
| Energy Loss ($F_3$) in kWh | 2,270.9 | 1,628.6 | 1,425.3 | 1,411.7 |
| Wind Curtailment (kWh) | 136.2 | 3.2 | 2.8 | 2.7 |
| Energy Storage Battery Degradation Cost ($) | 97.1 | 90.9 | 91.7 | 58.8 |
| Average Discharge Depth of Energy Storage Battery | 85% | 82% | 80% | 68% |
Analyzing these outcomes reveals several insights. In Scenario 2, introducing P2G reduces total cost by 2.3% and energy loss by 28.3% compared to Scenario 1, primarily due to better wind power utilization (curtailment drops from 136.2 kWh to 3.2 kWh). The P2G unit absorbs excess electricity, converting it to gas, which lowers grid purchases and pollution costs. However, without heat recovery, P2G’s energy efficiency remains suboptimal.
Scenario 3, which recovers P2G reaction heat for heating, further improves performance. Total cost decreases by 1.7% relative to Scenario 2, and energy loss is reduced by 12.5%. The recovered heat amounts to 192.5 kWh over the day, boosting P2G’s overall energy efficiency by approximately 12%. This alleviates the gas boiler’s burden, as seen in reduced gas consumption by 20.3 m³ during off-peak hours (e.g., 0-7 h and 22-24 h), translating to lower operational and environmental costs.
Scenario 4, incorporating the energy storage battery lifespan degradation cost, yields the best overall results. Compared to Scenario 1, total cost drops by 4.2%, and energy loss by 37.8%. Crucially, the energy storage battery’s average discharge depth decreases from 80% in Scenario 3 to 68%, and its degradation cost falls by 35.9% (from $91.7 to $58.8). This indicates that the optimization now intelligently manages discharge patterns to extend battery life while maintaining economic benefits. For instance, during periods of low electricity prices or high demand, the energy storage battery discharges more conservatively, avoiding deep cycles that accelerate aging. This balance between battery health and system economics underscores the value of explicitly modeling lifespan effects.
The scheduling results for P2G and gas boiler across scenarios further illustrate these dynamics. In Scenario 3, P2G operates more actively during high wind periods, producing both gas and heat. The gas boiler’s output is correspondingly reduced, especially when P2G heat is available. In Scenario 4, the energy storage battery’s SOC profile shows smoother transitions, with fewer deep discharges, compared to Scenario 3 where the battery often reaches near-full depletion. This prolongs the battery’s usable life and enhances system resilience.
Discussion on Algorithm Performance and System Implications
The AEMOPSO algorithm proved effective in solving the complex multi-objective problem. Its adaptive mechanisms allowed for faster convergence and better Pareto front coverage compared to traditional methods. In tests, AEMOPSO achieved a hypervolume improvement of 15% over MOPSO and 10% over NSGA-II for the IES problem, indicating superior diversity and convergence. This performance is critical for real-time or day-ahead scheduling where solution quality and speed are essential.
The integration of P2G heating significantly enhances system energy efficiency. By utilizing reaction heat, the overall efficiency of P2G rises from around 60% (electricity to gas) to over 70% (including heat), making it more competitive. This also reduces dependency on traditional heating sources, lowering carbon emissions. The environmental cost savings in Scenario 4 reach 8.7% compared to Scenario 1, contributing to greener operations.
The energy storage battery’s role is multifaceted. Without lifespan consideration, the battery is often exploited aggressively, leading to high degradation costs and shortened service life. By incorporating degradation costs into the objective, the optimization promotes healthier usage patterns, such as shallower discharges and optimal charging times. This not only saves costs in the long run but also ensures reliable backup power and grid support. For instance, the energy storage battery’s cycle life can be extended by up to 20% under Scenario 4’s dispatch strategy, based on typical lithium-ion battery degradation models.
Sensitivity analyses on key parameters reveal additional insights. Varying wind power uncertainty shows that the model robustly maintains low curtailment, with P2G and the energy storage battery absorbing fluctuations. Changes in energy prices affect the trade-offs between grid interaction and self-generation, but the multi-objective approach consistently balances costs and efficiency. The energy storage battery’s degradation parameters ($\mu_1$, $\mu_2$) have a notable impact; higher sensitivity to discharge depth leads to more conservative usage, further highlighting the importance of accurate battery modeling.
Conclusion
This study presents a comprehensive framework for optimizing Integrated Energy Systems with P2G heating and explicit consideration of energy storage battery lifespan. The multi-objective model simultaneously minimizes operational cost, environmental cost, and energy loss, incorporating realistic constraints and device models. The proposed AEMOPSO algorithm enhances solution quality through adaptive inertia weight and mutation, effectively handling the system’s complexity. Case study results demonstrate that integrating P2G with heat recovery reduces total costs by up to 4.2% and energy losses by 37.8%, while also cutting pollution treatment costs. Moreover, including energy storage battery degradation costs in the optimization leads to healthier battery usage, with average discharge depth reduced by 15.4% and degradation costs lowered by 35.9%. These findings underscore the value of holistic scheduling that balances economic, environmental, and technical factors. Future work could explore real-time adaptation under uncertainty, integration of other storage technologies, and scalability to larger grids. Ultimately, such approaches pave the way for more sustainable and resilient energy systems, where technologies like P2G and advanced energy storage batteries play pivotal roles in the transition to renewable energy.