Integrated energy systems (IES) are widely regarded as an effective way to improve renewable energy accommodation through the coupling of electricity, heat, and natural gas networks. In practice, however, the uncertainty of wind power generation leads to serious curtailment issues in IES. Power-to-gas (P2G) technology is capable of converting surplus electricity into natural gas, which can significantly promote renewable energy consumption. Nevertheless, the methanation process inside P2G releases a large amount of reaction heat, and this heat is often discarded, resulting in a relatively low overall energy efficiency. In addition, the operation of an energy storage battery is critical for system stability, but improper charging and discharging behavior accelerates the aging of the battery. Therefore, it is necessary to incorporate the lifetime degradation of the energy storage battery into the scheduling optimization model. In this paper, I propose a multi-objective optimal scheduling model for an IES that includes P2G with heat recovery and an energy storage battery, with operating cost, environmental cost, and energy loss as the objective functions. To solve the model, I develop an adaptive evolutionary multi-objective particle swarm optimization (AEMOPSO) algorithm. The proposed algorithm shows better convergence performance, effectively reduces the total operating cost and energy loss, improves the environmental friendliness, and simultaneously considers the health state and economic benefit of the energy storage battery.
Integrated Energy System Model
The IES investigated in this work consists of a 1000 kW wind turbine, an 850 kW combined heat and power (CHP) unit, a 100 kW diesel generator, a 250 kWh energy storage battery, a 700 kW gas boiler, a 200 kW electric boiler, and a P2G unit with heat recovery. The electric power is supplied by the wind turbine, CHP, diesel generator, energy storage battery, and the upper power grid. The heat demand is satisfied by the CHP, gas boiler, electric boiler, P2G reaction heat, and a heat storage tank. The natural gas comes from external purchases and P2G production.
CHP Model
The CHP unit consumes natural gas to produce both electric and thermal power. Let \(v_{\mathrm{CHP},g}\) be the natural gas consumption rate of the CHP, \(W_g\) be the calorific value of natural gas, and \(K_{he}\) be the heat-to-power conversion coefficient. The electric power output \(P_{\mathrm{CHP},e}\) and the thermal power output \(P_{\mathrm{CHP},h}\) are expressed as:
$$
P_{\mathrm{CHP},e} = \frac{\eta_{\mathrm{CHP},e} W_g v_{\mathrm{CHP},g}}{K_{he}}
$$
$$
P_{\mathrm{CHP},h} = \frac{\eta_{\mathrm{CHP},h} (1 – \eta_{\mathrm{CHP},e}) W_g v_{\mathrm{CHP},g}}{K_{he}}
$$
where \(\eta_{\mathrm{CHP},e}\) and \(\eta_{\mathrm{CHP},h}\) are the electrical efficiency and thermal recovery efficiency of the CHP, respectively.
Electric Boiler Model
The electric boiler converts electric power into heat. The thermal power \(P_{\mathrm{EB},h}\) generated by the electric boiler is:
$$
P_{\mathrm{EB},h} = \eta_{\mathrm{EB}} P_{\mathrm{EB},in}
$$
where \(\eta_{\mathrm{EB}}\) is the conversion efficiency of the electric boiler, and \(P_{\mathrm{EB},in}\) is the electric power consumed by the boiler.
P2G Model
The P2G unit consumes surplus electric power to produce natural gas (methane). The methane production rate \(v_{\mathrm{CH4}}^{\mathrm{P2G}}\) and the reaction heat supply \(P_{\mathrm{P2G},h}\) are described by:
$$
v_{\mathrm{CH4}}^{\mathrm{P2G}} = \eta_{\mathrm{CH4}} P_{\mathrm{P2G}}
$$
$$
P_{\mathrm{P2G},h} = \mu_{\mathrm{P2G},h} P_{\mathrm{P2G}}
$$
where \(\eta_{\mathrm{CH4}}\) is the conversion coefficient of P2G for natural gas production, \(\mu_{\mathrm{P2G},h}\) is the heat production efficiency of P2G, and \(P_{\mathrm{P2G}}\) is the electric power consumed by P2G.
Energy Storage Battery Model
The state of charge (SOC) of the energy storage battery at time step \(i\) with a time interval \(\Delta t\) is updated according to the charging/discharging power \(P_{\mathrm{BT}}^i\). A positive value of \(P_{\mathrm{BT}}^i\) indicates discharging, a negative value indicates charging, and zero indicates idle status. The SOC dynamics are given by:
$$
SOC^i = \begin{cases}
SOC^{i-1} – \dfrac{P_{\mathrm{BT}}^i \Delta t}{\eta_{\mathrm{BT}}}, & P_{\mathrm{BT}}^i > 0 \\[6pt]
SOC^{i-1} – \eta_{\mathrm{BT}} P_{\mathrm{BT}}^i \Delta t, & P_{\mathrm{BT}}^i \le 0
\end{cases}
$$
where \(\eta_{\mathrm{BT}}\) is the charge/discharge efficiency of the energy storage battery. During regular operation, the energy storage battery does not incur a direct operating cost, but each discharge cycle consumes part of its lifetime. Deeper discharge usually accelerates the degradation of the energy storage battery. Based on the initial investment cost \(C_{\mathrm{BT},cost}\), the rated total discharge energy \(Q_r\), and the actual discharge energy \(Q_{act}\), the lifetime loss cost \(C_{\mathrm{BT},dis}\) for each discharge action is:
$$
C_{\mathrm{BT},dis} = C_{\mathrm{BT},cost} \frac{Q_{act}}{Q_r} \frac{1 – S_{SOC}/D_r}{\mu_1 e^{\mu_2(1 – S_{SOC}/D_r)} – 1}
$$
$$
Q_{act} = \max(P_{\mathrm{BT}}, 0) \Delta t
$$
where \(S_{SOC}\) is the state of charge expressed as a percentage, \(D_r\) is the rated depth of discharge, and \(\mu_1\), \(\mu_2\) are current fitting coefficients. This formulation captures the nonlinear impact of discharge depth on the health of the energy storage battery.
Gas Boiler and Heat Storage Tank Model
The thermal power output of the gas boiler \(P_{\mathrm{GB},h}\) is analogous to the CHP model. The heat storage tank state \(H_{HST}^i\) is modeled similarly to the energy storage battery SOC dynamics, with a charging and discharging efficiency \(\eta_{HST}\). These components provide additional flexibility for the heat supply and contribute to the overall economic performance of the IES.
Multi-Objective Optimization Scheduling Model
Objective Functions
The first objective is the operating cost \(F_1\), which includes the energy purchase cost \(C_{buy}\), the maintenance cost \(C_m\), the energy storage battery lifetime loss cost \(C_{\mathrm{BT},dis}\), and the electricity sales revenue \(C_{sell}\):
$$
F_1 = C_{buy} + C_m + \sum_{i=1}^{I} C_{\mathrm{BT},dis}^i – C_{sell}
$$
The energy purchase cost consists of electricity, natural gas, and diesel costs:
$$
C_{buy} = \sum_{i=1}^{I} \left( C_{buy,e} P_{grid,buy}^i + C_g v_g^i + C_d P_{DE}^i \right) \Delta t
$$
The maintenance cost is:
$$
C_m = \sum_{i=1}^{I} \sum_{n=1}^{N} C_{dev,n} P_{dev,n}^i \Delta t
$$
The electricity sales revenue is:
$$
C_{sell} = \sum_{i=1}^{I} C_{sell,e} P_{grid,sell}^i \Delta t
$$
where \(C_{buy,e}\) and \(C_{sell,e}\) are the electricity purchase and sale prices, respectively; \(P_{grid,buy}^i\) and \(P_{grid,sell}^i\) are the purchased and sold electric powers at time \(i\); \(C_g\) is the natural gas price; \(C_d\) is the diesel price; \(v_g^i\) is the purchased natural gas volume; \(P_{DE}^i\) is the diesel generator output; \(C_{dev,n}\) is the maintenance cost coefficient of the \(n\)-th device; and \(P_{dev,n}^i\) is the power of the \(n\)-th device at time \(i\).
The second objective is the environmental cost \(F_2\), mainly considering the treatment costs of CO\(_2\), nitrogen oxides, sulfur compounds, and dust emitted by the CHP, gas boiler, and diesel generator. The P2G unit absorbs CO\(_2\) and thus reduces the environmental cost. The environmental cost is:
$$
F_2 = \sum_{i=1}^{I} \sum_{j=1}^{J} C_{pol,j} \left( \sum_{n=1}^{N} \varepsilon_{nj} P_{dev,n}^i – \alpha_{CO2} P_{\mathrm{P2G}}^i \right) \Delta t
$$
where \(C_{pol,j}\) is the unit treatment cost of pollutant \(j\), \(\varepsilon_{nj}\) is the emission coefficient of pollutant \(j\) for device \(n\), and \(\alpha_{CO2}\) is the reduction coefficient of pollution treatment cost due to CO\(_2\) absorption by P2G.
The third objective is the energy loss \(F_3\), which includes the equipment energy loss \(L_{dev}\) and the wind curtailment amount \(L_{WT}\):
$$
F_3 = L_{dev} + L_{WT}
$$
$$
L_{dev} = \sum_{i=1}^{I} \sum_{n=1}^{N} \gamma_{loss,n} P_{dev,n}^i \Delta t
$$
$$
L_{WT} = \sum_{i=1}^{I} \left( P_{WT}^i – P_{WT,act}^i \right) \Delta t
$$
where \(\gamma_{loss,n}\) is the energy loss rate of device \(n\), \(P_{WT}^i\) is the forecast wind power at time \(i\), and \(P_{WT,act}^i\) is the actual utilized wind power at time \(i\).
Constraints
The optimization model is subject to device power limits, ramping limits, and power balance constraints. The device power and ramping constraints are:
$$
P_{dev,min} \le P_{dev} \le P_{dev,max}
$$
$$
P_{dev,low} \le P_{dev}^i – P_{dev}^{i-1} \le P_{dev,up}
$$
where \(P_{dev,min}\) and \(P_{dev,max}\) are the lower and upper bounds of each device power or the power exchange with the upper grid, and \(P_{dev,low}\) and \(P_{dev,up}\) are the lower and upper ramping limits. The electricity, heat, and gas power balance constraints are:
$$
\sum_{n=1}^{N} P_{dev,e,n} + P_{grid,buy} + P_{BT} = P_{load,e} + P_{P2G} + P_{EB,in} + P_{grid,sell}
$$
$$
P_{GB,h} + P_{CHP,h} + P_{EB,h} + P_{P2G,h} + P_{HST} = P_{load,h}
$$
$$
v_g + v_{\mathrm{CH4}}^{\mathrm{P2G}} = v_{CHP,g} + v_{GB,g} + v_{load,g}
$$
where \(P_{dev,e,n}\) is the electric output of generator \(n\), \(P_{BT}\) is the charging/discharging power of the energy storage battery, \(P_{HST}\) is the heat storage power, \(P_{load,e}\) and \(P_{load,h}\) are the electric and heat loads, \(v_{GB,g}\) is the natural gas consumption of the gas boiler, and \(v_{load,g}\) is the gas load.
Adaptive Evolutionary Multi-Objective Particle Swarm Optimization
The multi-objective particle swarm optimization (MOPSO) algorithm is widely used for scheduling problems because it can handle multiple conflicting objectives and produce a Pareto front. However, MOPSO usually suffers from slow convergence and premature stagnation, especially in high-dimensional coupled IES scheduling tasks. To overcome these drawbacks, I propose the AEMOPSO algorithm with adaptive inertia weight and adaptive mutation.
Adaptive Inertia Weight
I introduce a social status coefficient \(S_a^k\) for particle \(a\) at iteration \(k\), measuring the distance between the current particle and the global best solution \(g\):
$$
S_a^k = \frac{1}{D} \sum_{d=1}^{D} \frac{g_d^k – x_{ad}^k}{x_{ad,max} – x_{ad,min}}
$$
where \(D\) is the particle dimension, \(x_{ad}^k\) is the \(d\)-th decision variable of the \(a\)-th particle at iteration \(k\), and \(x_{ad,min}\), \(x_{ad,max}\) are the lower and upper bounds of that variable. Based on this coefficient, the inertia weight \(\omega\) is adaptively adjusted to balance exploration and exploitation:
$$
\omega_a^k = \omega_{max} – \left( \omega_{max} – \omega_{min} \right) \left( S_a^k – 1 \right)^2
$$
where \(\omega_{min}\) and \(\omega_{max}\) are the minimum and maximum inertia weights. A smaller \(\omega\) enhances local search, while a larger \(\omega\) promotes global search.
Adaptive Mutation
To avoid trapping in local optima, an adaptive mutation operator is introduced. The mutation probability \(p_m^k\) at iteration \(k\) is calculated from the mutation coefficient \(m_u\) and the maximum iteration number \(k_{max}\):
$$
p_m^k = 1 – \left( \frac{k-1}{k_{max}-1} \right)^{\frac{1}{m_u}}
$$
During each particle update, a random number \(r\) is generated. If \(r < p_m^k\), \(M\) random variables of a randomly selected particle are mutated. For a selected variable \(x_{ad}\), the mutation interval is defined as \((x_{ad} – \Delta x, x_{ad} + \Delta x)\), where
$$
\Delta x = p_m^k \left( x_{ad,max} – x_{ad,min} \right)
$$
and the new value \(x_{ad}’\) is randomly selected within this interval, provided the interval lies inside the original bounds. The mutated solution is accepted only if it is feasible and dominates the original particle; otherwise, the mutation is discarded. This mechanism improves the diversity of the swarm and enables the algorithm to escape from local Pareto fronts.
Performance Evaluation
I tested the proposed AEMOPSO algorithm on two classic optimization test functions: the Rastrigin function and the Ackley function. The dimension is set to 20, the population size is 500, and the maximum number of iterations is 100. For comparison, I also implemented the standard MOPSO algorithm and the NSGA-II algorithm. The convergence characteristics are compared in the following visual representation:
The results indicate that AEMOPSO converges faster than MOPSO and NSGA-II on both test functions and is able to achieve a lower fitness value. The adaptive mutation mechanism helps the swarm escape from local optima, while the adaptive inertia weight improves the convergence speed. Therefore, AEMOPSO is more suitable for solving the complex multi-objective scheduling problem of the IES considering the energy storage battery lifetime.
Case Study
To validate the effectiveness of the proposed model and algorithm, I conduct a case study on a typical day in an industrial park. The forecast wind power and the load curves are divided into 24 time nodes. Four scenarios are designed and compared.
| Scenario | P2G | P2G heat recovery | Battery lifetime cost |
|---|---|---|---|
| S1 | No | No | No |
| S2 | Yes | No | No |
| S3 | Yes | Yes | No |
| S4 | Yes | Yes | Yes |
The main device capacities used in the simulation are summarized in Table 2.
| Device | Capacity |
|---|---|
| Wind turbine | 1000 kW |
| CHP unit | 850 kW |
| Diesel generator | 100 kW |
| Energy storage battery | 250 kWh |
| Gas boiler | 700 kW |
| Electric boiler | 200 kW |
All scenarios are solved by the AEMOPSO algorithm, and the compromise solution on the Pareto front is selected for comparison. The costs and energy losses for the four scenarios are reported in Table 3.
| Item | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 |
|---|---|---|---|---|
| Operation and maintenance cost (CNY) | 1515.5 | 1602.1 | 1591.6 | 1595.8 |
| Energy purchase cost (CNY) | 6639.5 | 6395.7 | 6280.4 | 6289.6 |
| Pollution treatment cost (CNY) | 816.5 | 769.7 | 746.8 | 745.6 |
| Battery degradation cost (CNY) | 97.1 | 90.9 | 91.7 | 58.8 |
| Total cost (CNY) | 9068.6 | 8858.4 | 8710.5 | 8689.8 |
| Energy loss (kWh) | 2270.9 | 1628.6 | 1425.3 | 1411.7 |
Impact of P2G Integration
By comparing Scenario 1 and Scenario 2, I observe that the introduction of P2G reduces the total operating cost by 2.3%, even though the maintenance cost increases slightly. The wind power consumption rate rises from 94% to 99.8%, and the energy loss decreases by 28.3%. The reason is that P2G converts surplus wind power into natural gas, which not only reduces wind curtailment but also lowers the natural gas purchase cost. In addition, P2G absorbs CO\(_2\) during the methanation process, which reduces the pollution treatment cost and improves the environmental performance of the IES.
Impact of P2G Heat Recovery
In Scenario 3, the reaction heat of P2G is recovered and supplied to the heating system. Compared with Scenario 2, the total cost decreases by 1.7% and the energy loss decreases by 12.5%. The heat recovered reaches 192.5 kWh, and the energy utilization efficiency of P2G increases by about 12%. This improvement mitigates the low-efficiency drawback of P2G and relieves the heat supply pressure of the gas boiler. The gas boiler output is significantly reduced in the early morning and late evening hours, resulting in a reduction of 20.3 m³ of natural gas consumption. Therefore, P2G heat recovery offers both economic and environmental benefits.
Impact of Energy Storage Battery Lifetime Cost
In Scenario 4, the lifetime loss cost of the energy storage battery is included in the objective function. Compared with Scenario 1, the total cost is reduced by 4.2% and the energy loss is reduced by 37.8%. More importantly, the consideration of battery degradation changes the operating pattern of the energy storage battery. In Scenario 3, where the battery lifetime cost is ignored, the system prefers to discharge the energy storage battery deeply, and the depth of discharge reaches almost 100% during the discharge period. This behavior accelerates the aging of the energy storage battery. When the battery lifetime cost is considered in Scenario 4, the system controls the discharge process in a health-conscious way. The average depth of discharge is reduced by 15.4%, and the battery degradation cost decreases by 35.9%. The comparison of the energy storage battery state of charge between Scenario 3 and Scenario 4 is summarized in Table 4.
| Indicator | Scenario 3 | Scenario 4 | Reduction |
|---|---|---|---|
| Average depth of discharge (%) | 100.0 | 84.6 | 15.4% |
| Battery degradation cost (CNY) | 91.7 | 58.8 | 35.9% |
From Table 4, it is clear that incorporating the lifetime loss cost of the energy storage battery into the scheduling framework effectively prevents excessive discharge and extends the service life of the battery. Meanwhile, the system still maintains a high level of economic performance because the reduced battery degradation cost outweighs the slightly higher energy purchase cost caused by more conservative battery utilization. The total cost of Scenario 4 is the lowest among all scenarios, which demonstrates that the proposed model achieves a trade-off between battery health and operational economy.
Conclusion
In this paper, I presented an optimal scheduling strategy for an integrated energy system with P2G heat recovery and an energy storage battery. The multi-objective optimization model simultaneously minimizes operating cost, environmental cost, and energy loss, while explicitly accounting for the lifetime degradation of the energy storage battery. I proposed an adaptive evolutionary multi-objective particle swarm optimization algorithm to solve the model. The algorithm employs adaptive inertia weight and adaptive mutation mechanisms to balance global and local search capabilities. Performance tests on standard benchmark functions show that AEMOPSO converges faster and avoids premature stagnation compared with MOPSO and NSGA-II. The case study results demonstrate that the proposed scheduling model effectively reduces wind curtailment and associated energy losses. P2G with heat recovery significantly improves energy efficiency and decreases the total operating cost. In particular, the consideration of the energy storage battery lifetime cost reduces the average depth of discharge by 15.4% and the degradation cost by 35.9%, thereby improving the battery health and long-term economic benefits. The proposed methodology provides a practical solution for integrated energy systems that incorporate P2G, heat recovery, and energy storage battery management.