Flexible dispatch strategy for electric grid integrating PEM electrolyzer for hydrogen generation

Abstract

Proton exchange membrane (PEM) electrolyzer (EL) is regarded as a promising technology for hydrogen generation, offering load flexibility for electric grids (EGs), especially those with a high penetration of renewable energy (RE) sources. This paper proposes a PEM-focused economic dispatch strategy for EG integrated with wind-electrolysis systems. Existing strategies commonly assume a constant efficiency coefficient to model the EL, while the proposed strategy incorporates a bottom-up PEM EL model characterized by a part-load efficiency curve, which accurately represents the nonlinear hydrogen production performance, capturing efficiency variations at different loads. To model this, it first establishes a 0D electrochemical model to derive the polarization curve. Next, it accounts for the hydrogen and oxygen crossover phenomena, represented by the Faraday efficiency, to correct the stack efficiency curve. Finally, it includes the power consumption of ancillary equipment to obtain the nonlinear part-load system efficiency. This strategy is validated using the PJM-5 bus test system with coal-fired generators (CFGs) and is compared with a simple EL model using constant efficiency under three scenarios. The results show that the EL modeling method significantly influences both the dispatch outcome and the economic performance. Sensitivity analyses on coal and hydrogen prices indicate that, for this case study, the proposed strategy is economically advantageous when the coal price is below 121.6 $/tonne. Additionally, the difference in total annual operating cost between using the efficiency curve anda constant efficiency to model becomes apparent when the hydrogen price ranges from 2.9 to 5.4 $/kg.

Graphical abstract

Keywords

proton exchange membrane electrolyzer (PEM EL) / nonlinear part-load efficiency / optimal dispatch strategy / hydrogen / wind energy

1 Introduction

The goal of achieving a carbon pollution-free grid has significantly accelerated the global deployment of renewable energy (RE) capacity [1]. However, due to the intermittent nature of RE, the mismatch between electricity demand and supply often leads to considerable RE curtailment. To maximize RE integration into the generation mix, a flexible and dynamic load is essential. Power-to-hydrogen systems, which follow the “electrons to chemical elements” pathway, can dynamically respond to fluctuations in renewable power by adjusting their power load [2]. Among three major types of EL, i.e., proton exchange membrane (PEM), alkaline, and solid oxide, the PEM technology is particularly well-suited for this application due to its wide load range, fast ramp rates, and quick startup times [3,4].

In recent years, several studies have focused on the optimal operation of RE-integrated multi-energy systems, in which subsystems are generally coupled by power-to-hydrogen based chemical processes. These studies vary in terms of the time horizon (e.g., 1 day, 1 month, 1 year), time resolution (e.g., 1 h, 15 min, or 10 min), energy conversion technologies (e.g., power-to-hydrogen, power-to-methane, or power-to-ammonia), and scenarios (e.g., electric grid (EG), energy hub, or off-grid community). For instance, Belderbos et al. [5] investigated the joint scheduling of electric power and natural gas systems through power-to-hydrogen and methanation processes, with a time horizon of 24 h and resolution of 1 h. They found that the Belgian gas network has a significant capacity to absorb the operational stress in the electric power system via power-to-gas technology. Xu et al. [6] developed a scheduling optimization model to determine the operational strategy for a hybrid refueling station in an off-grid region, relying solely on RE to provide hydrogen, electricity and synthetic natural gas. Clegg and Mancarella [7] analyzed the impact of power-to-gas technology on the operational strategies of electricity and gas transmission networks in terms of technical, environmental, and economic aspects. Two-stage models have also been formulated simultaneously to optimize operation and design problems [8,9]. Additionally, the other roles of power-to-X technology, such as providing spatial and temporal flexibility in redispatch [10] and decarbonizing future urban energy systems [11], have also been investigated.

Some dispatch optimization models also incorporate technical details of the electrolyzer (EL). For instance, Wei et al. [12] integrated the external characteristics of hydrogen generation in the day-ahead dispatch model to coordinate hydrogen production and heating power. Rabiee et al. [13] introduced ramp constraints for PEM ELs into a multi-period optimal power flow model for a large-scale power system, determining the hourly dispatch of generators and power-to-hydrogen units. Jiang et al. [14] considered hydrogen crossover effects when deriving the stack efficiency curve of PEM ELs, and integrated the stack efficiency curve into a scheduling optimization problem for a 20-node hydrogen network.

The above research provides valuable insights into the optimal scheduling of energy systems, with ELs functioning as an essential bonding in bridging various subsystems. However, there are several limitations in the modeling of PEM EL systems in current studies.

1) The energy conversion process from electricity to hydrogen via the EL is often simplified by assuming a constant efficiency, which fails to capture the optimization potential of the EL’s response as its efficiency varies with load. This simplified approach might lead to suboptimal dispatch results.

2) The current inefficiency caused by hydrogen and oxygen fluxes crossing the proton-exchange membrane is not considered, which leads to an overestimation of the EL’s hydrogen production and a resulting higher efficiency.

3) The power consumption of the auxiliary equipment is neglected, leading to a calculated stack efficiency higher than the actual system efficiency.

Both limitations 2) and 3) result in an underestimation of the total operating cost (TC) of the system.

A review of the existing literature on the optimal dispatch of EGs integrated with ELs has revealed that many dispatch strategies rely on simplified methods to model the EL, without properly characterizing its input-output relationship and fully exploiting its load modulation potential. In this context, the primary objective of this work is to formulate a dispatch optimization framework for a coal-wind power grid integrated with a PEM EL system, which is modeled using a real part-load efficiency curve. This paper focuses on assessing the impact of a detailed EL modeling method on the efficient utilization of wind energy and the reduction of TC. This paper contributes by developing a bottom-up PEM EL model that includes a 0D electrochemical model formulation, Faraday efficiency calculation, model calibration, and the computation of auxiliary power consumption. This model enables the derivation of part-load efficiency for any type of PEM EL, accurately capturing the energy conversion process from electricity to hydrogen under varying input power conditions. Additionally, it formulates a dispatch optimization problem for electricity-hydrogen systems to derive a flexible dispatch strategy. Special emphasis is placed on comparing the scheduling results and economic performance (i.e., marginal cost of hydrogen and TC) between a real efficiency curve derived in this paper and the most typical modeling method (i.e., constant efficiency) under different wind plant configurations and operating strategies.

2 Part-load efficiency of PEM EL

2.1 0D electrochemical model and model calibration

Under irreversible conditions, the operating voltage of an EL cell (V_op, in V) is estimated by adding three types of overvoltages to the voltage under reversible conditions (V_rev, in V) [15,16]

(1)V_{\mathrm{o}\mathrm{p}}=V_{\mathrm{r}\mathrm{e}\mathrm{v}}+V_{\mathrm{o}\mathrm{h}\mathrm{m}}+V_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}+V_{\mathrm{a}\mathrm{c}\mathrm{t}},

where V_ohm, V_conc, and V_act (in V) are the ohmic, concentration, and activation overvoltages, respectively. The derivation of the reversible voltage and three types of overvoltages are provided in the Electronic Supplementary Material (ESM).

After deriving the 0D electrochemical model, a fitting process is necessary to validate the model and determine the specific fitted parameters. This ensures that the simulated polarization curve best fits the experimental J–V curve. In this model, nine fitted parameters are selected, which are area-specific resistance related to electron transport, ASR_elec; membrane conductivity in reference condition, {\sigma }_{\mathrm{m}\mathrm{e}\mathrm{m}}^{\mathrm{r}\mathrm{e}\mathrm{f}}; activation energy required by proton transport through the membrane, E_{\mathrm{m}\mathrm{e}\mathrm{m}}^{\mathrm{o}\mathrm{h}\mathrm{m}}; charge transfer coefficient at anode and cathode, α_an and α_cat; activation energy required by the electrochemical reaction at the anode and cathode, E_{\mathrm{a}\mathrm{n}}^{\mathrm{a}\mathrm{c}\mathrm{t}} and E_{\mathrm{c}\mathrm{a}\mathrm{t}}^{\mathrm{a}\mathrm{c}\mathrm{t}}; and anodic and cathodic exchange current density at the reference temperature, J_{\mathrm{a}\mathrm{n}}^{0,\mathrm{r}\mathrm{e}\mathrm{f}} and J_{\mathrm{c}\mathrm{a}\mathrm{t}}^{0,\mathrm{r}\mathrm{e}\mathrm{f}}.

The model calibration is performed using experimental data of a Nafion 117-based membrane with a thickness of 183 μm and an area of 25 cm², collected at different temperatures (60, 70, and 80 °C) [17]. To calibrate the simulated model, the Sum of Square Error (SSE) between experimental and simulated operating voltage is minimized, which is expressed as [18]

(2){\sigma }_2=\sum _{i=1}^N{\left[V_i^{\mathrm{e}\mathrm{x}\mathrm{p}}\left(I_i,T_i,p_i\right)-V_i^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}\left(I_i,T_i,p_i,{\mathbf{v}}_{\mathrm{f}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{d}}{\mathbf{v}}_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}}\right)\right]}^2,

where V_i^{\exp } is the voltage of the ith experimental point, V_i^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}} is the simulated voltage derived from the electrochemical model under the same operating conditions of V_i^{\exp }, I (in A) is the current, v_fixed is the vector of the known fixed parameters, and v_fitted is the vector of the unknown fitted parameters.

The fixed parameters, along with the resulting fitted parameters and operating parameters for the PEM electrochemical model, are provided in ESM (Tables S1 to S3) [19]. Based on these parameters, the polarization curve (i.e., J–V curve) of the specific PEM EL cell is shown in Fig. 1(a) under the operating conditions of 60 °C and 30 bar. Additionally, the reversible voltage and the three different types of overpotentials as a function of current density are shown in Fig. 1(b).

2.2 Cell efficiency

At the cell level, efficiency is defined as the ratio of the output hydrogen power per unit cell area (in W/cm²) to the input DC power density (in W/cm²) [16]

(3){\eta }_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}=\frac{{\dot{N}}_{{\mathrm{H}}_2}\cdot \mathrm{L}\mathrm{H}\mathrm{V}}{P_{\mathrm{i}\mathrm{n},\mathrm{D}\mathrm{C}}},

where LHV (240 kJ/mol) is the lower heating value of hydrogen.

The hydrogen production rate per unit cell area (in mol/(s∙cm²)) is calculated by using the Faraday’s law [20]

(4){\dot{N}}_{{\mathrm{H}}_2}={\eta }_{\mathrm{F}}\cdot \frac{J}{2\mathrm{F}},

where F is the Faraday constant (96485 C/mol); η_F is the Faraday efficiency, which describes the deviation of hydrogen volume between actual and theoretical maximum production, computed using [16,21]

(5){\eta }_{\mathrm{F}}=1-\frac{2\cdot \mathrm{F}}{J}\cdot \left({\dot{N}}_{{\mathrm{H}}_2,\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{t}\mathrm{o}\mathrm{t}}+2{\dot{N}}_{{\mathrm{O}}_2,\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{t}\mathrm{o}\mathrm{t}}\right),

where {\dot{N}}_{{\mathrm{H}}_2,\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{t}\mathrm{o}\mathrm{t}} and {\dot{N}}_{{\mathrm{O}}_2,\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{t}\mathrm{o}\mathrm{t}} (in mol/(s∙cm²)) are the total hydrogen molar flux permeating the membrane from the cathode to the anode electrode and the total oxygen molar flux from the anode to the cathode electrode, respectively. These fluxes are both estimated by considering both the diffusive contribution driven by the concentration gradient and the convective contribution caused by the pressure gradient and electro-osmotic drag [22]. A comparison of the cell efficiency, with and without considering Faraday efficiency, is shown in Fig. 3.

2.3 Part-load efficiency

To transition from cell efficiency to part-load system efficiency, the power consumption of the necessary ancillary equipment must be considered. In this paper, the ancillaries are categorized into three contributions: power supply unit (PSU), which includes transformer and AC-DC converter; water pump; and other equipment, e.g., water purification, hydrogen purification and drying, and fans.

The input AC power density (P_in,AC,stack, in W/cm²) of the stack is calculated using [23]

(6)P_{\mathrm{i}\mathrm{n},\mathrm{A}\mathrm{C},\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}=\frac{P_{\mathrm{i}\mathrm{n},\mathrm{D}\mathrm{C},\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}}{{\eta }_{\mathrm{c}}},

where P_in,DC,stack (in W/cm²) is the input DC power density, which is calculated by multiplying the operating voltage and current density of the stack, and η_c is the conditioning efficiency which is expressed as [24]

(7)\begin{array}{rl}{\eta }_{\mathrm{c}}=& 99.3-3.14\frac{P_{\mathrm{i}\mathrm{n},\mathrm{D}\mathrm{C},\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}}{P_{\mathrm{n}\mathrm{o}\mathrm{m},\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}}-4.19{\left(\frac{P_{\mathrm{i}\mathrm{n},\mathrm{D}\mathrm{C},\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}}{P_{\mathrm{n}\mathrm{o}\mathrm{m},\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}}\right)}^2-\\ & \frac{0.96}{P_{\mathrm{i}\mathrm{n},\mathrm{D}\mathrm{C},\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}/P_{\mathrm{n}\mathrm{o}\mathrm{m},\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}},\end{array}

where P_nom,stack (in W/cm²) is the nominal power density of the stack.

The water pump provides the water for the electrochemical reaction and aids in thermal regulation for the stack. Its power consumption per unit cell area, P_wp (in W/cm²), varies nonlinearly with the operating point and is expressed as [25]

(8)P_{\mathrm{w}\mathrm{p}}=\frac{\dot{V}\cdot \mathrm{\Delta }{\mathrm{P}}_{\mathrm{w}\mathrm{p}}}{{\eta }_{\mathrm{w}\mathrm{p}}},

where ΔP_wp (in Pa) is the pump total head, η_wp is the efficiency of the water pump (which accounts for the power and mechanical inefficiency), and V̇ (in m³/(cm²∙s)) is the volumetric flow rate of water.

Within the modulation range (10%−100% of the nominal power), as the stack operates in exothermic conditions, the volumetric flow rate of water is calculated by subtracting the reversible heat from the irreversible heat associated with overvoltages [26], using

(9)\dot{V}=\zeta \cdot \frac{\left(V_{\mathrm{o}\mathrm{p}}-V_{\mathrm{t}\mathrm{n}}\right)\cdot J}{C_{\mathrm{p}}\cdot \mathrm{\Delta }\mathrm{T}},

where C_p (in J/(°C∙g)) is the specific heat capacity of water, ΔT (in °C) is the temperature change, which is stated in the range from 5 to 10 °C [24]; ζ is the unit converter from mass to volume; and V_tn (in V) is the thermoneutral voltage, which is assumed constant as it only decreases by 0.83% from 25 to 100 °C.

To sum up, the total input AC power density of the EL system (P_in,sys, in W/cm²), schematically described in Fig. 2, is calculated as [27]

(10)P_{\mathrm{i}\mathrm{n},\mathrm{s}\mathrm{y}\mathrm{s}}=P_{\mathrm{i}\mathrm{n},\mathrm{A}\mathrm{C},\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}+P_{\mathrm{w}\mathrm{p}}+P_{\mathrm{o}\mathrm{a}},

where P_oa (in W/cm²) is the power consumption per unit cell area of other auxiliaries (e.g., purification, dryer, and fans), which is assumed to be 3.5% of the rated power of the EL [28]. The parameters used to derive the power consumption of auxiliaries are provided in Table S4 in ESM.

The system efficiency is then calculated by correcting the cell efficiency equation (Eq. (3)) to account for the additional AC power consumption

(11){\eta }_{\mathrm{s}\mathrm{y}\mathrm{s}}=\frac{{\dot{N}}_{{\mathrm{H}}_2}\cdot \mathrm{L}\mathrm{H}\mathrm{V}}{P_{\mathrm{i}\mathrm{n},\mathrm{s}\mathrm{y}\mathrm{s}}}.

Figure 3 shows the system efficiency curve of the PEM EL, where the input AC power value is normalized relative to its nominal power. The resulting nominal specific energy consumption is 5.7 kWh/Nm³ (equivalent to an LHV efficiency of 52%), which falls within the range of 5.0−6.5 kWh/Nm³ reported by Butler and Spliethoff [29]. It can be observed that the EL system achieves its highest efficiency (around 0.58) at a partial load of approximately 0.38 (denoted as r^e,max). Below this partial load, the efficiency decreases rapidly due to the power consumption of the auxiliary system. Additionally, Faraday efficiency experiences a significant loss at low loads. For partial load exceeding r^e,max, the efficiency decreases as a result of the various overvoltage losses.

Overall, the part-load system efficiency curve has been derived by developing an electrochemical model of the EL cell, which includes model calibration, calculation of Faraday efficiency, and accounting for the power consumption of the auxiliary system. A comparison of the efficiency curves from cell level to the system level is shown in Fig. 3. The system efficiency curve is then approximated by a polynomial equation to be used in the optimal dispatch strategy.

3 Overall problem formulation

In this paper, an hourly cost-based dispatch model for an EG with wind-PEM EL systems is developed. Figure 4 depicts the layout of the electric power system under analysis, which consists of a 5-bus EG, 3 coal-fired generators (CFGs), a wind power plant (WPP), and a PEM EL. The bidirectional electricity flow between the wind-electrolysis system and the EG, along with the part-load characteristics of the PEM EL, enhances the scheduling flexibility of the electric power system. The hydrogen produced is sold to offset the fuel cost (FC) of the CFGs (with fuel referring to coal), thus minimizing the overall cost of the EG.

The overall structure of the proposed methodology is illustrated in Fig. 5. A detailed PEM EL model is developed by calculating a part-load system efficiency curve, which is then incorporated into the optimization model to derive dispatch strategies. The objective function of the optimization problem aims to minimize the daily TC (Section 3.1) and the constraints concerning the PEM EL, EG and WPP are presented in Section 3.2. The PEM EL model is implemented in MATLAB, while the nonlinear dispatch optimization problem is solved using GAMS.

3.1 Objective

The objective is to minimize the daily TC of the overall energy system while adhering to all technical constraints. The daily TC of the system is calculated as the FC of the three CFGs, minus the hydrogen revenue, plus the daily depreciation cost of the PEM EL (denoted as C_e).

(12)min\sum _{t=1}^{24}\left[\sum _{g=1}^3{C}_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}\cdot f\left(P_{g,t}\right)-C_{{\mathrm{H}}_2}\cdot {\dot{m}}_t\right]+C_{\mathrm{e}},

where C_coal (in $/tonne) is the coal price, f is a function to calculate the coal consumption related to the power generation of gth CFG at hour t; C_{{\mathrm{H}}_2} is the hydrogen price (in $/kg), ṁ_t (in kg/h) is the hydrogen produced at hour t and is calculated using Eq. (14); C_e (in $/day) is the daily depreciation cost of the PEM EL, which is calculated as

(13)C_{\mathrm{e}}=\frac{c_{\mathrm{i}}\cdot P^{\mathrm{e},\mathrm{n}\mathrm{o}\mathrm{m}}\cdot 1000-C_{\mathrm{r}}}{365\cdot Y}.

where c_i (in $/kW) is the specific investment cost of the PEM EL, P^e,nom (in MW) is the nominal power of the EL, and C_r (in $) is the salvage value, which is assumed to be 10% of the investment cost. The lifetime of the facility Y is assumed to be 10 years [30].

3.2 Constraints

The equality and inequality constraints for the EG and the wind-electrolysis system in the model are shown as follows:

The volume of hydrogen produced (in kg/h) at hour t is calculated using

(14){\dot{m}}_t=\frac{r_t^{\mathrm{e}}\cdot P^{\mathrm{e},\mathrm{n}\mathrm{o}\mathrm{m}}\cdot {\eta }_t^{\mathrm{e}}\cdot {\rho }_{{\mathrm{H}}_2}}{{\mathrm{L}\mathrm{H}\mathrm{V}}_{{\mathrm{H}}_2}},

where r_t^{\mathrm{e}} is the partial load factor of the EL at hour t; and {\eta }_t^{\mathrm{e}} is the EL efficiency at hour t, which is dependent on the partial load factor:

(15){\eta }_t^{\mathrm{e}}=f\left(r_t^{\mathrm{e}}\right),

where f is the polynomial function of the nonlinear EL efficiency curve.

The modulation range of the partial load factor is defined as

(16)\mathrm{m}\mathrm{p}\mathrm{l}\le r_t^{\mathrm{e}}\le 1.

where mpl is the allowable minimal partial load, assumed to be 0.1 in this paper [19]. The lower bound of the partial load is set to avoid low efficiency, instability, and safety issues during the operation of EL. Operating the EL within a certain range provides flexibility to the EG, as it can response to renewable generation and electric load variability by adjusting its input power based on the efficiency curve.

In this paper, the operation of the power system is determined using DC optimal power flow. Therefore, the nodal active power balance for each bus i is represented by

(17)\sum _{g\in {\mathrm{\Omega }}_G^i}{P}_{g,t}+P_{i,t}^{\mathrm{w}}=\sum _{j\in {\mathrm{\Omega }}_l^i}{P}_{ij,t}+L_{i,t}+P_{i,t}^{\mathrm{e}},

where P_g,t is the power generation of gth CFG at hour t, P_{i,t}^{\mathrm{w}} is the wind power integrated into the EG at hour t, L_i,t is the electric demand at hour t, P_{i,t}^{\mathrm{e}} is the power load of EL at hour t; P_{ij,t} is the active power flow of branch connecting bus i and j, which is calculated using

(18)P_{ij,t}=\frac{{\delta }_i-{\delta }_j}{x_{ij}},

where δ_i is the voltage angle of bus i and x_ij is the reactance of transmission line ij.

The operating limits and ramp rates of CFGs are defined as

(19)P_g^{min}\le P_{g,t}\le P_g^{max}.

(20)P_{g,t}-P_{g,t-1}\le R{U}_g.

(21)P_{g,t-1}-P_{g,t}\le R{D}_g.

The wind power curve of the wind turbine is fitted as [31]

(22)P_{i,t}^{\mathrm{w}\mathrm{a}}=\frac{P^{\mathrm{w},\mathrm{n}\mathrm{o}\mathrm{m}}}{1+{\mathrm{e}}^{(-(v-n)\cdot m)}},

where m and n are characteristic adjustable parameters for turbine Gamesa G90 2.0 MW [32], v is the wind velocity, P_{i,t}^{\mathrm{w}\mathrm{a}} is the available wind power, and P^w,nom is the nominal power of the wind turbine.

Wind curtailment is estimated by subtracting the integrated wind power from the available wind power, as expressed by Eq. (23). In addition, the integrated wind power and wind curtailment must not exceed the total available wind power, as constrained by Eq. (24).

(23)P_{i,t}^{\mathrm{w}\mathrm{c}}=P_{i,t}^{\mathrm{w}\mathrm{a}}-P_{i,t}^{\mathrm{w}}.

(24)0\le P_{i,t}^{\mathrm{w}},P_{i,t}^{\mathrm{w}\mathrm{c}}\le P_{i,t}^{\mathrm{w}\mathrm{a}}.

The parameters in this section are provided in Table S5 in ESM.

In summary, the optimization problem is formulated to address the day-ahead optimal scheduling of the EG with PEM EL systems, aiming to minimize the daily total TC of the EG (Eqs. (12) and (13)) while respecting the operational constraints of all components, including PEM EL (Eqs. (14) to (16)), EG (Eqs. (17) and (18)), CFGs (Eqs. (19) to (21)), and WPP (Eqs. (22) to (24)). These constraints collectively ensure the scheduling model to derive an economically optimized and technically feasible strategy for the EG while satisfying the electric demand. In particular, the part-load efficiency characteristic of the PEM EL (Eq. (15)) enables flexible adjustment of power consumption in response to variable electric load and fluctuating renewable generation. Additionally, by utilizing the part-load efficiency curve, the system achieves a lower marginal cost of hydrogen, that is, the EL produces more hydrogen per unit of electricity consumed. Accordingly, the integration of the PEM EL enhances the flexibility and efficiency of the electricity-hydrogen energy system, thereby minimizing the daily TC of the system.

4 Case studies

4.1 Test system and parameters

The day-ahead dispatch optimization is performed on a 5-node test system [33], with the one-line diagram of the network depicted in Fig. 6. The network data are taken from Li and Bo [33] with a few modifications. Three CFGs are installed at buses 1, 3, and 4, respectively, with their parameters listed in Table 1 [34] (where RU and RD represent ramp-up and ramp-down rate of generators). A 150-MW WPP and a 60-MW PEM EL are both installed at bus 5. The investment cost of the PEM EL is assumed to be 1443 $/kW [4], resulting in a depreciable cost of approximately $21307 calculated using Eq. (13). The price of coal and hydrogen are assumed to be 103.1 $/tonne [35] and 4.1 $/kg [36], respectively. The simulation has a time horizon of one day with an hourly resolution.

The wind and electric load variation within a year in a city located in northern China is represented by three typical days, characterized by normal wind speed (transition), low wind speed (summer), and high wind speed (winter). The total electric load of this test system is assumed to be 400 MW, which is distributed across buses 2, 3, and 4, with load share of 25%, 37.5%, and 37.5%, respectively, and the same load variation pattern. The data for wind speed, electric load variation (pu), and the probability of these three typical days (i.e., 0.5, 0.2, and 0.3) are taken from Gao et al. [37] and shown in Fig. 7.

To assess the economic benefits that the EL brings to the renewable EG and compare the differences between using constant efficiency (an approximation) versus a real efficiency curve, four cases are conducted and analyzed:

• Case 1: No EL is integrated, which establishes a baseline to show the performance of the EG without the inclusion of the EL.

• Case 2: The EL is modeled with a constant efficiency of 52% (the nominal efficiency when the normalized input AC power of the EL equals 1, as shown in Fig. 3). The EL operates only during periods of wind curtailment. This case reflects a common dispatch strategy where the EL is used solely to absorb curtailed wind energy, demonstrating basic benefits but limited flexibility.

• Case 3: The EL is modeled with a constant efficiency of 52% (i.e., the nominal efficiency when the normalized input AC power of EL equals 1, as shown in Fig. 3), but is not restricted to operate only during wind curtailment. This case demonstrates the additional operational flexibility of the EL, but still without reflecting actual efficiency variations under different loads.

• Case 4 (the strategy presented in this paper): The EL is modeled using a real efficiency curve, which accurately represents the input–output relationship of the EL. This case allows for more precise and cost-effective dispatch decisions, reflecting the actual efficiency variations of the EL and optimizing the integration of renewable energy.

4.2 Dispatch results and economic performance analysis

The optimal dispatch results for the four cases outlined above are presented for each typical day, with the focus on wind curtailment, the utilization factor of the EL (measured as the total hours spent in production mode), the bidirectional electricity flow between the EG and the wind-electrolysis system, and the marginal cost of hydrogen (λ_x, in $/kg). The marginal cost of hydrogen is defined as the ratio of the increased FC (in $) to the total volume of hydrogen produced (in kg) during the day, where x corresponds to Case 3 or Case 4).

(25){\lambda }_x=\frac{{\mathrm{F}\mathrm{C}|}_x-{\mathrm{F}\mathrm{C}|}_{\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e}1}}{{{\displaystyle \sum _{t=1}^{24}{\dot{m}}_t}|}_x}.

The daily TC is compared across each typical day for all four cases. Finally, the annual TC of each case is calculated by multiplying the probability of each typical day, the number of days in a year, and the corresponding daily TC. Based on these calculations, the yearly economic performance of each case is analyzed. The economic results are presented in Fig. 11.

4.2.1 Typical day 1

The optimal dispatch results for typical day 1 (transition) are shown in Figs. 8(a)−8(d). Figure 8(a) depicts the power generation from the three CFGs and the integrated wind power of the WPP in Cases 1 and 2, and the power load of the EL in Case 2. The total electric load for the day is 3963.2 MWh, with 77.7% covered by the CFGs and the remainder by the WPP. The period from hours 22 to 6 has low electricity demand and high wind generation. Constrained by the lower operating limits of the CFGs, about 368.5 MWh of wind energy (orange box) is curtailed in Case 1. In Case 2, curtailment is instead exploited to produce 5685 kg of hydrogen via the EL, which operates at a constant efficiency. Therefore, the integrated wind power in Case 2 (dark green box) is the sum of the integrated wind energy in Case 1 (light green box) and the curtailed wind power. The hydrogen revenue from this case is $23435, which offsets the daily TC by 9.6%, but barely covers the daily depreciable cost of the EL. As a result, the daily TC of Case 2 is only 0.88% lower than that of Case 1 (Fig. 11). Additionally, the utilization factor of the EL is around 37.5%, as it operates only during wind curtailment periods.

The optimal scheduling results for Cases 3 and 4 are depicted in Figs. 8(b) and 8(c), respectively. Unlike Case 2, the EL operates continuously throughout the day in both cases. In these cases, only 6 MWh of wind energy is curtailed at hour 3, as the electric load reaches its minimum, three CFGs approach their lower output limits, and the EL reaches its upper operating limit.

In Case 3 (Fig.8 (b)), the EL operates as an adjustable load. For instance, during the period from hours 6 to 13, the EL (orange box) consumes more electricity than the WPP can provide (green box), so 142.3 MWh of electricity is absorbed from the EG. During peak load intervals (hours 14 to 17), the EL consumes less power than the wind generation, allowing 71.4 MWh of wind energy to be supplied to the EG. Overall, the EG provides 175 MWh of the electricity to the EL, while 222.9 MWh of wind generation is supplied to the EG during the day. However, in Case 3, the EL operates relatively inflexibly, maintaining its maximum load for long periods, i.e., hours 20 to 24 and hours 1 to 10.

Figure 8(c) shows the optimal dispatch results for Case 4. In this case, integrating the EL into the EG reduces the wind curtailment (compared to Case 1) and increases the utilization factor of the EL (compared to Case 2). Consequently, the daily TC of Case 4 is 3.58% and 2.73% lower than Cases 1 and 2, respectively. Compared to Case 3, the electricity supplied from the EG to the EL decreases by 66.56%, while the supply of wind generation from the WPP to the EG increases by 18.94%, both of which contribute to a lower generation of CFGs and leads to a lower daily TC of Case 4 (1.28%). Moreover, the marginal cost of hydrogen in Case 4 is 2.35 $/kg, which is 12% lower than in Case 3 (2.67 $/kg). This economic improvement is due to the EL operating in higher efficiency than its nominal value over a wide range of partial loads. This is particularly advantageous during periods of high load or low wind power. As shown in Fig. 8(c), for example, at hour 15, the optimal power load of the EL is 31.6 MW (corresponding to the load factor of 0.53) with an efficiency of nearly 0.58 (Fig. 8(d)), close to the maximum efficiency of the EL.

4.2.2 Typical day 2

The optimal dispatch results for typical day 2 (summer), characterized by low wind speeds, are shown in Figs. 9(a)−9(d). The total electric load for the day is 4784.8 MWh. In both Cases 1 and 2, 82.5% of the load is covered by CFGs, while the remainder is supplied by the WPP. Due to inadequate wind energy, wind curtailment occurs only during hours 4 and 6, totaling 22.6 MWh (as shown in Fig. 9(a)). This implies that in Case 2, the EL operates for only two hours during the day. As a result, the utilization factor of the EL further decreases to 8.3%, and the total hydrogen production drops to 348.6 kg. The corresponding hydrogen revenue ($1437.9) is even lower than the daily depreciable cost of the EL. Therefore, the daily TC of Case 2 is instead 6.3% higher than that of Case 1.

In Cases 3 and 4, there is no wind energy being curtailed. In Case 3 (Fig. 9(b)), approximately 168.2 MWh of the electricity is provided to the EL from the EG, while 127.1 MWh of wind generation is supplied to the EG. In contrast, the electricity supplied from the EG to the EL in Case 4 increases by 15.5%. This can be explained by the fact that during peak hours with low wind energy, such as the interval from hours 10 to 18 as shown in Fig. 9(c), instead of decreasing the power load of the EL to the minimum limit (Case 3, as shown in Fig. 9(b)), the strategy in Case 4 instead schedules additional electricity from the EG for the EL to maintain high efficiency (greater than 0.58, as shown in Fig. 9(d)). Meanwhile, the wind generation supplied to the EG in Case 4 increases by 26.9% compared to Case 3. Consequently, the daily TC of Case 4 is 1.14% lower than Case 3. The marginal cost of hydrogen in Case 4 is 3.52 $/kg, which is 5.88% lower than in Case 3 (3.74 $/kg).

It should also be noted that on typical day 2, the best case is Case 1 (i.e., without the EL) due to the insufficient wind energy. The daily TC in Case 4 is 3.99% higher than in Case 1 and 2.38% lower than in Case 2. This indicates that high levels of wind generation are more favorable for integrating ELs into the EG.

4.2.3 Typical day 3

The optimal dispatch results on typical day 3 (winter), characterized by high wind speeds, are shown in Figs. 10(a)−10(d). The total electric load is 4808.1 MWh. In Cases 1 and 2, 62.5% of the load is covered by CFGs, with the remainder supplied by the WPP. In Case 1 (Fig. 10 (a)), the total wind curtailment increases up to 425.5 MWh during the interval from hours 20 to 6. Consequently, the utilization factor of the EL in Case 2 rises to 45.8%, and the total hydrogen production increases to 6563 kg. The hydrogen revenue ($27068.3) reduces the FC by 11.3%, leading to a 2.4% reduction in the daily TC for Case 2 compared to Case 1.

In Cases 3 and 4, approximately 20.8 MWh of wind energy is curtailed at hours 1 and 24. In Case 3 (Fig. 10(b)), the majority of the electricity consumed by the EL comes from the WPP (99.1%), with only 10.6 MWh of electricity drawn from the EG during hours 16 and 17. Meanwhile, around 1054.5 MWh of wind generation is supplied to the EG. In contrast, in Case 4, the electricity absorbed from the EG by the EL decreases by 9.4%, while the wind generation supplied to the EG increases by 6.97%. As a result, the daily TC of Case 4 is 1.09% lower than Case 3. The marginal cost of hydrogen in Case 4 is 2.27 $/kg, which is 7.3% lower than in Case 3 (2.45 $/kg). Additionally, compared to Cases 1 and 2, the daily TC in Case 4 is reduced by 4.76% and 2.48%, respectively.

4.3 Economic performance analysis

The economic results, including daily TC, fuel cost, hydrogen revenue and annual TC, for each typical day in the four cases are shown in Fig. 11. Specifically, the daily TC is calculated as the FC minus hydrogen revenue, plus the fixed depreciable cost of the EL (approximately $21307). The annual TC is computed by multiplying the probability of each typical day, the number of days in a year, and the corresponding daily TC, which is then used to assess the economic performance of the four cases over the course of a year. As discussed in Section 4.2, the daily TC is compared in detail across the four cases for each typical day. In this section, the focus is on analyzing the annual economic performance.

Among four cases, Case 2 ($928.13 × 10⁵) has the highest annual TC, while Case 4 ($904.42 × 10⁵) has the lowest. On typical days 1 and 3, the daily TC of Case 2 (second highest) is only 0.82% and 2.5% lower than that of Case 1, respectively. However, on typical day 2, the daily TC of Case 2 is significantly higher than that of Case 1 by 6.3%, leading to the highest the annual TC for Case 2. In contrast, Case 4 benefits from the modulation of the EL’s power load based on its efficiency curve, resulting in a lower daily TC on all three typical days compared to Case 3. As a result, the annual TC of Case 4 is 1.2% lower than that of Case 3. Furthermore, the economic performance of Case 4 is superior to both Cases 1 and 2, with the annual TC for these cases being 2.1% and 2.6% higher, respectively, than that of Case 4.

4.4 Sensitivity analysis

Two groups of sensitivity analyses regarding coal and hydrogen prices on the annual TC of the four cases are conducted. The sensitivity of annual TC with respect to the relative coal price (coal price divided by the base price of 103.1 $/tonne) is depicted in Fig. 12. It is observed that the advantage of Case 4 remains consistent across a wide range (from 0.4 to 1.1). Specifically, the annual TC in Case 4 is approximately $15.9 million and $16.3 million lower than that of Cases 1 and 2, when the relative coal price is 0.4 (equivalent to a coal price of 41.2 $/tonne). However, when the coal price exceeds a certain threshold, specifically 121.6 $/tonne in this paper, Case 1 (where the EL is not installed) becomes more advantageous due to the extremely high price of coal.

As constant efficiency is commonly used in the existing dispatch model, the impact of hydrogen price on the difference in annual TC between Cases 3 and 4 is explored. The results are shown in Fig. 13, in which the vertical axis is the value of the annual TC of Case 3 minus Case 4, and the horizontal axis shows the relative hydrogen price (with the base price of 4.1 $/kg). The results indicate that, in the range of 0.7 to 1.3, modeling the EL using a constant efficiency leads to an overestimation of the annual TC. This overestimation is particularly significant (approximately $1.09 million) when the relative hydrogen price is close to 1.0 (i.e., a hydrogen price of 4.1 $/kg). When the relative hydrogen price falls outside this range, the annual TC of Cases 3 and 4 converges, and the difference becomes negligible.

5 Conclusions

This paper presents a flexible dispatch strategy for EG integrated with WPP-PEM EL systems. The PEM EL system is modeled using an efficiency curve derived from a bottom-up electrochemical model, and its performance is compared with a simplified model using a constant efficiency. The optimal dispatch strategy is tested across three typical days to evaluate the economic performance of four scenarios, which differs in their operation schemes and modeling methods of the EL. Additionally, the sensitivity of the annual TC with respect to coal and hydrogen prices is analyzed. The key findings are summarized as follows.

1) The dispatch strategy that employs an efficiency curve to model the EL (Case 4) produces different dispatch results compared to constant efficiency model (Case 3), leading to a 1.2% reduction in the annual TC. The power modulation provided by the EL enhances load flexibility for the EG.

2) The annual TC for Case 4 is 2.1% and 2.6% lower than Case 1 (without EL) and Case 2 (where the EL operates only during wind curtailment with a constant efficiency), respectively. Integrating the EL into the EG reduces wind curtailment in Case 4 compared to Case 1 and increases the utilization factor of the EL compared to Case 2.

3) When the coal price is below 121.6 $/tonne, the proposed strategy (Case 4) maintains its economic advantage over Cases 1 and 2. However, if the coal price exceeds 121.6 $/tonne, the optimal shifts to Case 1, where the EL is not installed.

4) When the hydrogen price ranges from 2.9 to 5.4 $/kg, the difference in annual TC between Cases 3 and 4 becomes significant. When the hydrogen price is around 4.1 $/kg, the greatest discrepancy (approximately $1.09 million) occurs.

The proposed schedule strategy provides a realistic and efficient decision tool for future EG systems with PEM EL integration. Though the grid used in this paper is relatively small, the strategy is applicable to larger-scale grids.

Future work will focus on optimizing the schedule of electricity–hydrogen systems with three states (i.e., on, off, standby) of the PEM EL at a higher resolution (15-minute intervals). Additionally, the incorporation of environmental and technical indicators will help evaluate the overall performance of the schedule strategy.

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