Regarding the electrolytic cells, there are mainly alkaline electrolytic cells, proton exchange membranes and solid oxide electrolytic cells. Alkaline electrolyzers are the most mature and widely used model, so this paper uses alkaline electrolyzers as an example. The waste heat generated by the electrolyzer is recycled, and the refined model of the hydrogen-heat cogeneration of the electrolyzer is proposed. Figure 2 shows the structure diagram of the refined utilization of the electrolytic cell. Electrolytic hydrogen production includes green electrolysis link, red heat transfer link and purple hydrogen energy utilization link.

Electrolysis link: During electrolysis, electrical energy is converted into hydrogen and heat. The relationship between the alternating current input to the electrolytic bath, the actual direct current used, and the hydrogen energy produced, and the heat energy is shown in (1–3). P e c , t = η e c P e c , i n , t = U e c i e c , t , T e c , t i e c , t = P e c , o u t , t + Q e c , t (1) P e c , o u t , t = i e c , t U t n T e c , t = η h 2 P e c , t (2) Q e c , t = i e c , t U e c i e c , t , T e c , t − U t n T e c , t = η h P e c , t (3) where P _ec,t and P _ec,in,t are the input DC power and AC power to the electrolyzer; η _ec is electrolyzer AC/DC conversion efficiency; i _ec,t is current of the electrolyzer; T _ec,t is temperature of the electrolyzer; U _ec (i _ec,t , T _ec,t ) is voltage function of electrolyzer; U _tn (T _ec,t ) is thermoneutral voltage function of the electrolyzer; P _ec,out,t and Q _ec,t are hydrogen and heat production power of the electrolyzer; η h 2 and η _h are hydrogen production efficiency and heat production efficiency of the electrolyzer.

In (2, 3), a non-linear relationship can be obtained between the thermal energy generated by the electrolyzer and the hydrogen energy and the temperature of the electrolyzer. Q e c , t = P e c , o u t , t U t n T e c , t U e c P e c , o u t , t U t n T e c , t , T e c − U t n T e c , t (4)

Due to the non-linear nature, it cannot be directly used in the optimal operation of E-H IES. The final model of the electrolysis link is obtained by linearizing it based on (Li et al., 2019; Pan et al., 2020). P e c , o u t , t = μ 1 P e c , i n , t + ν 1 δ e c , t T e c , t (5) Q e c , t = μ 2 P e c , i n , t + ν 2 T e c , t (6) where μ ₁, μ ₂, ν ₁, ν ₂ denote the linearized parameters for the hot hydrogen co-production operating area of the electrolyzer; δ _ec,t is operating status of electrolyzer.

Heat transfer link: Some of the heat generated by the electrolyzer is lost and most of the rest goes to the heat exchanger. The heat entering the heat exchanger can either feed the heat load or optionally flow to the electrolyzer itself, maintaining the temperature of the electrolyzer itself. The electrolyzer can also choose to supply heat to the thermal load. The relationship between the final thermal energy supplied to the heat load and the thermal energy entering the heat exchanger is shown in (7). Quasi-steady state model of the electrolyzer temperature is expressed in (8). The thermal power lost by the electrolytic bath is shown by (9). Q h e , t = η h e Q h e , i n , t (7) T e c , t + Δ t = T e c , t + Δ t C e c Q e c , t − Q s l , t − Q h e , i n , t (8) Q s l , t = T e c , t − T out,t / R e c (9) where Q _he,t is heat energy supplied to the heat load by the electrolyzer; η _he is conversion efficiency of heat energy in heat exchanger; Q _he,in,t is hermal energy of electrolyzer entering heat exchanger; Q _sl,t is thermal energy lost in the electrolyzer; C _ec is lumped heat capacity of electrolyzer; T _out,t is outdoor temperature; R _ec is lumped thermal resistance of electrolyzer. If Q _ec,t − Q _sl,t = Q _he,in,t , almost all of the waste heat is supplied to the heat load and the temperature of the electrolyzer is basically unchanged. If Q _ec,t − Q _sl,t > Q _he,in,t , some of the waste heat flows to the electrolyzer and the temperature of the electrolyzer rises. If Q _ec,t − Q _sl,t < Q _he,in,t , the electrolyzer releases some heat to the heat load and the corresponding temperature of the electrolyzer decreases.

In addition, the operation of the electrolyzer also needs to meet the start-stop constraints and operational constraints. ∑ t = k k + M e c , o n , min − 1 α e c , o n , t ≤ 1 (10) ∑ t = k k + M e c , o n , min − 1 α e c , o n , t + α e c , o f f , t ≤ 1 (11) ∑ t = k k + M e c , o f f , min − 1 α e c , o f f , t ≤ 1 (12) ∑ t = k k + M e c , o f f , min − 1 α e c , o n , t + α e c , o f f , t ≤ 1 (13) α e c , o n , t + α e c , o f f , t ≤ 1 (14) α e c , o n , t − α e c , o f f , t = δ e c , t − δ e c , t − 1 (15) ∑ t = 1 T α e c , o n , t ≤ α e c , o n , max (16) ∑ t = 1 T α e c , o f f , t ≤ α e c , o f f , max (17) δ e c , 0 = δ e c , T (18) ζ e c , min δ e c , t C apa ≤ P e c , i n , t ≤ ζ e c , max δ e c , t C apa (19) P e c , i n , t − P e c , i n , t − 1 ≤ Δ P e c , max (20) T e c , min ≤ T e c , t ≤ T e c , max (21) where M _ec,on, min and M _{ec,off, min} denote minimum operating time and minimum downtime for electrolyze; α _ec,on,t and α _ec,off,t denote 0–1 variables for electrolyzer start-up and shutdown actions; α _ec,on, max and α _{ec,off, max} denote the upper limit of the daily start and stop actions of the electrolyzer; ζ _ec, min and ζ _ec, max are minimum/maximum load rate of the electrolyzer; C _apa is electrolyzer capacity; ΔP _ec, max is the maximum ramping power of the electrolyzer; T _ec, min and T _ec, max are the minimum and maximum temperature of the electrolyzer. (10–13) limit the minimum working time and minimum downtime of the electrolyzer. (14) shows that the start-up and shutdown of an electrolytic cell cannot take place concurrently. (15) expresses the relationship between the state of the electrolytic cell switch and the start/stop action. (16) and (17) limit the number of start-ups and shutdowns of an electrolyzer per day. (18) ensures that the state of the electrolyzer before work is consistent with the state after work. (19) represents the relationship between the input electric power and the capacity of the electrolyzer. (20) and (21) represent the climbing constraints and the upper and lower temperature constraints for electrolyzer, respectively.

The HFC supplies the electrical energy generated by the converter to the electrical load, and recycles the thermal energy generated by the stack reaction. The heat exchanger heats the return water of the circulating thermal system to supply the heat load and can also supply the fuel cell stack itself. HFC itself can absorb and release thermal energy. The relationship diagram of HFC cogeneration can be drawn, as shown in Figure 3. There is a clear proportional relationship between the electrical and thermal power generated by HFC. Considering that the loss of heat energy taken away by air flow does not affect the electrical output, the operating interval is shifted down overall from the ideal case. In the operating interval where air flow is considered, the highest electricity production efficiency is 0.6, corresponding to a heat production efficiency of 0.35, and the highest heat production efficiency is 0.53, corresponding to an electricity production efficiency of 0.33. According to the combined heat and power operating range of HFC, the operating model of the external characteristics of HFC can be summarized. P hfc,t = η hfc,e,t P hfc,in,t (22) Q hfc,t = η hfc,h,t P hfc,in,t (23) η hfc,e,min ≤ η hfc,e,t ≤ η hfc,e,max (24) η hfc,h,min ≤ η hfc,h,t ≤ η hfc,h,max (25) η hfc,h,t ≤ k h , max η hfc,e,t (26) η hfc,h,t ≥ k e , max η hfc,e,t (27) Q hfc,sl,t = T hfc,t − T out,t / R hfc,ec (28) P hfc,in,t = P hfc,t + Q hfc,t + Q hfc,sl,t (29) Q hfc,he,t = η h e Q hfc,he,in,t (30) T hfc,t+Δt = T hfc,t + Δ t C hfcc Q hfc,t − Q hfc,he,in,t (31) ζ hfc,min δ hfc,t C apb ≤ P hfc,in,t ≤ ζ hfc,max δ hfc,t C apb (32) P hfc,in,t − P hfc,in,t−1 ≤ Δ P hfc,max (33) T hfc,min ≤ T hfc,t ≤ T hfc,max (34) where P _hfc,t and Q _hfc,t denote electricity and thermal energy generated by HFC; P _hfc,in,t is hydrogen power input HFC; η _hfc,e,t and η _hfc,h,t denote electricity generation efficiency and heat generation efficiency of HFC; η _hfc,e,min and η _hfc,e,max are minimum and maximum efficiency of electricity generation; η _hfc,h,min and η _hfc,h,max are minimum and maximum efficiency of heat generation; k _h, max and k _e, max are the slope of maximum thermal efficiency and electrical efficiency operating boundary; Q _hfc,sl,t is the heat energy lost by HFC; T _hfc,t is the temperature of the HFC; R _hfc,ec is lumped thermal resistance of HFC; Q _hfc,he,t is the heat energy supplied by HFC to the heat load; Q _hfc,he,in,t is the heat energy entering the heat exchanger; η _he is the heat efficiency; C _hfcc is lumped heat capacity of HFC; δ _hfc,t is perating status of HFC; ζ _hfc,min and ζ _hfc,max are the minimum and maximum load rate of the HFC; C _apb is HFC Capacity; ΔP _hfc,max is the maximum ramping power of the HFC; T _hfc,min and T _hfc,max are the minimum and maximum temperature of HFC. (22) and (23) express the relationship between the power generation and heat generation and the input hydrogen energy. (24) and (25) place limits on the efficiency of heat production and electricity production respectively. (26) and (27) ensure that HFC operates within the allowable range. The thermal power lost by HFC is shown by (28). (29) ensures that the input and output energy of HFC is conserved. (30) represents the relationship between the thermal energy supplied to the heat load and the thermal energy entering the heat exchanger. Eq. 31 is a quasi-steady-state model of HFC temperature. (32) represents the relationship between the input hydrogen power and the capacity of HFC. (33) and (34) represent the climbing constraint and the upper and lower temperature constraints for HFC, respectively.

There are non-linear terms in (22, 23) for the product of two continuous variables. To simplify the computation, the following linearized transformation of the non-linear terms is performed using the binary method and the big M method. W x = W min x + y ∑ n = 0 p 2 n δ n x (35) 0 ≤ y ∑ n = 0 p 2 n δ n x ≤ W max x − W min x (36) δ n x ∈ 0,1 (37) m W x = m W min x + y ∑ n = 0 p 2 n τ n x (38) m − M 1 − δ n x ≤ τ n x ≤ m + M 1 − δ n x (39) − M δ n x ≤ τ n x ≤ M δ n x (40) where m and W ^x are continuous variables. δ n x is a binary variable. τ n x is an auxiliary variable. M is a very large number. W max x and W min x are the maximum and minimum values of the continuous variable W ^x , respectively. p is the number of digits in the binary. y is the resolution.

In this paper, the ideal gas equation of state is used to describe the relationship between the hydrogen mass and the pressure in HST. P HST,t V HST = n HST,t R T H (41) n HST,t = m HST,t M H 2 (42) P HST,min ≤ P HST,t ≤ P HST,max (43) m HST,t+1 = m HST,t + m HST,in,t − m HST,out,t (44) 0 ≤ m HST,in,t ≤ B HST,in,t m HST,in,max (45) 0 ≤ m HST,out,t ≤ B HST,out,t m HST,out,max (46) B HST,in,t + B HST,out,t ≤ 1 (47) where P _HST,t is the pressure of the HST; V _HST is the volume of the HST; n _HST,t is amount of hydrogen substance; R is ideal gas constant; T _H is gas temperature; m _HST,t is mass of hydrogen; M H 2 is molar mass of hydrogen; P _HST,min and P _HST,max are the minimum and maximum pressure of HST; m _HST,in,t and m _HST,out,t are hydrogen storage and release in HST; m _HST,in,max and m _HST,out,max are the maximum hydrogen storage and release rate; B _HST,in,t and B _HST,out,t are HST hydrogen storage and release 0–1 variable. (43–47) shows the operating model of HST. (43) is the pressure safe operating range of HST. (44) is the relationship between the hydrogen mass in HST at two adjacent moments. (45, 46) are the maximum constraints of hydrogen storage and hydrogen release. (47) is the constraint of hydrogen storage state and hydrogen release state to ensure that the HST does not store and release hydrogen at the same time.

Electric energy storage model including energy model (48–53) and life model (54–58). E t = E t − Δ t − P dis,t Δ t / η d + η c P c h , t Δ t (48) 0 ≤ P c h , t ≤ B c h , t P c h , max (49) 0 ≤ P dis,t ≤ B dis,t P dis,max (50) B c h , t + B dis,t ≤ 1 (51) E min ≤ E t ≤ E max (52) E 0 = E T (53) Γ R = L R D R C R (54) d eff = k 0 1 − k 1 S O C t + k 2 S O C t 2 (55) k 0 = e u 1 − 1 + 1 D R d R / D R u 0 (56) k 1 = u 0 + u 1 / D R (57) k 2 = u 0 u 0 − 1 / 2 + u 0 u 1 / D R + u 1 2 / 2 D R 2 (58) where E _t is the amount of electricity contained in the EES; P _ch,t and P _dis,t are charging and discharging power; η _c and η _d are charging and discharging efficiency; P _ch, max and P _dis,max are the maximum charging and discharging power; B _ch,t and B _dis,t are charging and discharging 0–1 flag variable; E _min and E _max are the minimum and maximum energy of EES; E ₀ and E _T are the initilal and the end energy of EES; Γ _R represents the rated usage Ampere hours of the electric energy storage; L _R is the rated cycle life; d _R is rated Ampere hours consumed by single discharge; D _R is rated depth of discharge; C _R is rated ampere-hour capacity. d _eff represents the ampere-hours consumed by a single use of electrical energy storage; SOC is the state of charge of the EES; k ₀, k ₁, k ₂ are constants. u ₀, u ₁ are the parameter values obtained from the test during the simulation process.

Although the renewable energy forecasting technology is always evolving and the forecasting accuracy is constantly improving, the optimized operation results derived by considering only the day-ahead renewable energy and load output may differ from the actual operation results.

Model predictive control (MPC) has the characteristics of rolling optimization and feedback correction (Ye et al., 2022), which can update the state of the system in real time and reduce the system error caused by inaccurate prediction. MPC mainly includes three steps: model prediction, rolling optimization and feedback correction (Vasilj et al., 2019).

The structure diagram of the E-H IES is shown in Figure 1. Load, photovoltaic and wind power forecast information are shown in Figures 5, 6, respectively. The parameters of some devices are shown in Table 1.

The World Meteorological Organization divides weather forecasts into three categories based on their duration: very short-range weather forecasting (next 12 h), short-range weather forecasting (next 3 days) and medium-range weather forecasting (next 10 days). In order to consider both the long-term operation of hydrogen energy and the accuracy of the forecast, this paper chooses to predict the information of the next 3 days. In addition, renewable energy and load forecasting are not the focus of this paper.

The comparison cases in this paper are as follows.

Case 1: Traditional day-ahead optimization.

To see the transfer effect of hydrogen energy more intuitively, a 4-day optimization operation is conducted. In Case 1 and Case 2, the energy changes of electric and hydrogen energy storage are shown in Figure 7. The relationship between thermal energy and temperature in the electrolyzer and HFC is shown in Figures 8, 9, respectively.

Figure 6 depicts the weather conditions over the next 4 days. The changes in the quality of hydrogen in the HST in cases 1 and case 2 are significant. In the traditional day-ahead optimization, the energy of hydrogen returns to its initial state at the end of each day, which can only meet the daily energy demand. In day-ahead optimization that considers multi-day forecast information, hydrogen is stored while renewable energy is abundant and released when it is scarce. The proposed method maximizes the use of renewable energy, resolves the problem of uneven distribution of renewable energy, and implement the inter-day energy transfer for hydrogen energy. Under the traditional optimal operation strategy, the electric energy storage is frequently charged and discharged and the SOC of electric energy storage is at a low level. The day-ahead optimal operation strategy proposed in this paper shows that the frequency of charging and discharging of electric energy storage is reduced and the SOC is at a high level, which has a certain mitigation effect on the life depreciation of electric energy storage. The proposed day-ahead long-time-scale optimal operation strategy achieves the inter-day energy transfer of hydrogen energy storage and the intra-day energy transfer of electric energy storage, and realizes the coordinated control of multiple types of energy storage.

Renewable energy is abundant in the system on the first and third days, when the electrolyzer produces a large amount of hydrogen with a large amount of heat. In case 1, the electrolyzer must continue to operate on the second and fourth days to maintain a balance between hydrogen energy and heat energy. However, in case 2, the HST plays a significant role on the second and fourth days, and the HFC can use this portion of the free resources to generate heat. In the day-ahead optimization considering multi-day forecast information, it is evident that the electrolyzer can reduce some workload when wind and solar resources are insufficient, thereby preventing damage to the equipment from excessive use. Based on the heat generation and the heat supply to the heat load, the temperature of the electrolyzer changes accordingly.

In case 1, on the first day when renewable energy is abundant, HFC employs hydrogen to generate a great deal of heat to prevent excessive abandonment of wind power and photovoltaics. When renewable energy is scarce, hydrogen is provided primarily to the hydrogen load, with any excess going to the HFC. Thus, the output of the HFC is relatively low. In case 2, when the renewable energy is sufficient, the HFC uses the hydrogen generated by the electrolyzer to work, as the HFC cannot work with the hydrogen stored in the HST, and the output of the HFC remains stable.

In the day-ahead stage, the power balances of different energy sources in case 1 and case 2 are shown in Figures 10, 11, respectively. In case 1, when the renewable energy is sufficient, E-H IES chooses to sell electricity to the upper power grid to make a profit. Under the penalty of abandoning wind power and photovoltaics, part of the renewable energy is converted into hydrogen through the electrolyzer, and HFC uses this part of the hydrogen to generate heat for sale. When there is a shortage of renewable energy, the system can only choose to buy electricity from the upper-level grid to meet the system demand, and the system does not sell heat to the outside. The traditional day-ahead optimization may obtain greater benefits in some cases, but in the long-term development, it is not a smart development model.

In case 2, from the electricity-hydrogen-thermal coupling point of view, the HST is almost in the storage state on the first and third days and is released for system usage on the second and fourth days. When renewable energy is in excess, the system also sells power and heat. When there is a shortage of renewable energy, the system not only greatly reduces the purchase of electricity from the upper grid, but also has a small profit from heat sales. The system’s overall economic value is then significantly enhanced.

Comparisons of case 1 and case 2 are shown in Table 2. It can be calculated from Table 2 that the average cost of case 1 is −24,435.8 yuan, while the average cost of case 2 is −32,421.7 yuan. The average life of the EES in case 1 is 5.80605 years, and the average life of the EES in case 2 is 6.1829 years. It can be seen that the profitability of case 1 is slightly higher than that of case 2 on the first and third days, but it is much lower than that of case 2 on the second and fourth days. From the perspective of long-term development of the system, day-ahead optimization considering multi-day forecast information can tackle the problem of uneven distribution of renewable energy. The overall economic effect of the system is better, and it also has a certain mitigation effect on the loss of EES, reducing the abandonment of wind power and photovoltaics in the system.

After the day-ahead plan is obtained, MPC is used to track and correct the day-ahead plan within the day. Since the steps of intra-day rolling are the same, this paper only analyses the intra-day optimization on the first day. According to the different dispatch periods of electricity-heat-hydrogen, Figure 12 shows the comparison results of the load day-ahead forecast and the intra-day forecast, respectively.

In the heat energy dispatching stage, the output adjustment of related heat equipment and thermal storage tank (TST) is mainly determined by the heat load prediction inaccuracy. Since the heat energy dispatching cycle is the longest, as the first stage of intra-day dispatching, it is sufficient to follow the system’s day-ahead plan and analyze the system operating economy and the output fluctuation of each component under the premise of meeting the load requirements. In the second stage of intra-day hydrogen scheduling, in addition to following the system’s day-ahead output plan, it also needs to follow the adjustment plan of the thermal energy scheduling stage. In the hydrogen scheduling stage, the output adjustment of the relevant hydrogen energy equipment is mostly determined by the hydrogen load prediction inaccuracy. Electric energy dispatching has the shortest cycle and is also the last stage of intra-day dispatching. On the basis of following the system’s day-ahead output plan, it is also necessary to meet the output adjustment plans of each component in the first two stages. In the electric energy dispatching stage, the output adjustment of related electric energy equipment and EES is mainly carried out according to the electric load and the forecast error of wind power photovoltaic. Figure 13 shows intra-day heat balance, hydrogen balance, and electrical power balance, respectively.