Off-grid wind/hydrogen systems with multi-electrolyzers: Optimized operational strategies

Optimized operation of wind/hydrogen systems can increase the system efficiency and further reduce the hydrogen production cost. In this regard, extensive research has been done, but there is a lack of detailed electrolyzer models and effective management of multiple electrolyzers, considering their physical restrictions. This work proposes electrolyzer models that integrate the efficiency variation caused by load level change, start–stop cycle (including hot and cold start), thermal management, and degradation caused by frequent starts. Based on the proposed models, three operational strategies are considered in this paper: two traditionally utilized methods, simple start–stop and cycle rotation strategies, and a newly proposed rolling optimization-based strategy. The results from daily operation show that the new strategy results in a more balanced load level among the electrolyzers and a more stable temperature. Besides, from a yearly operation perspective, it is found that the proposed rolling optimization method results in more hydrogen production, higher system efficiency, and lower LCOH. The new method leads to hydrogen production of 311297 kg, compared to 289278 kg and 303758 kg for simple start–stop and cycle rotation methods. Correspondingly, the system efficiencies for the new, simple start–stop and cycle rotation methods are 0.613, 0.572, and 0.587. The resulting LCOH from the new method is 3.89 e /kg, decreasing by 0.35 e /kg and 0.21 e /kg compared to the simple start–stop and cycle rotation methods. Finally, the proposed model is compared with two conventional models to show its effectiveness in revealing more operational details and reliable results.


Introduction
Electrolytic hydrogen has gained significant attention worldwide for its potential to facilitate deep decarbonization and enhance modern power system operations [1].However, the high cost of producing electrolytic hydrogen remains a major obstacle to its widespread adoption [2].For instance, the levelized cost of producing grey hydrogen currently ranges from 1-1.4 e/kg, whereas green hydrogen generated through wind-powered water electrolysis costs approximately 3.75-5.11e/kg [3].To address this issue, optimizing the control and operation of electrolyzer systems by considering their complex dynamic properties is crucial.Such optimization typically involves making decisions related to power scheduling [4], state control [5], and thermal management [6] of the electrolyzers.Adopting efficient operational strategies that maximize the potential of electrolyzers' flexibility can significantly enhance the system's efficiency.
Electrolytic hydrogen systems can be categorized as either on-grid or off-grid based on grid accessibility [7].On-grid systems provide a stable power supply, high electrolyzer load factors, and revenue stacking opportunities in power markets [8].However, they cannot ensure the renewable origin of the electricity used and hydrogen produced.Furthermore, on-grid configuration is less feasible for remote regions that encounter difficulties in connecting to the grid [9].On the other hand, off-grid systems that rely on renewable power for electrolysis ensure the cleanliness of hydrogen, but necessitate electrolyzers to respond quickly to fluctuating power inputs, potentially reducing load factors.Effective operational optimization is crucial for both configurations.Additionally, an electrolytic hydrogen system can consist of either a large electrolyzer or multiple small electrolyzers [10], with the latter requiring more efficient operational strategies.This paper focuses on an off-grid wind/hydrogen system with multiple electrolyzers, but the presented methods can be readily applied to other configurations as well.
There have been numerous studies aimed at optimizing the operation of electrolytic hydrogen systems.A study [10] explored a wind/multi-electrolyzer system with two configurations and two control strategies.The authors found that four 0.5 MW electrolyzers led to a higher energy absorption rate but lower overall hydrogen production https://doi.org/10.1016/j.enconman.2023 They also demonstrated that the improved strategy based on cycle rotation resulted in less electrolyzer imbalance.One such study [11] focused on a grid-connected hybrid wind/electrolyzer/supercapacitor system with hourly resolution.The authors proposed three strategies to operate the five electrolyzers within the system, and found that the segment start strategy significantly reduced the switching times of electrolyzers and increased hydrogen production.Another study [12] proposed a segmented fuzzy control approach to allocate electric power to 100 electrolyzers.The authors defined five fuzzy sets to characterize the difference of input power and the theoretically optimal power of electrolyzer groups.Electric power was then distributed to electrolyzers based on which fuzzy sets the difference belonged to, leading to a higher hydrogen production efficiency compared to a simple start-stop strategy.A study [13] investigated an on-grid wind/hydrogen system using ten electrolyzers to regulate the reference line power of wind generators.The authors adopted a first-in-first-out switching strategy to control the electrolyzer group, which made each electrolyzer operate at full load and potentially improve lifespan and efficiency.
To summarize, existing research on electrolyzer operational strategies has highlighted the importance of efficiency variation, load range, and start-stop properties.Theoretically, electrolyzer efficiency rises as current density decreases, meaning that lower hydrogen production leads to lower energy consumption per kg of hydrogen [14].Also, electrolyzers have a preferred working range, typically between 25% to 100% of their nominal capacity, and cannot operate below their lower bound or risk overload [15].This constraint can lead to frequent start-stop cycles, which has been extensive discussed.Several rules for electrolyzer start-stop management have been proposed.
Despite the existence of previous research on electrolyzer operation and control, there are still gaps that require further investigation.Firstly, the start-stop processes are oversimplified, as electrolyzers have three states: production, standby, and off states [14,16].Hot start and cold start transitions refer to the shift from standby to production and from off-state to production, respectively.However, the different start types have yet to be considered.Furthermore, hot and cold starts take a long time to complete, creating important limitations on electrolyzer operation [17].To properly address the start process, a smaller time resolution is required.Most current research is based on an hourly resolution, which conceals many operational details, leading to less reliable results.Secondly, operational strategies that consider the efficiency variation limits and start-stop processes must be developed to increase system efficiency and reduce hydrogen production costs.Finally, the long-term impacts of operational patterns are not adequately addressed.For instance, a strategy may generate high short-term efficiency but severe degradation due to dynamic operation [18].Hence, it is essential to include the long-term influence of operational strategies when evaluating their effectiveness.
To address these research gaps, this paper contributes to the stateof-the-art in the following ways: 1. We propose a detailed electrolyzer model that considers efficiency variation and start processes.The model is used to optimize the operation of an off-grid wind/hydrogen system with a time resolution of 5 min.
2. Three operational strategies are introduced to optimize the allocation of wind power to the multiple electrolyzers.In addition to the updated simple start-stop and cycle rotation strategies, we propose a rolling optimization strategy that considers the complex constraints from electrolyzer states.3. We calculate the levelized cost of hydrogen, taking into account the voltage degradation caused by frequent start-stop cycles.This provides a deeper view to evaluate operational strategies.
The remainder of this paper is organized as follows: Section 2 describes the operation management problem of the hybrid wind/hydrogen system, followed by Section 3, which details the modelling of electrolyzer efficiency, thermal management, start-stop processes, and voltage degradation.Section 4 introduces the operational indicators and the proposed strategies.Section 5 compares the operational indicators from different strategies on daily and yearly basis.The difference in using different electrolyzer models is also discussed.Finally, in Section 6, we conclude this work and suggest future research directions.

Problem description
This study aims to optimize the operation of an off-grid hybrid wind/hydrogen system, as illustrated in Fig. 1.The system consists of four 1 MW electrolyzer stacks that are connected to the DC bus and powered by a 4.2 MW wind turbine.The hydrogen and oxygen produced by the electrolyzers are separated and collected in separate containers, while the lye circulates throughout the system and exchanges heat with cooling water via external heat exchangers to control the electrolyzer temperature.The main challenge is to allocate wind power from the turbine to the electrolyzers in a way that maximizes the system's efficiency.During the investigated time horizon, it is necessary to determine the power consumption of each electrolyzer at each time step while taking into account the constraints imposed by the electrolyzers.For instance, if an electrolyzer is shut down, it requires a considerable amount of time to restart, and during this period it is unable to produce hydrogen.

Modelling of electrolyzer stack
This section introduces the mathematical models for electrolyzer stacks, which can be used for all three types of electrolyzers: alkaline, proton exchange membrane and solid oxide electrolyzers [19].Naturally, the parameters in these models are distinct, given the specific electrolyzer type.For instance, solid oxide electrolyzers bear higher efficiency while proton exchange membrane electrolyzers can operate more dynamically [20].Alkaline electrolyzers are generally more cost-competitive [21].

Hydrogen production
Electrolyzers split water into hydrogen and oxygen using electricity.In terms of a system-level study, it is necessary to reflect on the relationship between power input and hydrogen output.While numerous first-principle models and empirical mathematical models are available for alkaline electrolyzer modelling, this work uses a simplified model to quantify hydrogen production from the perspective of engineering usage.Considering an electrolyzer stack with  single cells in series, we first focus on an electrolysis cell.The hydrogen production rate ṁ (g/s) can be calculated as: In a simple electrolyzer model, hydrogen conversion rate (HCR)  (g/J) is deemed as a constant, leading to a linear relationship between ṁ and input power  (W) [22].However, this method cannot fully reflect the nonlinear hydrogen-power relationship.In fact,  is a monotonically increasing function of  and ṁ is a nonlinear function of .This work derives the expression of ṁ by fitting the available data.Provided that the hydrogen production rate at nominal power   , half of nominal power  0.5 and a quarter of nominal power  0.25 are known, we can get the fitting curve for ṁ() using these data as well as (0, 0) on the ṁ− surface, as shown in Fig. 2. In general, the HCRs at partial load factor are reported by the manufacturer.Fig. 2 again suggests that the ṁ() is not a linear function.Finally, we assume that each cell within the stack is working in the same situation, and the stack power and hydrogen production can be obtained by multiplying  with  and ṁ, respectively.This engineering method can also be used to obtain the electrical variables.Faraday's law describes the relationship between cell current  (A) and hydrogen production ṁ (g/s): in which F is the Faraday constant (C/mol).M  2 is the molar mass of hydrogen gas (g/mol), and  denotes the mole of the transferred electrons for every mole hydrogen production (mol/mol).The cell voltage U (V) is estimated by:

Thermal modelling
This section introduces the thermal modelling of an electrolyzer stack.Electrolyzer thermal models are required to capture the temperature variation and heat generation.As shown in Fig. 1, electrolyzer stacks are cooled by the lye flow, which exchanges heat with the cooling water through a heat exchanger.Fig. 3 further illustrates the heat exchange within an electrolyzer stack.Using a zero-dimensional model, we have the following difference equations describing the electrolyzer thermal properties: where  th is the electrolyzer heat capacity (MJ/ • C) and  ℎ, is the stack temperature ( • C).On the right side of the equation are the heat flows: heat generation from overvoltage Qgen, (MW), heat transferred to the cycling electrolyte Qliq, (MW) and the heat loss to the environment Qloss (MW).These three items are calculated as: Qloss, =  ℎ, −    th (7) Eq. ( 5) quantifies the heat caused by overvoltage, with  ℎ being the thermal neutral voltage.Eq. ( 6) calculates the heat transferred to the flowing liquid (electrolyte), where ṁliq is the mass flow rate (kg/s);   liq is the specific heat capacity of (MJ/kg/ • C); and  ℎ, −  liq, is the temperature difference between stack and inlet electrolyte.Heat loss depends on the temperature difference of the electrolyzer and environment  ℎ, −   , and the thermal resistance  th (MW/ • C), as denoted in Eq. ( 7).
Before entering the electrolyzer stack, the electrolyte exchanges heat with cooling water, as presented in Fig. 3 (B).Considering an ideal parallel flow heat exchanger, we have: Equations in (8) describe the energy balance between the hot and cold flows in the heat exchanger. he,t is the heat power transferred from hot flow to cold flow (MW). cw ,  ,, and  ,, are the specific heat capacity (MJ/kg/ • C), outlet and inlet temperature ( • C) of the cooling water.Eq. ( 9) calculates the power exchange from the perspective of heat transfer, where  he and  he are the heat exchanger coefficient (MW/m 2 / • C) and heat transfer area (m 2 ).
Generally, the electrolyzer temperature is remained in a suitable range by controlling the inlet temperature  cw,in, and mass flow rate of cooling water ṁcw, .This paper uses a simple feedback control of the temperature to focus our attention on multiple electrolyzer modelling.The controlled variable here is the mass flow rate of cooling water ṁcw, , which follows [23]: in which  ref is the reference temperature of the electrolyzer stack.  and   are proportional and integral coefficients.

Electrolyzer state
During the operation of an electrolyzer, the electrolyzer may be shut down or put into a standby state.In a multi-stack system, electrolyzers have distinct states, resulting in a complex operational concern.In this work, we propose two state variables  , and  , to describe the states regarding standby and off state of electrolyzer  at time , which are described in Table 1.A combination of  , and  , indicate a sole state of an electrolyzer.For example, if  , = RTH  and  , = RTC  , the electrolyzer is in the normal operational state, producing hydrogen.If  , = RTH  and  , = −1, the electrolyzer is in the off state and has to go through cold-start before producing hydrogen.RTH denotes the required time steps to finish a hot start, defined as: Given a hot start time of 500 s and a time resolution of 300 s, RTH would be 2, meaning that the hot start takes two time steps to be completed.RTC is defined in a similar way.Also note that an electrolyzer cannot be in two states simultaneously, implying the values of  , and  , are not always compatible.State variables update at every time step and impose restrictions on electrolyzers' power consumption and hydrogen production.Detailed descriptions of every state can be found in our previous work and other research [14].State variables can also be set externally.For example, if a shutdown signal is received at time ,  , would be set to −1.Fig. 4 presents the updating process of state variables, the real power consumption given a power input signal and the external actions that can be implemented.
Fig. 4(A) shows the response of electrolyzer  given a power input signal  to,ele , at time .The electrolyzer model calculates the real power consumption and updates its state variables depending on its current state at time .If it is on off state, the real power consumption  ele , would be zero, and the standby power consumption  sb , takes zero too.It is also expected to maintain off state, i.e.  ,+1 = −1.An external signal of cold start is required to put the electrolyzer in the cold start process by setting  ,+1 = 0 at the next time step  + 1.If the electrolyzer is during the cold start process,  ,+1 would be increased by one compared to  , , indicating that cold start is in progress.If the electrolyzer is not in an off-state or cold-start process, then  , is checked to determine the electrolyzer state.For the standby state or hot-start process, the  sb , is non-zero, implying the power consumption to maintain standby.Finally, the simplest situation is that electrolyzer  is in production state ( , = RTH  and  , = RTC  ), which means it can produce hydrogen.However, the real power consumption  ele , is still influenced by the power at previous time −1, the ramping up/down rates  , and  , and the capacity of the electrolyzer .The good news is that the ramping limits can generally be ignored if the time resolution is much larger than the ramping time [24].
Fig. 4(B) defines the functions that can be called to change the state variables of electrolyzer , which can be viewed as control signals in reality.For example, once the shut down hot() function is called,  , is manually set to −1 to indicate that electrolyzer enters standby at time .This function, however, cannot be called if the electrolyzer is in the off state or during the cold start process.Shut down cold() function puts electrolyzers in the off state.Unlike shut down hot(), this manipulation can be done even if the electrolyzer is in the standby state.In reality, one can stop a standby electrolyzer but not put an off-state electrolyzer on standby.Also, note that we assume electrolyzers can be shut down or put on standby instantaneously to simplify their modelling.The hot start() and cold start() functions set the corresponding variables to zero without causing any incompatibility of  , and  , .

Electrolyzer degradation caused by frequent start
Frequent start-stop operations of electrolyzers result in fluctuating temperatures.It may also cause electrode corrosion and catalyst degradation.This work assumes that every hot and cold start cause a slight voltage increase and further decreases the HCR.Although the degradation is a continuous process, we focus on the yearly change of overvoltage and HCR because degradation caused by a single start is minor.According to [25,26], for an electrolyzer cell, the voltage (V) at year  + 1 has the following form: where NCS  and NHS  are the number of cold and hot starts during year , and   and  ℎ represents the resulting voltage change.Thus, for a specific current  of an electrolyzer cell (A), the power consumption is (W): with   the electrolyzer cell power at year .According to Faraday's law [27], we calculate the hydrogen mass production rates as: This equation implies ṁ is measured by (g/s).Furthermore, we have the HCR at year  + 1: Eq. ( 16) implies that given the number of hot/cold starts and the HCR during year , we could estimate the HCR at +1.To use Eq. ( 16), the  should be measured with (g/J), which is inconvenient.For  measured by (kg/kWh), we reformulate the equation as: While the above analysis is based on an electrolyzer cell, we assume it also applies to electrolyzer stacks.Meanwhile, since  is selected arbitrarily, the above equation is applicable to hydrogen conversion rates at every load level.

Performance assessment indicators
Table 2 introduces the indicators we use to evaluate the performance of the system operation.In particular, the overall hydrogen production, system efficiency and LCOH are critical indicators to assess the operational performance.As the model provides operational details, we are also able to investigate the indicators that have not been well discussed in the state-of-the-art, such as NHS and NCS.
While most indicators are clearly explained in Table 2, we further define the LCOH considering the electrolyzer degradation caused by hot/clod starts.For the studied system, the LCOH is defined as [2]: where the numerator sums the CAPEX of all components and their discounted OPEX. represents the interest rate, and  is the set of the operational years.The denominator calculates the discounted hydrogen production, where  , is the annual hydrogen production from electrolyzer  without considering degradation.Due to the degradation of electrolyzers, more power is required to produce same amount of hydrogen, implying that the actual hydrogen production is less than  , .In this work, we assume the same annual operation for the whole project lifetime and estimate the hydrogen production by comparing the HCR at full load level during year  ( , ) and its counterpart during initial year ( 1 ).LCOH (e/kg) Levelized cost of hydrogen, defined as the overall cost divided by overall hydrogen production during life-cycle

Operational strategy
This section introduces the operational management strategy for the investigated wind/hydrogen system with multiple electrolyzer stacks.Here, we consider a 4.2 MW wind turbine [28] and four 1 MW electrolyzers.The primary function of the operational algorithm is to allocate the wind power to each electrolyzer considering the operational limits.The proposed electrolyzer models are implicitly involved into each manipulation in the operational strategies.Table 3 introduces a simple start-stop method to manage the system.The wind power is first allocated to electrolyzer 1, followed by the other electrolyzers.Initially, all the electrolyzers are working on production states.If the power input of electrolyzer  is less than its lower power limit, it will be put into standby by calling shut down hot() function in Fig. 4(B), as indicated by lines 6 and 7.If electrolyzer  has been on standby for more  ℎ (taking 6 in this work) time steps, it will be shut down.For electrolyzers that are already shut down or standby, they will be started again if there is sufficient power input (see lines 11 to 14).Line 16 calculates the actual power consumption of electrolyzer  according to Fig. 4(A) and the input power signal  to,ele , .Eq. ( 1) is then used to calculate the hydrogen production rate ṁ, as well as the hydrogen production.The unused wind power would be updated, taking into account the real power consumption of electrolyzer  (line 18).For example, if the wind power   1 = 0.5 MW, electrolyzer 1 will work at 0.5 MW, but the others are put on standby.Then if wind power   2 = 3.5 MW, electrolyzer 1 will work at 1 MW.The other electrolyzers will be started.However, depending on the selected time resolution, there may not be hydrogen production from other electrolyzers because they are still undergoing a hot start.The real power consumption would be zero of these electrolyzers.Another example is if the wind turbine output remains at zero for an extended period from the start, all electrolyzers will first enter standby mode and then shut down.
A particular concern of operational strategy S1 is that electrolyzers 1 and 2 are frequently used, but electrolyzers 3 and 4 are seldom used.This will lead to unbalanced load factors of electrolyzers and a disordered maintenance schedule in the future.Strategy S2 solves this problem by simply renumbering the electrolyzers every   (taking 24 in this work) time steps, as shown in lines 3 and 4 in Table 4, where a rotational numbering is adopted.
Strategies S1 and S2 provide effective methods to allocate wind power.However, these strategies may only partially utilize the flexibility of electrolyzers, and the system efficiency can be improved by adopting advanced algorithms.Strategy S3 provides a method based on rolling optimization to optimize the operation further.The key to this strategy lies in line 11 of Table 5, along with other external control logic.Fig. 5 depicts the process of the rolling optimization.The optimizer here dispatches the wind power    at the current time step  given the status of all electrolyzers at time step  − 1 denoted as vector  −1 .The electrolyzer status are then adjusted according to the control signal  tl,ele , .This optimization is again solved at time  + 1 with updated constraints from electrolyzer status at time .
The objective of the optimizer is to maximize the overall hydrogen production rate Note that a second-order function is utilized to characterize the nonlinear relationship between hydrogen production rate ṁ, and power input of electrolyzers  to,ele

𝑖,𝑡
. The coefficients  (⋅) are obtained by curve fitting using the available data.Subscript  in this optimization problem can be viewed as a fixed parameter, and the optimization is solved repeatedly within the loop.
This wind power dispatch problem is constrained by the following limits: The first constraint ensures that the overall power consumption is less than the power supply.Besides, according to the electrolyzer status at the previous time step  − 1, the following constraints should be satisfied: The above limitations state that the power consumption of electrolyzer  is bounded by different limits.If it is not on standby or off state and not during hot-start or cold-start processes, the power consumption is limited by its ramping rates; otherwise, the power consumption is forced to be zero.Besides, the  ele , should be non-negative and lower than the capacity of electrolyzer : The nature of electrolysis usually results in a conic function in the objective (19).Given that all the constraints are linear, the optimization problem is expressed as quadratic programming (QP), which can be efficiently solved by solvers such as Gurobi.The problem scale mainly depends on the number of electrolyzers, i.e., |E|.Since only four electrolyzers are considered in this work, the QP can be solved within one second.This feature ensures the computational feasibility of the rolling optimization.

System parameters
The parameters of the investigated wind turbine and electrolyzers are shown in Table 6, 7 and 8. Here, four identical electrolyzers with 1 MW capacity are utilized to produce hydrogen to focus our attention on operation instead of the system design.However, the proposed modelling and operational strategies can be used for electrolyzers with distinct capacities and dynamic properties.For example, we can consider four electrolyzers with capacities of 2 MW, 1 MW, 0.5 MW and 0.5 MW, respectively.The sizing of electrolyzers can also be optimized to maximize hydrogen production given a specific operational strategy, and we can even use more electrolyzers with smaller capacities to adopt wind power.While these design problems are imperative, we highlight the operational-level details in this paper.
The wind turbine parameters are obtained from specifications of V150 wind turbine [28].Part of the electrolyzer parameters are from [29], but hot and cold start time are not reported.These two parameters are estimated according to [20,30].The parameters of voltage degradation  ℎ and   , caused by start/stop cycles, are seldom reported by technical reports or scientific papers.Thus, we made initial estimates of the order of magnitude of these parameters in light of [31].The electrolyzer heat capacity is estimated according to [32].Other thermal parameters are estimated based on the test from our experiment of a smaller alkaline electrolyzer.

Operational details
To fully reveal the operational performance led by the proposed models and active strategies, we pay attention to a specific daily operation of the hybrid wind/hydrogen system and show the power trajectory and state transitional processes of each electrolyzer.For the investigated day, the wind power output from the turbine is presented in Fig. 6, which is obtained from converting wind speed into power using classical wind turbine models [33].Note that this study's time resolution is 5 min instead of one hour that may obscure operational details.
The operational details are shown from Figs. 7 to 9 towards different operational strategies.Fig. 7(A) illustrates the power allocations resulting from strategy S1 among the four 1 MW electrolyzers at every time step.It is firstly observed that electrolyzer 1 is frequently used to produce hydrogen because it has a high priority to consume the wind power given strategy S1.Other electrolyzers are only used if there is sufficient wind power, e.g., from hours 5 to 6. Fig. 7(B) further displays the electrolyzer states.While electrolyzer 1 mostly works on production states, other electrolyzers are frequently put on standby or shut down.Electrolyzer 4 are seldom utilized, leading to a low load factor.
There are two more details worth mentioning: the transition from standby to off state and the cold start processes.Strategy S1 claims that if an electrolyzer is operated in standby for a long time, it will be shut down.The maximal standby time is set as 30 min in this calculation.
To see this, we can observe that the red blocks in Fig. 7(B) will not last for more than 30 min and every red block lasting 30 min is followed by a long blue block, which means the transition from standby to off state.It should also be highlighted that a cold start cannot be finished instantaneously, and it takes one hour for the selected electrolyzers.This can be demonstrated by the observation of blue blocks.The end of the blue blocks with a blue gradient indicates the cold start processes.Electrolyzers during clod start cannot produce hydrogen, which should be reflected by its operational models.
Fig. 8 displays the operational details of the electrolyzer group when strategy S2 is utilized.The only difference of S1 and S2 is that strategy S2 renumbers the electrolyzers every 24 time steps, i.e., 2 h in this case.The small change leads to distinct operational properties.The electrolyzer power curves show that the wind power is evenly distributed to each electrolyzer and no electrolyzer dominates the hydrogen production.In parallel, no electrolyzer is shut down since all electrolyzers are frequently used.Fig. 8 shows a periodic hydrogen    production.During the first two hours, electrolyzer 1 is leading the hydrogen production.However, it will be electrolyzer 2 that dominates hydrogen production in the next two hours.Given strategy S2, the utilization rates of the four electrolyzers would be similar, although more hot starts are required implying more frequent state transitions between production and stand-by states.
Fig. 9 depicts the performance of strategy S3.It is seen that electrolyzer 1 is more frequently used, as what happens in strategy S1.Fewer state transitions are observed in Fig. 9(B).The key contribution of strategy is the optimal allocation of wind power to each electrolyzer.
We summarize the short-term operational performance, as shown in Table 9.The resulting daily hydrogen production increases from strategy S1 to strategy S3, highlighting the importance of strategical operation.Similarly, the system efficiency is also raised using more effective strategies.Strategy S3 outperforms the other two due to the introduction of optimal dispatch of the wind power, which demonstrates the effectiveness of the rolling optimization.
The temperature variation of the electrolyzer stacks is shown in Fig. 10.The flow rate of the cooling water is adjusted according to the control algorithm in (10) to maintain the temperature within safe range.The result for strategy S1 shows that the temperature variation of each electrolyzer significantly differs.The strategy uses electrolyzer 1 more frequently, and its temperature is much higher than that of the others.The current lye flow rate is not fast enough to cool down the electrolyzer 1, leading to a higher temperature than the reference temperature, 80 • C. The temperature of other electrolyzers declines due to heat loss to the environment when they are shut down.Strategy S2 results in a more stable temperature variation for all the four electrolyzers.The temperature is controlled to be within the range of 80 • C to 90 • C. As each electrolyzer is used in a similar way, no significant temperature difference is observed among the electrolyzers.It is found that strategy S3 has the most stable temperature performance due to the relatively uniform load distribution among the four electrolyzers.
The above results demonstrate that mode switch among production, standby and off states play a central role in the management of multiple electrolyzers.The temperature variation is also significantly influenced.These findings have not been well discussed in the existing literature such as [10,11].The proposed model enables us to investigate these operational details to support more realistic and reliable operational strategies.

Strategy performance for a yearly operation
This section extends the operational time to a year and calculates the operational performance indicators.Fig. 11 summarizes the key indicators for the electrolyzers given different operational strategies.It is found from Fig. 11(A) that the number of hot start in strategy S1 increases from electrolyzer 1 to electrolyzer 4. Number of hot start in strategy S2 is similar in terms of each electrolyzer due to the symmetric nature of the electrolyzer operation.Strategy S3 results in a fewer hot start number, and electrolyzer 1 is the most frequently put into standby, different from what observed in strategy S1.Cold starts are only observed in strategy S1 since only in S1 can an electrolyzer standby for a long time.Fig. 11(C) depicts the load factors of the four electrolyzers.We found that strategy S1 leads to an unbalanced utilization of the electrolyzers.The load factor of electrolyzer 1 reaches around 80%, but electrolyzer 4 only has load factor of 15%.Strategy S2, as expected, results in same load factors for each electrolyzer, at around 40%.Finally, in strategy S3, electrolyzer 1 is the one mostly utilized but the difference of load factor among the four electrolyzers is less, compared to strategy S1.Fig. 11(D) illustrates the yearly energy consumption to sustain standby for each electrolyzer.In strategy S1 and S3, electrolyzer 4 is frequently put into standby, thereby consuming the most energy for standby.Strategy 2 exhibits similar energy consumption for each electrolyzer.Also note that the strategy S1 has the least overall energy consumption because the electrolyzers are shutdown to avoid long-time standby.
Table 10 displays the performance indicator for the yearly operation.The overall hydrogen production and system efficiency are increased from strategy S1 to S3. Strategy S3 results in a system  efficiency of 0.613, which is 4.1% higher than that of strategy S1.Correspondingly, the LCOH in S3 reaches 3.89 e/kg, which is 0.35 e/kg less than LCOH resulting from S1.There are two reasons for the lower LCOH in S3: first, more efficient hydrogen production; and second, less degradation.As electrolyzers bear higher efficiency when it is working at partial load, S3 optimally allocate the wind power to the electrolyzers to maximize the hourly hydrogen production rate.In parallel, compared to S1 and S2, S3 leads to fewer number of hot/cold starts, which alleviates the electrolyzer degradation and increases the overall efficiency in the long run.These results demonstrate the   effectiveness of the proposed rolling optimization scheme to support more efficient operation of the hybrid wind/hydrogen system.

Model comparison
To further understand the characteristics of the proposed model, in this section, we recalculate the cases in Sections 5.2 and 5.3 using different models and compare the results.Electrolyzers can be modelled in multiple ways, depending on the specific sub-models of describing the hydrogen production, thermal properties, state transitions and degradation.Table 11 introduces two other models except the proposed model, namely model M1.Model M2 is used in a research regarding the integration of alkaline electrolyzers in district heating system.Compared to the proposed model M1, this model simplifies the heat transfer and temperature control and the state transitions.Model M3 is a even simpler model, which is used in the design of wind-powered hydrogen electrolyzer hubs.Such a model is in fact widely used in feasibility, economic assessment of systems involving electrolyzers.
To make the comparison concise, we focus on the operational details when strategy S1 is applied.Fig. 12 presents the operational details of the four electrolyzers, and Fig. 7 is repeated here for convenience.The first observation is that electrolyzers are more frequently used when model M2 and M3 are utilized in the calculation.Model M2 only shows the state transition between standby and production while model M3 does not consider any state constraints.Due to the lack of physical constraints (e.g.start up time), model M2 and M3 result in higher daily hydrogen production as well as system efficiency, as shown in Table 12.The temperature variation is also not well revealed from model M2 and M3.Model M2 assumes a free variable called waste heat, which conceals the heat transfer process and cannot reveal information of temperature of inlet lye flow etc. Model M3 assumes that temperature is always controlled at a fixed value, representing an ideal case of temperature control.The detailed temperature profile is illustrated in Fig. 13.Table 13 presents the yearly operational results from different models.Model M2 and M3 lead to higher hydrogen production and lower LCOH, presenting an overestimate assessment due to lack of model details.The LCOH difference could be as large as 0.6 e/kg, highlighting the importance of selecting proper models.It should be noted that model M1 and M2 use different ways of dealing with degradation.However, the effectiveness of the models should be further evaluated using real-life data.

Conclusions and further research
This work optimizes the operation of an off-grid wind/hydrogen system with four electrolyzers considering the efficiency variation, hot/cold starts, thermal management, and degradation of each electrolyzer.Three operational strategies are proposed to allocate the wind power to the four electrolyzers, given the physical restrictions.The objective of the operation is to increase the system efficiency and bring down the hydrogen production costs, i.e., LCOH, in this paper.
Strategy S1 prioritizes the use of electrolyzer 1, and other electrolyzers are frequently put on standby or shut down.The detailed daily operation shows that such a strategy causes imbalanced load distribution and temperature, and the yearly operation shows lower system efficiency of 0.572 and higher LCOH of 4.24 e/kg due to less efficient load level and more degradation.
Strategy S2 uses the four electrolyzers in turn, leading to a balanced load level and more stable temperature variation, as shown by the daily operation.The annual system efficiency reaches 0.587, and the LCOH is 4.10 e/kg.
Strategy S3 is based on step-by-step optimization.Although such optimization may not lead to a theoretically optimal solution, the annual system efficiency rises to 0.613, and the LCOH declines to 3.89 e/kg.These improvements are mainly from more efficient allocation of wind power and less degradation.The electrolyzer temperature resulting from this strategy is also more stable.
To show the necessity of the proposed electrolyzer model, we compare it with two cutting-edge models by using them in the same case studies.The results show that the proposed model provides much more operational details, thereby leading to a more reasonable basis for operational decision making.
Further research can be conducted from the perspective of more realistic modelling and operational strategies.The degradation model in this paper can be further improved by using physical models and validated by the real operational data.Besides, it is assumed that the temperature has no influence of the hydrogen conversion rate of the electrolyzers in this paper.If the temperature of the electrolyzer is in a larger range, the coupling of electrochemical and thermal models has to be taken into account.In terms of operational strategies, model predictive control can be an effective way to optimize the system operation.Also, instead of using a rule-based operational strategy, one can optimize the system operation using mathematical programming.However, the non-convex and non-linear properties of the electrolyzers and temperature feedback control can be foreseeable challenges.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 2 .
Fig. 2. Tested hydrogen production rates at several power levels and the corresponding fitting curve for ṁ().

Fig. 3 .
Fig. 3. Thermal model of an electrolyzer stack.(A) Heat exchanges occurring within an electrolyzer stack (B) Heat exchange between lye and cooling water (in heat exchanger).

Fig. 4 .
Fig. 4. (A) State updates and real power consumption calculation given a power input signal of electrolyzer  at time .(B) External functions that can be called to manipulate the electrolyzer state.

Fig. 5 .
Fig.5.Rolling optimization used in strategy 3. Note that the parameters in the optimization problems is dependent on the electrolyzer status at the previous time step.

Fig. 10 .
Fig. 10.Temperature variation during the daily operation given different operational strategies.

Fig. 11 .
Fig. 11.Operational performance indicators for an yearly operation (A) Number of hot starts (B) Number of cold starts (C) Load factors (D) Energy consumption in standby state of each electrolyzer.

Fig. 13 .
Fig. 13.Temperature variation of the electrolyzer stacks resulted from different models.

Table 1
Definition of state variables for electrolyzers. , < RTH  During hot-start process 0 ≤  , < RTC  During cold-start process  , = RTH  Hot start finished  , = RTC  Cold start finished

Table 2
Key assessment indicators for evaluating the system operation.

Table 3
Simple start-stop strategy.

Table 4
Cycle rotation strategy.

Table 5
Rolling optimization-based strategy.

Table 6
Parameters for the wind turbine.

Table 7
General parameters for the single 1 MW electrolyzer.

Table 8
Thermal parameters for the single 1 MW electrolyzer and related heat exchanger.

Table 9
Performance assessment indicators for a daily operation.

Table 10
Performance assessment indicators for a yearly operation.

Table 11
Cutting-edge models for electrolyzers.

Table 12
Results from different models for the daily operation.

Table 13
Model performance for the yearly operation.