Optimal Operation and Market Integration of a Hybrid Farm with Green Hydrogen and Energy Storage: A Stochastic Approach Considering Wind and Electricity Price Uncertainties

where SOH is the state of health in per unit (pu), $Ca{p}_{Av}$ is the currently available capacity, and $Ca{p}_{Nom}$ is the nominal capacity, both in MWh.

, for a new BESS can be estimated by utilizing a given DOD. This is calculated for each cycle c. This is illustrated in Figure 2. The variation in state of health ($SO{H}_{BESS}$), measured in per unit, that results from this can be translated into a cost through the use of the units-of-production method described in [22].

• The BESS cannot be charged above its nominal power, as represented by the equation

  $$P_{ch}$$

  (3)

  where $P_{ch}$

  is the charging power during hour t in MW, $P_{nom}$ is the nominal power in MW, and ND(t) is a boolean variable that is equal to 1 when the BESS is not discharging at time t.

• The BESS cannot discharge above its nominal power, as represented by the equation

  $$P_{dis}$$

  (4)

  where $P_{dis}$

  is the discharging power during hour t in MW, $P_{nom}$ is the nominal power in MW, and NC(t) is a boolean variable that is equal to 1 when the BESS is not charging at time t.

• Charging power is always positive, as represented by the equation
• Discharging power is also always positive, as represented by the equation

  $$P_{dis}$$

  (6)

• The BESS cannot be charged and discharged simultaneously, as represented by the equation

  $$NC\left(t\right)+ND\left(t\right)\le 1.$$

  (7)

• The resulting BESS power at period t is computed for convenience as

  $$P_{Bat}$$

  (8)

• The energy stored at the end of the period equals the amount stored at the end of the previous hour and the charged/discharged energy considering efficiencies, as represented by the equation

  $$E$$

  ξ D i s ,

  (9)

  where E(t) is the stored energy at the end of hour t in MWh, ${\xi }_{Ch}$ is the charging efficiency as a percentage, and ${\xi }_{Dis}$ is the discharging efficiency as a percentage.

• The SOC at any given time must be between the boundaries $SO{C}_{min}$ and $SO{C}_{max}$.

• The SOC at $t=0$, $SO{C}_{init}$, is set by the user for the first day of simulation, but it inherits its value from the final value of the previous day for multiple-day simulations.

• The BESS daily degradation cost is computed as follows:

  $$De{g}_{cost,BESS}=\Delta SO{H}_{BESS}·K_{BESS},$$

  (10)

  where $De{g}_{cost,BESS}$ is the daily BESS degradation cost in EUR, and $K_{BESS}$ is the BESS replacement cost in EUR. The use of $K_{BESS}$ for computing a cost from battery degradation creates an optimization problem for amortizing a future BESS replacement.

where:

• $Lif{e}_{AEL}$

  : AEL remaining useful lifetime t (pu);

• Hours(t): Hours of operation during time period t;

• $EO{L}_{hours}$: Lifetime hours;

• Cycles(t): Start/stop cycles performed during time period t;

• $EO{L}_{cycles}$: Start/stop lifetime cycles.

where:

• On mode: The AEL is producing hydrogen and operating at nominal conditions.

• Idle mode: Nominal conditions are maintained, but the AEL is not producing hydrogen.

• Off mode: The AEL is not generating hydrogen, and no energy is being applied to maintain nominal conditions, resulting in a gradual cooling until reaching a fully cold state.

• The total electrolyzer power output for each time period is derived from the sum of the various operating modes:

  $$P_{on}\left(t\right)+P_{idle}\left(t\right)+P_{start}\left(t\right)=P_{AEL}\left(t\right),$$

  (13)

  where $P_{on}\left(t\right)$ is the power employed in producing hydrogen during hour t in MW, $P_{idle}\left(t\right)$ is the power consumed in the Idle mode during hour t in MW, $P_{start}\left(t\right)$ is the cold starting power consumed during time period t in MW, and $P_{AEL}\left(t\right)$ is the total power consumption during hour t in MW.

• Binary variables are used to determine the AEL operation mode at each hour, and only one mode is allowed at a time:

  $${\pi }_{on}$$

  (14)

  where ${\pi }_{on}$

  is the binary variable for On mode, ${\pi }_{off}$

  is the binary variable for Off mode, and ${\pi }_{idle}$

  is the binary variable for Idle mode. Each binary variable takes value 1 when its respective mode is enabled.

• An additional mode is used to define the cooling process, which appears only when the AEL is off:

  $$Cool\left(t\right)\le {\pi }_{off}$$

  (15)

  where $Cool\left(t\right)$ is a binary variable that defines Cool mode, in which AEL temperature and pressure slowly decrease since it is shut down. It equals 1 when its temperature and pressure state has become colder.

• The pressure and temperature state of the electrolyzer is updated at each hour as follows:

  $$\tau \left(t\right)=\tau \left(t-1\right)+\upsilon$$

  (16)

  where $\tau \left(t\right)$ is the state at the end of hour t; this variable is limited between 0 and $t_{off}$. The variable $t_{off}$ is the number of hours it takes to get fully cold when off, and $\upsilon \left(t\right)$ is the fraction of the starting power required for a cold start.

• Whenever the electrolyzer is off, its temperature and pressure decrease until they reach the fully cold state:

  $$\left\{\begin{array}{c}Cool\left(t\right)\le \frac{\tau \left(t-1\right)}{t_{off}}+1\hfill \end{array}\right.$$

  C o o l ( t ) ≥ τ t − 1 t o f f − 1 + π o f f ( t )

  ,

  (17)

  where the value 1 in the first term prevents the binary variable $Cool\left(t\right)$ from reaching a value of 0 until $\frac{\tau \left(t-1\right)}{t_{off}}$ is 0.

• Idle mode’s hourly power consumption is defined as:

  $$P_{idle}={\pi }_{idle}$$

  (18)

  where $P_{idle}$ is the hourly power consumption during Idle mode in MW.

• On mode’s hourly power consumption is limited as:

  $$P_{on,min}·{\pi }_{on}$$

  (19)

  where $P_{on,min}$ and $P_{on,max}$ are the minimum and maximum AEL On mode’s active powers, respectively, in MW.

• The cold start power factor is defined as:

  $$\upsilon$$

  t o f f .

  (20)

  This factor, bounded between 0 and 1, represents how far the AEL is from reaching an operational state. If 1, it needs the full cold start power.

• The cold start power consumption is defined as:

  $$P_{start}\left(t\right)=\upsilon$$

  (21)

  where $P_{start}\left(t\right)$ is the AEL starting power in MW, $t_{start}$ is the cold start time in hours, and $AE{L}_{Pmax}$ is the maximum electrolyzer power in MW. This equation models the starting power consumption and considers that less power is required if the AEL is not completely cold when a cold start takes place.

• The hydrogen generation process is defined as:

  $$Q\left(t\right)=P_{on}·\frac{AE{L}_{eff}}{HHV},$$

  (22)

  where $Q\left(t\right)$ is the hydrogen flow during hour t in kg, $AE{L}_{eff}$ is the AEL efficiency in pu, and $HHV$ is the hydrogen high heating value in kg/MWh.

• The operation cost of the AEL, defined by the degradation models described above, is formulated as:

  $$De{g}_{Cost,AEL}$$

  E O L h o u r s + π o f f ( t − 1 ) + π o n ( t ) + π i d l e ( t ) − 1

  E O L c y c l e s · K ,

  (23)

  where $De{g}_{Cost,AEL}$

  is the degradation cost during hour t in EUR, and K is the electrolyzer replacement cost in EUR. The first term of the sum represents the degradation caused by operational hours of usage, and the second term is the degradation caused by on/off operational cycles.

• The oxygen contamination process is formulated as:

  $$imp\left(t\right)=imp\left(t-1\right)+cont\left(t\right)-2·purg$$

  (24)

  where $imp\left(t\right)$ is the oxygen impurity level at the end of hour t in %, $cont\left(t\right)$ is the oxygen contamination during hour t in %, and $purg\left(t\right)$ is a binary variable that defines the AEL purging.

• The Purging mode is enabled when no $H_2$ is being produced:

  $$purg\left(t\right)\le {\pi }_{off}$$

  (25)

  An inequality constraint is applied to allow for the possibility of Off and Idle modes to be non-zero when no purge is in progress. This scenario occurs, for instance, when the AEL is in a clean state and does not intend to generate hydrogen, enabling the Off or Idle binary variables to have a value of 1.

• When oxygen contamination reaches 2%, the AEL is shut down:

  $$imp\left(t-1\right)-{\pi }_{off}$$

  (26)

• Purging mode is disabled when the contamination value reaches 0:

  $$purg\left(t\right)\le imp$$

  (27)

• A piecewise constraint is used to regulate the contamination process, derived from (12):

  $$cont$$

  < 0.3

  3.5 − 5 · P o n ( t ) A E L P m a x

  if 0.3 ≤ P o n ( t ) A E L P m a x

  ≤ 0.7

  0 if P o n ( t ) A E L P m a x

  > 0.7

  .

  (28)

• The power generated by the wind farm is defined as:

  $$P_{gen}\left(t\right)=P_{WTG,BESS}$$

  (29)

  where $P_{gen}\left(t\right)$ is the generated power during hour t in MW, $P_{WTG,BESS}$

  is the generated power used by the BESS during hour t in MW, $P_{WTG,Grid}$

  is the generated power sent to the grid during hour t, $P_{WTG,AEL}$

  is the generated power sent to the AEL during hour t in MW, and $P_{WTG,Curt}$

  is the curtailed power during hour t in MW.

• The BESS discharging power is divided as follows:

  $$P_D\left(t\right)=P_{BESS,Grid}\left(t\right)+P_{BESS,AEL}\left(t\right),$$

  (30)

  where $P_{BESS,Grid}\left(t\right)$ is the BESS power sent to the grid during hour t in MW, and $P_{BESS,AEL}\left(t\right)$ is the BESS power sent to the AEL during hour t.

• The AEL power flow is defined as follows:

  $$P_{AEL}\left(t\right)=P_{BESS,AEL}$$

  (31)

where ${\pi }_{el,fore}\left(t\right)$ is the hourly forecasted electricity price in EUR/MWh, and ${\pi }_{H2}$ is the hydrogen price in EUR/kg. As can be seen, the objective function has two terms: one for the operation in the electricity market and the other for the operation in the hydrogen market.

where $P_{Com,DM}\left(t\right)$ is the day-ahead market power commitment for hour t in MW, ${\pi }_{el,real}\left(t\right)$ is the real electricity price for hour t in EUR/MWh, and $\lambda$ is the deviation cost factor for hour t. The objective function of this optimization problem is formulated as follows:

where $Cost{(i,j)}_x$ represents the second-stage costs under wind speed scenario j and electricity price scenario i for candidate solution x in EUR. $De{v}_{Costs}$

are the deviation costs in EUR, and $\Delta E_{Ben}$

is the difference between expected and real electricity market benefits in EUR.

where $E_1$

are the expected benefits for candidate solution S in EUR, and $CVa{R}_{1-\beta }\left(E_2\left(S\right)\right)$ is the conditional value-at-risk of the second-stage costs for candidate solution S in EUR for the confidence interval $\beta$, which has been set to 95%, a common choice in stochastic problems.

where $CVa{R}_{1-\beta }\left(E_2\left(\overline{S}\right)\right)$ is the conditional value-at-risk of the second-stage cost means for candidate solution S in EUR for the confidence interval $\beta$. A 95% confidence interval is used in this case.

is appropriate for forecasting such prices, and the same order is used here. The previous 100 days of hourly electricity prices are used as training data.

. Wind speed data from the previous five days are used as training data.

• Ideal scenario: The plant receives perfect wind speed and electricity price forecasts.

• Stochastic Monte Carlo scenario: The Monte Carlo algorithm is used for day-ahead commitment generation.

• Stochastic Bootstrap scenario: The proposed bootstrap algorithm is used for day-ahead commitment generation.

• Deterministic scenario: A direct approach that considers deterministic forecasts.