Electrolyzer Model¶

The electrolyzer model implements power-to-hydrogen (P2H2) conversion [49], coupling the electrical power system with hydrogen production for sector coupling applications [50]. It is implemented in electrolyzer.jl.

1. Overview¶

Electrolyzers consume electricity to produce hydrogen via water electrolysis (splitting water molecules into hydrogen and oxygen). In power system planning, electrolyzers serve dual roles:

Flexible load: Electrolyzers can modulate their power consumption to absorb surplus renewable generation, providing demand-side flexibility.
Sector coupling: Hydrogen output can satisfy non-electric energy demands (industrial feedstock, transport fuel, heating) or be stored for later reconversion to electricity via fuel cells.

The model supports:

Capacity investment decisions for new electrolyzer installations
Variable efficiency modeling based on operating load
Ramp rate constraints reflecting physical limitations
Coupling with both the power system (electricity consumption) and the primary energy module (hydrogen output)
Water consumption tracking

2. Technology Background¶

2.1 Electrolyzer Types¶

The model is technology-agnostic but parameterized to represent common electrolyzer technologies:

Technology	Typical Efficiency	Ramp Rate	Capital Cost	Lifetime
Alkaline (AEL)	60--70%	Moderate	Lower	20--30 years
PEM (PEMEL)	55--70%	Fast	Higher	10--20 years
Solid Oxide (SOEL)	70--85%	Slow	Highest	5--10 years

The efficiency values represent the ratio of hydrogen energy output (HHV) to electrical energy input.

2.2 Energy Content of Hydrogen¶

The energy required to produce hydrogen depends on the thermodynamic pathway. Key reference values:

Property	Value
Lower Heating Value (LHV)	33.33 kWh/kg
Higher Heating Value (HHV)	39.41 kWh/kg
Thermodynamic minimum	39.41 kWh/kg (reversible, HHV)
Practical consumption	50--60 kWh/kg (including system losses)

The model parameter energy_per_kg_h2 captures the total system-level energy consumption per kilogram of hydrogen produced.

3. Decision Variables¶

Variable	Domain	Units	Description	Julia Name
$I^{elz}_n$	$\mathbb{R}_+$	MW	Electrolyzer capacity investment at node $n$	`elz_investment[n]`
$P^{elz}_{n,t}$	$\mathbb{R}_+$	MW	Power consumption at node $n$, hour $t$	`electrolyzer_power[n,t]`
$H_{n,t}$	$\mathbb{R}_+$	kg/h	Hydrogen production at node $n$, hour $t$	`hydrogen_production[n,t]`

Built by build_electrolyzer_variables!().

4. Constraints¶

ELZ-1: Power Capacity Limit¶

Electrolyzer power consumption is bounded by the sum of existing rated capacity and new investment:

\[ P^{elz}_{n,t} \leq \bar{P}^{elz}_n + I^{elz}_n \quad \forall n, t \tag{ELZ-1} \]

where:

$\bar{P}^{elz}_n$ is the existing rated electrolyzer capacity at node $n$ (MW)
$I^{elz}_n$ is the investment variable (MW), bounded by invest_max_power[n]

This constraint ensures that power consumption never exceeds total installed capacity. The investment variable is bounded by:

\[ 0 \leq I^{elz}_n \leq \bar{I}^{elz}_n \qquad \forall n \]

where $\bar{I}^{elz}_n$ is the maximum allowable investment at node $n$.

ELZ-2: Hydrogen Production¶

Hydrogen production is proportional to power consumption via an average efficiency:

\[ H_{n,t} = P^{elz}_{n,t} \times \frac{1000 \cdot \bar{\eta}^{elz}}{E_{H_2}} \quad \forall n, t \tag{ELZ-2} \]

where:

$\bar{\eta}^{elz} = (\eta^{rated} + \eta^{min}) / 2$ is the average efficiency over the operating range
$E_{H_2}$ is the energy per kilogram of hydrogen (kWh/kg), from energy_per_kg_h2
The factor 1000 converts MW to kW for unit consistency

Derivation of the average efficiency. Electrolyzer efficiency varies with load: it is typically highest at partial load and decreases at rated power due to increased overpotentials. The true efficiency curve $\eta(P)$ is nonlinear. Rather than introducing a piecewise-linear approximation (which would add complexity), the model uses the arithmetic mean of the efficiencies at rated power and minimum operating point:

\[ \bar{\eta}^{elz} = \frac{\eta^{rated} + \eta^{min}}{2} \]

For example, with $\eta^{rated} = 0.70$ and $\eta^{min} = 0.65$, the average efficiency is 0.675.

Efficiency Approximation

The model uses a constant average efficiency rather than a piecewise-linear efficiency curve. This linearization avoids additional variables and constraints while providing a reasonable approximation for capacity planning studies. For detailed operational studies where load-dependent efficiency significantly impacts results, a PWL efficiency curve could be implemented using the same segment decomposition approach used for generator fuel cost curves (see Operational Dispatch -- PWL).

ELZ-3: Ramp Rate Constraints¶

Power consumption changes between consecutive time steps are limited by ramp rates, reflecting the physical inertia of the electrolysis process (thermal management, gas handling, membrane conditioning):

\[ P^{elz}_{n,t} - P^{elz}_{n,t-1} \leq (\bar{P}^{elz}_n + I^{elz}_n) \cdot r^{up}_{n} \quad \forall n, t > 1 \tag{ELZ-3a} \]

\[ P^{elz}_{n,t-1} - P^{elz}_{n,t} \leq (\bar{P}^{elz}_n + I^{elz}_n) \cdot r^{down}_{n} \quad \forall n, t > 1 \tag{ELZ-3b} \]

where:

$r^{up}_n \in [0,1]$ is the maximum ramp-up rate (fraction of total capacity per hour)
$r^{down}_n \in [0,1]$ is the maximum ramp-down rate (fraction of total capacity per hour)

Typical ramp rates range from 0.1 (solid oxide, slow thermal response) to 1.0 (PEM, can ramp from zero to full power in under a minute, effectively unconstrained at hourly resolution).

Implemented in add_electrolyzer_constraints!().

ELZ-4: Minimum Operating Point (Optional)¶

When configured, the electrolyzer has a minimum stable operating point below which it must shut down:

\[ P^{elz}_{n,t} \geq \underline{P}^{elz}_n \cdot u^{elz}_{n,t} \quad \forall n, t \]

where $\underline{P}^{elz}_n$ is the minimum power fraction and $u^{elz}_{n,t}$ is a binary on/off status. In the LP formulation, this constraint is relaxed or omitted to maintain linearity.

5. Objective Terms¶

The electrolyzer contributes the following cost terms to the objective function, implemented in get_electrolyzer_objective_terms():

\[ C^{elz} = \sum_n \left[ C^{elz,inv}_n + C^{elz,fix}_n + C^{elz,var}_n + C^{elz,water}_n \right] \tag{ELZ-OBJ} \]

5.1 Annualized Investment Cost¶

Investment cost is annualized over the electrolyzer lifetime and prorated to the optimization window:

\[ C^{elz,inv}_n = \frac{c^{inv,elz}_n}{\tau^{elz}_n \cdot 8760} \cdot I^{elz}_n \cdot T \]

where:

$c^{inv,elz}_n$ is the total investment cost ($/MW)
$\tau^{elz}_n$ is the lifetime (years)
$T$ is the number of time steps in the optimization window
The factor $8760$ converts annual cost to hourly

5.2 Fixed O&M Cost¶

Fixed operations and maintenance cost, proportional to total installed capacity:

\[ C^{elz,fix}_n = c^{fix,elz}_n \cdot (\bar{P}^{elz}_n + I^{elz}_n) \cdot T \]

where $c^{fix,elz}_n$ is the fixed O&M cost rate ($/MW/hour).

5.3 Variable O&M Cost¶

Variable cost proportional to electricity consumed:

\[ C^{elz,var}_n = \sum_t c^{var,elz}_n \cdot P^{elz}_{n,t} \]

where $c^{var,elz}_n$ is the variable O&M cost ($/MWh).

5.4 Water Cost¶

Water consumption cost proportional to hydrogen produced:

\[ C^{elz,water}_n = \sum_t c^{water} \cdot H_{n,t} \]

where $c^{water}$ is the water cost per kilogram of hydrogen ($/kg H2). Water consumption for electrolysis is approximately 9 liters per kg H2.

5.5 Cost Summary Table¶

Term	Formula	Units	Description
Annualized investment	$\frac{c^{inv}}{\tau \cdot 8760} \cdot I \cdot T$	$	Capital cost spread over lifetime
Fixed O&M	$c^{fix} \cdot (\bar{P} + I) \cdot T$	$	Capacity-proportional maintenance
Variable O&M	$\sum_t c^{var} \cdot P_t$	$	Energy-proportional operating cost
Water	$\sum_t c^{water} \cdot H_t$	$	Feedstock cost

6. Power System Coupling¶

The electrolyzer is coupled to the power system as an additional electrical load. Electrolyzer power consumption appears on the demand side of the power balance equation:

Single-bus (PB-1):

\[ \sum_g P_{g,n,t} + \ldots = D_{n,t} + P^{elz}_{n,t} + \sum_b P^{ch}_{b,n,t} + \ldots \]

Multi-bus (DC-1):

\[ \text{net\_inj}_{n,t} = \sum_g P_{g,n,t} - D_{n,t} - P^{elz}_{n,t} - \ldots = \sum_\ell K_{n,\ell} f_{\ell,t} \]

This coupling is handled by couple_electrolyzer_to_power_system!(). The electrolyzer power variable $P^{elz}_{n,t}$ is added to the demand-side expressions at the node where the electrolyzer is located.

7. Hydrogen Demand Coupling¶

When the primary energy module is active, hydrogen produced by the electrolyzer can satisfy hydrogen demand requirements:

\[ \sum_t H_{n,t} \cdot \Delta t \geq D^{H_2}_{n} \quad \forall n \]

where $D^{H_2}_n$ is the hydrogen demand at node $n$ over the optimization window (kg). This constraint links the electrolyzer to the broader energy system, ensuring that hydrogen production meets industrial, transport, or other non-electric hydrogen demands.

If hydrogen storage is available, the temporal profile of production can differ from the demand profile, allowing the electrolyzer to operate flexibly:

\[ V^{H_2}_{n,t+1} = V^{H_2}_{n,t} + H_{n,t} \cdot \Delta t - W^{H_2}_{n,t} \]

where $V^{H_2}_{n,t}$ is the hydrogen inventory and $W^{H_2}_{n,t}$ is the hydrogen withdrawal for end-use.

8. Operational Strategies¶

The optimizer naturally selects the electrolyzer operating strategy that minimizes total system cost:

Strategy	Condition	Behavior
Load following RE	High RE penetration	Electrolyzer absorbs surplus RE, reducing curtailment
Baseload	Low electricity prices	Continuous operation at high capacity factor
Peak shaving	High peak demand	Electrolyzer reduces consumption during peaks
Idle	High electricity prices	Electrolyzer shuts down, hydrogen demand unmet (penalized)

The flexibility of the electrolyzer is bounded by ramp rate constraints (ELZ-3) and minimum operating point (if configured). Faster-ramping technologies (PEM) can better follow variable renewable generation.

9. Configuration¶

Electrolyzer parameters are specified per node in the system configuration:

electrolyzers:
  alkaline:
    name: Alkaline Electrolyzer
    rated_power: [10.0]           # MW per node
    eff_at_rated: [0.70]          # Efficiency at rated power
    eff_at_min: [0.65]            # Efficiency at minimum power
    energy_per_kg_h2: 55.0        # kWh per kg H2 (system-level)
    ramp_up: [0.5]                # Fraction of capacity per hour
    ramp_down: [0.5]
    invest_cost: [1500000.0]      # $/MW total investment
    invest_max_power: [50.0]      # MW maximum investment per node
    fixed_cost: [30000.0]         # $/MW/year fixed O&M
    variable_cost: [2.0]          # $/MWh variable O&M
    water_cost: 0.05              # $/kg H2
    life_time: [20]               # years

9.1 Parameter Reference¶

Parameter	Symbol	Units	Description
`rated_power`	$\bar{P}^{elz}_n$	MW	Existing installed capacity per node
`eff_at_rated`	$\eta^{rated}$	--	Efficiency at rated power
`eff_at_min`	$\eta^{min}$	--	Efficiency at minimum operating point
`energy_per_kg_h2`	$E_{H_2}$	kWh/kg	System energy consumption per kg H2
`ramp_up`	$r^{up}$	1/h	Maximum ramp-up rate (fraction of capacity)
`ramp_down`	$r^{down}$	1/h	Maximum ramp-down rate (fraction of capacity)
`invest_cost`	$c^{inv,elz}$	$/MW	Total investment cost
`invest_max_power`	$\bar{I}^{elz}$	MW	Maximum investment per node
`fixed_cost`	$c^{fix,elz}$	$/MW/yr	Annual fixed O&M cost
`variable_cost`	$c^{var,elz}$	$/MWh	Variable O&M cost
`water_cost`	$c^{water}$	$/kg H2	Water feedstock cost
`life_time`	$\tau^{elz}$	years	Equipment lifetime

10. Implementation Reference¶

Julia Function	Purpose	File
`build_electrolyzer_variables!()`	Create investment, power, and production variables	`electrolyzer.jl`
`add_electrolyzer_constraints!()`	Add ELZ-1 through ELZ-3 constraints	`electrolyzer.jl`
`get_electrolyzer_objective_terms()`	Compute objective cost contributions	`electrolyzer.jl`
`couple_electrolyzer_to_power_system!()`	Add electrolyzer power to demand side of balance	`electrolyzer.jl`
`extract_electrolyzer_solution()`	Extract power, production, and investment results	`electrolyzer.jl`

References¶

The power-to-hydrogen conversion modeling and electrolysis technology review follow Buttler and Spliethoff [49]. The economics of renewable hydrogen production are analyzed by Glenk and Reichelstein [50].

See the full bibliography for complete citation details.

Variable	Domain	Units	Description	Julia Name
\(I^{elz}_n\)	\(\mathbb{R}_+\)	MW	Electrolyzer capacity investment at node \(n\)	`elz_investment[n]`
\(P^{elz}_{n,t}\)	\(\mathbb{R}_+\)	MW	Power consumption at node \(n\), hour \(t\)	`electrolyzer_power[n,t]`
\(H_{n,t}\)	\(\mathbb{R}_+\)	kg/h	Hydrogen production at node \(n\), hour \(t\)	`hydrogen_production[n,t]`

Term	Formula	Units	Description
Annualized investment	\(\frac{c^{inv}}{\tau \cdot 8760} \cdot I \cdot T\)	$	Capital cost spread over lifetime
Fixed O&M	\(c^{fix} \cdot (\bar{P} + I) \cdot T\)	$	Capacity-proportional maintenance
Variable O&M	\(\sum_t c^{var} \cdot P_t\)	$	Energy-proportional operating cost
Water	\(\sum_t c^{water} \cdot H_t\)	$	Feedstock cost

Parameter	Symbol	Units	Description
`rated_power`	\(\bar{P}^{elz}_n\)	MW	Existing installed capacity per node
`eff_at_rated`	\(\eta^{rated}\)	--	Efficiency at rated power
`eff_at_min`	\(\eta^{min}\)	--	Efficiency at minimum operating point
`energy_per_kg_h2`	\(E_{H_2}\)	kWh/kg	System energy consumption per kg H2
`ramp_up`	\(r^{up}\)	1/h	Maximum ramp-up rate (fraction of capacity)
`ramp_down`	\(r^{down}\)	1/h	Maximum ramp-down rate (fraction of capacity)
`invest_cost`	\(c^{inv,elz}\)	$/MW	Total investment cost
`invest_max_power`	\(\bar{I}^{elz}\)	MW	Maximum investment per node
`fixed_cost`	\(c^{fix,elz}\)	$/MW/yr	Annual fixed O&M cost
`variable_cost`	\(c^{var,elz}\)	$/MWh	Variable O&M cost
`water_cost`	\(c^{water}\)	$/kg H2	Water feedstock cost
`life_time`	\(\tau^{elz}\)	years	Equipment lifetime