Storage and coupled carriers: modelling power, hydrogen and heat together

Add a battery, an electrolyzer, a hydrogen tank and a heat store to one plant, and find where coupling pays

Modeler lesson M4 of Energy Systems, Step by Step: write one linear program for a wind plant behind a grid limit, add a battery’s state of charge, an electrolyzer with a part-load limit, a hydrogen tank with a minimum level and a heat store behind a heat-network limit, compare configurations in a winter and a summer week to see where coupling earns value, and read each carrier’s marginal value; a Python notebook rebuilds the model.
  • Modeler · L3
  • 40 min guided · 45 min lab · about 90 min practice
  • After M1 and A8

One wind plant, two weeks of 2025: what each step of coupling saves against separate systems

Configuration Winter week (9–15 Dec) Summer week (14–20 Jul)
+ electrolyzer and tank $61.3k $30.5k
+ heat pump and heat store $82.3k $33.2k
+ waste heat from the electrolyzer $83.6k $35.6k

Separate systems: the plant with its battery, every kg of hydrogen trucked in, all heat from a gas boiler. Each saving is a difference of optimal weekly costs, the same for every best plan.

LP relaxation: in these runs the electrolyzer may run below its 15% minimum load.

With JavaScript on, you add each device yourself and solve the plant's linear program in the browser.

Illustrative plant and prices. 100 MW of wind behind a 50 MW connection, trading at Prairie Town's cost of one more MWh ($0, $40 or $100, lesson 9). Trucked hydrogen $2.50 per kg; boiler heat $15 per MWh.

Real shapes. Wind and the town's demand follow the Southwest Power Pool's 2025 hours (EIA-930); wind is scaled to 33.8%, the U.S. wind fleet's 2025 capacity factor (EIA). Heat demand follows the town's shape to a 15 MW winter peak.

Efficiencies. Battery round trip 0.8259 (DOE 2022 assessment); alkaline minimum load 15% (IRENA 2020); heat pump 3 MWh of heat per MWh (DOE). Electrolyzer 57 kWh per kg: DOE's PEM figure, illustrative here, inside IRENA's 50–78 for alkaline systems.

About 40 guided minutes in eight steps. Solver: HiGHS (open source), loaded on your first solve (about 3.5 MB, once).

Build it in Python · notebook lab, about 45 minutes, then 90 minutes of practice

The same plant in scipy.optimize.linprog(method="highs"), and its exact on/off version with scipy.optimize.milp. assert cells check your costs, savings and prices against the lesson's reference block.

Schedules are printed, not asserted: other plans of the same cost can run the devices differently.

  1. Download the notebook and the four data files into one folder (in Colab, upload them with the folder icon).
  2. Open it in Google Colab (File → Upload notebook) or locally with Python 3, NumPy and SciPy 1.9 or later.
  3. Run all cells (a few seconds), then try the practice tasks at the end.

Notebook (.ipynb) Lesson data Town Year 2025 Capacity factors

What you just did, as diagrams

One LP, three carriers
A balance row each hour for power, hydrogen and heat: 2,520 columns and 1,176 rows for a week.
Storage links the hours
Battery, tank and heat store each carry a level from hour to hour and end the week where they start.
Conversion couples the balances
Coupling: the electrolyzer turns 1 MWh into about 17.5 kg (1,000 ÷ 57); the heat pump turns it into 3 MWh of heat.
Where coupling pays
The value of coupling: heat coupling saves $21.0k in the winter week, $2.7k in summer. Conversion and storage need cheap power and heat demand to connect; summer has little heat demand.
Coupled prices
When the heat pump sets heat's price, heat is worth power's price ÷ 3: $40 ÷ 3 = $13.33 per MWh of heat.
LP relaxation
These costs, savings and prices let the electrolyzer run below its 15% minimum; the exact on/off cost is shown only for the winter hydrogen case.
Go deeper · ModelerThe plant's LP in algebra

For each hour \(t = 0 \dots 167\) (MW over the hour is MWh), with \(t-1\) read cyclically so every level ends the week where it starts. Names as in the code and the notebook:

\[\min \sum_t \Big( p_t\,(\mathit{imp}_t - \mathit{exp}_t) + 2.50\,\mathit{tr}_t + 15\,\mathit{boil}_t \Big)\]

\[ \begin{aligned} \mathit{exp}_t + \mathit{bc}_t + \mathit{ez}_t + \mathit{hp}_t + \mathit{cur}_t - \mathit{imp}_t - \mathit{bd}_t &= w_t && \text{(power)}\\ k\,\mathit{ez}_t + \mathit{tr}_t + L_{t-1} - L_t &= 150 && \text{(hydrogen)}\\ \mathit{COP}\,\mathit{hp}_t + \mathit{rec}_t - \mathit{hin}_t + \mathit{hout}_t + \mathit{boil}_t &= H_t && \text{(heat)}\\ \mathit{COP}\,\mathit{hp}_t + \mathit{rec}_t - \mathit{hin}_t + \mathit{hout}_t &\le 12 && \text{(network)}\\ s_t = s_{t-1} + \eta\,\mathit{bc}_t - \mathit{bd}_t, \quad Q_t &= (1-\lambda)\,Q_{t-1} + \mathit{hin}_t - \mathit{hout}_t && \text{(levels)}\\ 400 \le L_t \le 4{,}000, \quad \mathit{rec}_t \le 0.2\,\mathit{ez}_t, \quad 0 \le \mathit{ez}_t &\le 20 && \text{(limits)} \end{aligned} \]

Symbol Meaning Unit
\(\mathit{exp}_t\), \(\mathit{imp}_t\) sold to and bought from the grid, each at most 50 MW
\(p_t\) Prairie Town’s price $/MWh
\(w_t\), \(\mathit{cur}_t\) wind, and wind curtailed MW
\(\mathit{bc}_t\), \(\mathit{bd}_t\), \(s_t\) battery charge, discharge (at most 20) and state of charge (at most 80) MW, MWh
\(\mathit{ez}_t\) electrolyzer power MW
\(L_t\), \(\mathit{tr}_t\) hydrogen in the tank, hydrogen trucked in kg
\(\mathit{hp}_t\) heat pump power (at most 5) MW
\(\mathit{rec}_t\), \(\mathit{boil}_t\), \(H_t\) heat recovered from the electrolyzer, from the boiler, and demanded MW of heat
\(\mathit{hin}_t\), \(\mathit{hout}_t\), \(Q_t\) heat store charge, discharge and level (at most 60) MW, MWh
\(k\) \(1000 / 57\), about 17.54: kg of hydrogen per MWh kg/MWh
\(\eta\) the battery’s round trip, 0.8259 (DOE’s 82.59%) share
\(\mathit{COP}\) the heat pump’s coefficient of performance, 3 MWh of heat per MWh
\(\lambda\) the heat store’s loss, \(1 - 0.98^{1/24}\) (2% a day) per hour
  • Duals. The power row’s dual, with its sign flipped, is power’s marginal value in that hour; the hydrogen and heat rows’ duals are the marginal costs of one more kg and one more MWh of heat.
  • Under this model’s assumptions: fixed sizes, the modelled prices, and the LP relaxation of the minimum load.
  • Unique or not. Step 7 offers only hours where each dual is unique; the test checks that raising and lowering the hour’s demand by 0.01 changes the cost at the same rate.
  • Size. 2,520 columns and 1,176 rows; HiGHS solves it in tens of milliseconds.
Go deeper · ModelerPart-load limits: two LPs and a mixed-integer program
  • The exact rule. Each hour the electrolyzer is on, between 15% and 100% of its size, or off: \(3\,z_t \le \mathit{ez}_t \le 20\,z_t\) with \(z_t \in \{0, 1\}\), a mixed-integer program (M2).
  • Two LP stand-ins. Drop the binary and allow any load (a relaxation, never dearer), or force \(z_t = 1\) (always on, never cheaper).
  • This winter week: relaxed −$178.6k, exact −$178.6k, forced on −$172.7k. The exact cost lies between the two, as it must.
  • No prices from the MILP. Its duals do not exist in the ordinary sense (§9.2); prices come from an LP with the on/off choices fixed.
Go deeperWhat this model leaves out
  • Sizes are given: choosing them is capacity expansion (M3), with a year of weeks linked as M3 links them.
  • Investment costs: the week's cost counts only buying, selling, trucks and the boiler.
  • Perfect foresight: the solver knows the whole week's wind and prices in advance.
  • Price-taking: the plant's sales do not move Prairie Town's price.
  • Electrochemistry, heat-network hydraulics and the tank's pressure: devices enter as efficiencies and limits.

Check yourself

Myth or fact? “A heat pump on a wind plant saves about the same in every season.”

  • Myth
  • Fact
Answer

Myth. Here heat coupling saves $21.0k in the winter week and $2.7k in the summer week: in summer there is little heat to make.

Both runs are LP relaxations that let the electrolyzer run below its 15% minimum.

Myth or fact? “Forcing an electrolyzer to stay on above its minimum load is the safe way to write the limit in an LP, and it costs nothing.”

  • Myth
  • Fact
Answer

Myth. It keeps the plan feasible but costs $6.0k more this winter week than the exact on/off model, which may switch the unit off.

The electrolyzer uses 57 kWh per kg. What coefficient links the power and hydrogen balances, in kg per MWh?

  • 57
  • 17.5
  • 0.057
  • 1.75
Answer

17.5: 1,000 ÷ 57 = 17.54 kg per MWh, about 17.5.

57 is kWh per kg, 0.057 is MWh per kg (the inverse), and 1.75 drops a factor of ten.

Numbers, sources and what’s next

Numbers to remember

Number What it means Source
17.5 kg per MWh Hydrogen from 1 MWh at 57 kWh per kg (1,000 ÷ 57 = 17.54, rounded): the coefficient that couples the two balances. DOE Hydrogen Program Record 24005 · 20 May 2024
15% to 100% The load range of an alkaline electrolyzer; PEM units go down to 5%. IRENA, Green Hydrogen Cost Reduction, Table 6 · December 2020
up to 3 times The heat a heat pump gives against the electricity it uses. DOE, Energy Saver Guide, p. 16 · July 2022

Next

M5 · Size and run a microgrid that survives an outage

Related lab: a SciPy/HiGHS notebook that rebuilds the plant model and asserts the browser's numbers, about 45 minutes, then 90 minutes of practice.

In the real world

Compared with wind and a battery alone, adding an electrolyzer reduced total system cost by about 7%, and using hydrogen as the storage medium for fuel cells reduced it by about 26%.

Holds when: one planned West Texas wind farm, ERCOT prices, costs from 2020 to 2023. Leaves out: final federal hydrogen credit rules.

Try it yourself

Optional, about 30 minutes; needs Python on your computer (lesson B2).

  • Run examples/configurations.py: four configurations of wind, battery, electrolyzer, tank and fuel cell on a synthetic site, sized and run hour by hour.
  • Predict first, then switch the electrolyzer to a constant rate (ModelOptions(electrolyzer_operation="constant")): which sizes change, and how much does flexibility save?

reopt-hydrogen-framework: public example repository · MIT licence · synthetic data.

Sources and further reading

Spotted an error, or stuck? Write to hlli@ksu.edu. Corrections are made in place and dated here.

Text licensed CC BY 4.0, code MIT: teachers and students may reuse and adapt it, with credit to Honglin Li.