Heat System with Storage¶

Thermal storage shifts gas consumption from expensive peak to cheap off-peak hours.

New concepts: Storage (buffer with efficiency & self-discharge) · Time-varying prices · Load shifting

In [1]:

import math
from datetime import datetime, timedelta

import plotly.graph_objects as go
import plotly.io as pio
from plotly.subplots import make_subplots

from fluxopt import Carrier, Converter, Effect, Flow, Port, Storage, optimize

pio.renderers.default = 'notebook_connected'

import math from datetime import datetime, timedelta import plotly.graph_objects as go import plotly.io as pio from plotly.subplots import make_subplots from fluxopt import Carrier, Converter, Effect, Flow, Port, Storage, optimize pio.renderers.default = 'notebook_connected'

Input Data — 48 h with day/night price spread¶

In [2]:

n = 48
timesteps = [datetime(2024, 1, 15) + timedelta(hours=h) for h in range(n)]
demand = [50 + 40 * max(0.0, math.sin(math.pi * ((h % 24) - 6) / 12)) for h in range(n)]
price = [0.04 if (h % 24) < 6 or (h % 24) >= 22 else 0.08 for h in range(n)]

fig = make_subplots(
    rows=2,
    cols=1,
    shared_xaxes=True,
    vertical_spacing=0.15,
    subplot_titles=('Heat Demand', 'Gas Price'),
)
fig.add_trace(go.Bar(x=timesteps, y=demand, marker_color='#ef553b', showlegend=False), row=1, col=1)
fig.add_trace(go.Scatter(x=timesteps, y=price, line_shape='hv', line_color='#636efa', showlegend=False), row=2, col=1)
fig.update_layout(height=350, margin={'l': 50, 'r': 20, 't': 30, 'b': 20}, template='plotly_white')
fig.update_yaxes(title_text='kW', row=1, col=1)
fig.update_yaxes(title_text='EUR/kWh', row=2, col=1)
fig

n = 48 timesteps = [datetime(2024, 1, 15) + timedelta(hours=h) for h in range(n)] demand = [50 + 40 * max(0.0, math.sin(math.pi * ((h % 24) - 6) / 12)) for h in range(n)] price = [0.04 if (h % 24) < 6 or (h % 24) >= 22 else 0.08 for h in range(n)] fig = make_subplots( rows=2, cols=1, shared_xaxes=True, vertical_spacing=0.15, subplot_titles=('Heat Demand', 'Gas Price'), ) fig.add_trace(go.Bar(x=timesteps, y=demand, marker_color='#ef553b', showlegend=False), row=1, col=1) fig.add_trace(go.Scatter(x=timesteps, y=price, line_shape='hv', line_color='#636efa', showlegend=False), row=2, col=1) fig.update_layout(height=350, margin={'l': 50, 'r': 20, 't': 30, 'b': 20}, template='plotly_white') fig.update_yaxes(title_text='kW', row=1, col=1) fig.update_yaxes(title_text='EUR/kWh', row=2, col=1) fig

System¶

Gas Grid ──► [gas] ──► Boiler η=92% ──► [heat] ◄──► Tank 500 kWh
 time-varying €                            │
                                         Office

In [3]:

max_d = max(demand)

result = optimize(
    timesteps=timesteps,
    carriers=[Carrier('gas'), Carrier('heat')],
    effects=[Effect('cost', unit='EUR', is_objective=True)],
    ports=[
        Port(
            'gas_grid',
            imports=[
                Flow('gas', size=500, effects_per_flow_hour={'cost': price}),
            ],
        ),
        Port(
            'office',
            exports=[
                Flow('heat', size=max_d, fixed_relative_profile=[d / max_d for d in demand]),
            ],
        ),
    ],
    converters=[
        Converter.boiler(
            'boiler',
            thermal_efficiency=0.92,
            fuel_flow=Flow('gas', size=300),
            thermal_flow=Flow('heat', size=150),
        ),
    ],
    storages=[
        Storage(
            'tank',
            capacity=500,
            charging=Flow('heat', size=100),
            discharging=Flow('heat', size=100),
            eta_charge=0.98,
            eta_discharge=0.98,
            relative_loss_per_hour=0.005,
            cyclic=True,
        )
    ],
)

max_d = max(demand) result = optimize( timesteps=timesteps, carriers=[Carrier('gas'), Carrier('heat')], effects=[Effect('cost', unit='EUR', is_objective=True)], ports=[ Port( 'gas_grid', imports=[ Flow('gas', size=500, effects_per_flow_hour={'cost': price}), ], ), Port( 'office', exports=[ Flow('heat', size=max_d, fixed_relative_profile=[d / max_d for d in demand]), ], ), ], converters=[ Converter.boiler( 'boiler', thermal_efficiency=0.92, fuel_flow=Flow('gas', size=300), thermal_flow=Flow('heat', size=150), ), ], storages=[ Storage( 'tank', capacity=500, charging=Flow('heat', size=100), discharging=Flow('heat', size=100), eta_charge=0.98, eta_discharge=0.98, relative_loss_per_hour=0.005, cyclic=True, ) ], )

Running HiGHS 1.13.1 (git hash: 1d267d9): Copyright (c) 2026 under MIT licence terms
LP linopy-problem-qifsnofb has 871 rows; 439 cols; 1404 nonzeros
Coefficient ranges:
  Matrix  [4e-02, 1e+00]
  Cost    [1e+00, 1e+00]
  Bound   [0e+00, 0e+00]
  RHS     [5e+01, 5e+02]
Presolving model
96 rows, 192 cols, 336 nonzeros  0s
Dependent equations search running on 96 equations with time limit of 1000.00s
Dependent equations search removed 0 rows and 0 nonzeros in 0.00s (limit = 1000.00s)
96 rows, 192 cols, 336 nonzeros  0s
Presolve reductions: rows 96(-775); columns 192(-247); nonzeros 336(-1068) 
Solving the presolved LP
Using dual simplex solver
  Iteration        Objective     Infeasibilities num(sum)
          0     0.0000000000e+00 Pr: 48(3007.66) 0.0s
        133     1.8811941584e+02 Pr: 0(0) 0.0s

Performed postsolve
Solving the original LP from the solution after postsolve

Model name          : linopy-problem-qifsnofb
Model status        : Optimal
Simplex   iterations: 133
Objective value     :  1.8811941584e+02
P-D objective error :  0.0000000000e+00
HiGHS run time      :          0.00

/home/runner/work/fluxopt/fluxopt/.venv/lib/python3.12/site-packages/linopy/common.py:491: UserWarning: Coordinates across variables not equal. Perform outer join.
  warn(

Results¶

In [4]:

print(f'Total cost: {result.objective:.2f} EUR  |  Avg: {result.objective / sum(demand) * 100:.2f} ct/kWh')

print(f'Total cost: {result.objective:.2f} EUR | Avg: {result.objective / sum(demand) * 100:.2f} ct/kWh')

Total cost: 188.12 EUR  |  Avg: 6.25 ct/kWh

In [5]:

times = result.flow_rates.coords['time'].values

fig = make_subplots(
    rows=2,
    cols=1,
    shared_xaxes=True,
    vertical_spacing=0.12,
    subplot_titles=('Storage Flows', 'Storage Level'),
)

for fid in result.flow_rates.coords['flow'].values:
    if str(fid).startswith('tank('):
        fig.add_trace(
            go.Scatter(x=times, y=result.flow_rate(str(fid)).values, name=str(fid), line_shape='hv'),
            row=1,
            col=1,
        )

level = result.storage_level('tank')
fig.add_trace(
    go.Scatter(
        x=level.coords['time'].values,
        y=level.values,
        name='level',
        fill='tozeroy',
        line_shape='hv',
        showlegend=False,
    ),
    row=2,
    col=1,
)

fig.update_layout(height=400, margin={'l': 50, 'r': 20, 't': 30, 'b': 20}, template='plotly_white')
fig.update_yaxes(title_text='kW', row=1, col=1)
fig.update_yaxes(title_text='kWh', row=2, col=1)
fig

times = result.flow_rates.coords['time'].values fig = make_subplots( rows=2, cols=1, shared_xaxes=True, vertical_spacing=0.12, subplot_titles=('Storage Flows', 'Storage Level'), ) for fid in result.flow_rates.coords['flow'].values: if str(fid).startswith('tank('): fig.add_trace( go.Scatter(x=times, y=result.flow_rate(str(fid)).values, name=str(fid), line_shape='hv'), row=1, col=1, ) level = result.storage_level('tank') fig.add_trace( go.Scatter( x=level.coords['time'].values, y=level.values, name='level', fill='tozeroy', line_shape='hv', showlegend=False, ), row=2, col=1, ) fig.update_layout(height=400, margin={'l': 50, 'r': 20, 't': 30, 'b': 20}, template='plotly_white') fig.update_yaxes(title_text='kW', row=1, col=1) fig.update_yaxes(title_text='kWh', row=2, col=1) fig

In [6]:

fig = make_subplots(
    rows=2,
    cols=1,
    shared_xaxes=True,
    vertical_spacing=0.12,
    subplot_titles=('All Flow Rates', 'Hourly Cost'),
)

for fid in result.flow_rates.coords['flow'].values:
    fig.add_trace(
        go.Scatter(x=times, y=result.flow_rate(str(fid)).values, name=str(fid), line_shape='hv'),
        row=1,
        col=1,
    )

cost = result.effects_temporal.sel(effect='cost')
fig.add_trace(go.Bar(x=times, y=cost.values, name='cost', showlegend=False), row=2, col=1)

fig.update_layout(height=400, margin={'l': 50, 'r': 20, 't': 30, 'b': 20}, template='plotly_white')
fig.update_yaxes(title_text='kW', row=1, col=1)
fig.update_yaxes(title_text='EUR', row=2, col=1)
fig

fig = make_subplots( rows=2, cols=1, shared_xaxes=True, vertical_spacing=0.12, subplot_titles=('All Flow Rates', 'Hourly Cost'), ) for fid in result.flow_rates.coords['flow'].values: fig.add_trace( go.Scatter(x=times, y=result.flow_rate(str(fid)).values, name=str(fid), line_shape='hv'), row=1, col=1, ) cost = result.effects_temporal.sel(effect='cost') fig.add_trace(go.Bar(x=times, y=cost.values, name='cost', showlegend=False), row=2, col=1) fig.update_layout(height=400, margin={'l': 50, 'r': 20, 't': 30, 'b': 20}, template='plotly_white') fig.update_yaxes(title_text='kW', row=1, col=1) fig.update_yaxes(title_text='EUR', row=2, col=1) fig

Insights¶

• Off-peak charging: boiler overproduces at night (cheap gas) to fill storage
• Peak discharge: storage supplements boiler during expensive hours
• Efficiency vs spread: round-trip loss (96%) + self-discharge means shifting only pays when the price gap exceeds the loss cost