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"source": [
"# Combined Heat and Power\n",
"The CHPAsset class represents an asset that can generate both heat and power at the same time and models starting procedures, minimum loads, ramping and a minimum runtime."
]
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"source": [
"## Technical Details\n",
"### Input Parameters\n",
"Possible input parameters of the CHPAsset are\n",
"\n",
"- Ramp $r$\n",
"- Conversion efficiency $\\alpha$ from heat to power.\n",
"- Maximum share heat $\\beta$\n",
"- Minimum capacity $l_t$ for every timestep $t \\in \\left\\{0,\\dots, T-1\\right\\}$\n",
"- Maximum capacity $u_t$ for every timestep $t \\in \\left\\{0,\\dots, T-1\\right\\}$\n",
"- Dispatch power $x_{-1}$ from the previous timestep\n",
"- Dispatch heat $w_{-1}$ from the previous timestep\n",
"- Number $k_{on}$ of timesteps the asset has already been running\n",
"- Number $k_{off}$ of timesteps the asset has already been of\n",
"- Minimum runtime $m_{run}$\n",
"- Minimum downtime $m_{down}$\n",
"- Start ramp specification:\n",
" - Minimum capacity $l^{start}_i$ for timestep $i \\in \\left\\{0,\\dots, k_{start}\\right\\}$ after starting\n",
" - Maximum capacity $u^{start}_i$ for timestep $i \\in \\left\\{0,\\dots, k_{start}\\right\\}$ after starting\n",
"- Shutdown ramp specification:\n",
" - Minimum capacity $l^{shutdown}_i$ for timestep $i \\in \\left\\{0,\\dots, k_{shutdown}\\right\\}$ before shutting off\n",
" - Maximum capacity $u^{shutdown}_i$ for timestep $i \\in \\left\\{0,\\dots, k_{shutdown}\\right\\}$ before shutting off"
]
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"### Output variables\n",
"The following variables are computed for a CHPAsset that runs for $T$ timesteps:\n",
"\n",
"- Dispatch power $x_t \\in \\mathbb{R}$, for every timestep $t \\in \\left\\{0,\\dots, T-1\\right\\}$\n",
"- Dispatch heat $w_t \\in \\mathbb{R}$, for every timestep $t \\in \\left\\{0,\\dots, T-1\\right\\}$\n",
"- \"On\"-variable $a_t \\in \\left\\{0,1\\right\\}$, for every timestep $t \\in \\left\\{0,\\dots, T-1\\right\\}$\n",
"- \"Start\"-variable $s_t \\in \\left\\{0,1\\right\\}$, for every timestep $t \\in \\left\\{0,\\dots, T-1\\right\\}$\n",
"- \"Shutdown\"-variable $d_t \\in \\left\\{0,1\\right\\}$, for every timestep $t \\in \\left\\{0,\\dots, T-1\\right\\}$\n",
"\n",
"When they are not needed to enforce a minimum runtime, start costs or similar, the \"on\" and \"start\" variables are removed."
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"### Variable bounds\n",
"The output variables are within the following bounds in every timestep $t \\in \\left\\{0,\\dots, T-1\\right\\}$:\n",
"\n",
"- $0 \\leq x_t \\leq u_t$\n",
"- $0 \\leq w_t \\leq \\beta u_t$\n",
"- $0 \\leq a_t \\leq 1$\n",
"- $0 \\leq s_t \\leq 1$\n",
"- $0 \\leq d_t \\leq 1$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Constraints\n",
"The following constraints are used in the asset\n",
"\n",
"- **Minimum and maximum capacity**: The virtual dispatch $\\tilde{x}_t = x_t + \\alpha w_t$ is 0 when $a_t = 0$, i.e. the asset is \"off\", it is bounded by the start or shutdown specifications during the start and shutdown, and otherwise it is between minimum and maximum capacity, i.e.\n",
"\\begin{align*} \n",
"\\tilde{x}_t &\\geq& l_t \\left[ a_t - \\sum_{i=0}^{k_{shutdown}-1}d_{t+i+1} - \\sum_{i=0}^{k_{start}-1}s_{t-i} \\right] + \\sum_{i=0}^{k_{start}-1}l^{start}_i s_{t-i} + \\sum_{i=0}^{k_{shutdown}-1}l^{shutdown}_i d_{t+i+1} \\\\\n",
"\\tilde{x}_t &\\leq& u_t \\left[ a_t - \\sum_{i=0}^{k_{shutdown}-1}d_{t+i+1} - \\sum_{i=0}^{k_{start}-1}s_{t-i} \\right] + \\sum_{i=0}^{k_{start}-1}u^{start}_i s_{t-i} + \\sum_{i=0}^{k_{shutdown}-1}u^{shutdown}_i d_{t+i+1}\n",
"\\end{align*} \n",
"for $t \\in \\left\\{0,\\dots, T-1\\right\\}$. It should be noted that the term $\\sum_{i=0}^{k_{start}-1}s_{t-i}$ is 1 if timestep $t$ is part of the start ramp and 0 otherwise. Analogously, $\\sum_{i=0}^{k_{shutdown}-1}d_{t+i+1}$ is 1 if timestep $t$ is part of the shutdown ramp and 0 otherwise. \n",
"If $k_{on} = 0$, i.e. previous state $a_{-1}$ is \"off\", we need the additional condition \n",
"\\begin{align*}s_0 \\geq a_0.\\end{align*}\n",
"\n",
"- **Start and shutdown constraints**: (Only applicable when start or shutdown ramp are defined)\n",
"Ensure that the start and shutdown variables are set correctly, i.e\n",
"\\begin{align*}\n",
"s_t - d_t &=& a_t - a_{t-1} \\\\\n",
"s_t + d_t &\\leq& 1\n",
"\\end{align*}\n",
"for $t \\in \\left\\{0,\\dots, T-1\\right\\}$.\n",
"\n",
"- **Minimum runtime**: Ensure that the asset follows the minimum runtime $m_{run}$ and is on during the start and shutdown ramp, i.e.\n",
"\\begin{align*}a_t \\geq s_{t-i}\\end{align*} \n",
"for $t \\in \\left\\{1,\\dots, T-1\\right\\}$ and $i \\in \\left\\{1, \\dots, m_{run}+k_{start}+k_{shutdown}-1\\right\\}$. "
]
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"source": [
"node_power = eao.assets.Node('node_power')\n",
"node_heat = eao.assets.Node('node_heat')\n",
"node_gas = eao.assets.Node('node_gas')\n",
"\n",
"timegrid = eao.assets.Timegrid(dt.date(2021, 1, 1), dt.date(2021, 1, 2), freq='15min', main_time_unit='h')\n",
"simple_contract_power = eao.assets.SimpleContract(name='SC_power', price='price_power', nodes=node_power,\n",
" min_cap=-30., max_cap=20)\n",
"simple_contract_heat = eao.assets.SimpleContract(name='SC_heat', price='price_heat', nodes=node_heat,\n",
" min_cap=-30., max_cap=20.)\n",
"simple_contract_gas = eao.assets.SimpleContract(name='SC_gas', price='price_gas', nodes=node_gas,\n",
" min_cap=-30., max_cap=20.)\n",
"\n",
"\n",
"chpasset = eao.assets.CHPAsset(name='CHP', \n",
" nodes=[node_power, node_heat, node_gas],\n",
" min_cap=5., \n",
" max_cap=30.,\n",
" extra_costs=0,\n",
" conversion_factor_power_heat=0.5, \n",
" max_share_heat=0.5,\n",
" ramp=10, \n",
" start_ramp_lower_bounds=[1, 2],\n",
" shutdown_ramp_lower_bounds=[2, 3],\n",
" ramp_freq=\"15min\",\n",
" start_costs=0., \n",
" running_costs=0., \n",
" min_runtime=0., \n",
" time_already_running=0., \n",
" min_downtime=0.,\n",
" time_already_off=1.,\n",
" last_dispatch=0., \n",
" start_fuel=0)\n",
"\n",
"prices = {'price_power': np.sin(0.5*np.pi * np.arange(timegrid.T)*np.pi/10)+2,\n",
" 'price_heat': np.cos(0.5*np.pi * np.arange(timegrid.T)*np.pi/10)+2, \n",
" 'price_gas': np.ones(timegrid.T)*2, \n",
" }\n",
"\n",
"portfolio = eao.portfolio.Portfolio([chpasset, simple_contract_power, simple_contract_heat, simple_contract_gas])\n",
"eao.network_graphs.create_graph(portf = portfolio)"
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"### Performing the Optimization"
]
},
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"cell_type": "code",
"execution_count": 3,
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"source": [
"# Define the grid size to display the figures:\n",
"num_imgs_per_row = 3\n",
"num_nodes = len(portfolio.nodes)\n",
"num_img_rows=int(np.ceil((num_nodes+2)/num_imgs_per_row))\n",
"fig, ax = plt.subplots(num_img_rows, num_imgs_per_row, tight_layout = True, figsize=(18,5*num_img_rows))\n",
"ax = ax.reshape(-1)\n",
"\n",
"# Plot the dispatch for each node:\n",
"current_ax=0\n",
"dispatch_per_node = {node: [] for node in portfolio.nodes}\n",
"for mycol in out['dispatch'].columns.values:\n",
" for node in dispatch_per_node:\n",
" if '(' + node + ')' in mycol:\n",
" dispatch_per_node[node].append(mycol)\n",
"for node in dispatch_per_node:\n",
" out['dispatch'][dispatch_per_node[node]].plot(ax=ax[current_ax], style='-x')\n",
" ax[current_ax].set_title(\"Dispatch \" + node)\n",
" current_ax+=1\n",
"\n",
"# Plot internal variables:\n",
"for v in out['internal_variables']:\n",
" out['internal_variables'][v].plot(ax=ax[current_ax], style='-x', label=v)\n",
"if out['internal_variables'].shape[1] == 0:\n",
" ax[current_ax].text(0.4, 0.5, \"No bool variables present\")\n",
"ax[current_ax].set_title('Bool Variables')\n",
"ax[current_ax].legend()\n",
"current_ax+=1\n",
"\n",
"# Plot prices\n",
"for key in out['prices']:\n",
" out['prices'][key].plot(ax=ax[current_ax], style='-x', label=key)\n",
"ax[current_ax].set_title('Prices')\n",
"ax[current_ax].legend()\n",
"\n",
"# Remove empty axes:\n",
"for i in range(current_ax+1, len(ax)):\n",
" fig.delaxes(ax[i])"
]
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