.. SPDX-FileCopyrightText: PyPSA-Earth and PyPSA-Eur Authors .. .. SPDX-License-Identifier: CC-BY-4.0 ===================================== Optimization Models in Energy Systems ===================================== The aim of this section is to provide an overview of the most common optimization approaches used in energy system models. In particular, it covers capacity expansion models and economic dispatch models, and it distinguishes between deterministic and stochastic formulations. Before addressing these categories, we introduce some general definitions. In general, an optimization problem can be expressed as: .. math:: \begin{aligned} \min_{\mathbf{x}} \quad & f(\mathbf{x}) \quad \text{(objective function)} \end{aligned} subject to: .. math:: \begin{aligned} & g_i(\mathbf{x}) \le 0 \quad \forall i \quad \text{(inequality constraints)} \\ & h_j(\mathbf{x}) = 0 \quad \forall j \quad \text{(equality constraints)} \end{aligned} where: - :math:`f(\mathbf{x})` is the objective function, i.e., the function to be minimized (or maximized). - :math:`\mathbf{x}` are the *decision variables*, i.e., the variables you can choose to minimize (maximize) the objective function. - :math:`g_i` and :math:`h_j` represent the set of inequality and equality constraints. .. _sec-simple-linear-optimization-model: Example: Simple linear optimization problem =========================================== Consider the following problem: .. math:: \begin{aligned} \min_{x_1, x_2} \quad & x_1 + x_2 \label{eq:obj_simple} \end{aligned} subject to: .. math:: \begin{aligned} & x_1 \geq 0 \quad \label{eq:const1} \\ & x_2 \geq 0 \quad \label{eq:const2} \\ & x_1 + 2x_2 = 4 \quad \label{eq:const3} \end{aligned} **Explanation:** - The *objective function* aims to minimize the sum :math:`x_1 + x_2`. - The *decision variables* are :math:`x_1` and :math:`x_2`. - Constraints impose that both variables must be non-negative. - Constraint :math:`x_1 + 2x_2 = 4` requires that the combination of :math:`x_1` and :math:`x_2` must satisfy a given requirement (e.g., total production must meet demand). **Optimal solution:** Since :math:`x_1` is less ‘expensive’ in the constraint `x_1 + 2x_2 = 4` (it counts 1 unit toward the total while :math:`x_2` counts 2), the optimal choice is to use as much :math:`x_2` as needed to meet the constraint with minimal total :math:`x_1 + x_2`. Solving: .. math:: x_1 = 0, \quad 2x_2 = 4 \quad \Rightarrow \quad x_2 = 2 gives an objective value :math:`0 + 2 = 2`. .. _sec-capacity-expansion-problem: Capacity expansion problem =========================== Let us assume a single-year timeframe, with the aim of minimizing the *total cost* of the system under technical and policy constraints. We consider a system composed of: - A photovoltaic (PV) plant with capacity :math:`P_{\text{PV}}` [MW]. - A gas generator with capacity :math:`P_{\text{gas}}` [MW]. - A fixed hourly electricity demand :math:`D(t)` [MW] for :math:`t=1,\dots,T`. The optimization problem can be formulated as: .. math:: \begin{aligned} \min_{\substack{P_{\text{PV}},\, P_{\text{gas}}, \\ p_{\text{PV}}(t),\, p_{\text{gas}}(t)}} & C_{\text{PV}}^{\text{cap}} \cdot P_{\text{PV}} + C_{\text{gas}}^{\text{cap}} \cdot P_{\text{gas}} \nonumber \\ & + \sum_{t=1}^T c_{\text{gas}} \cdot p_{\text{gas}}(t) \cdot \Delta t \label{eq:obj_energy}\end{aligned} subject to: .. math:: \begin{aligned} & p_{\text{PV}}(t) + p_{\text{gas}}(t) = D(t) \quad \forall t \label{eq:bal} \\ & 0 \leq p_{\text{PV}}(t) \leq \min\{G(t) \cdot \eta_{\text{PV}},\; P_{\text{PV}}\} \quad \forall t \label{eq:pv_lim} \\ & 0 \leq p_{\text{gas}}(t) \leq P_{\text{gas}} \quad \forall t \label{eq:gas_lim} \\ & P_{\text{PV}} \geq 0, \quad P_{\text{gas}} \geq 0 \label{eq:cap_nonneg}\end{aligned} **Interpretation:** - The first line of the objective function represents the *design cost* (capital expenditure), depending on installed capacities. - The second line of the objective function represents the *operational cost* (fuel cost for the gas generator). - The equality constraint enforces the energy balance, i.e. the sum of generation must match the demand. - The inequality constraints enforce the technical limits. The gas production is bounded by the gas generator size, while the PV production is bounded by the PV size and irradiance availability. Both productions must be positive (generators cannot act like loads). Since capacities are part of the decision variables, this is a capacity expansion model: it optimizes both design and operation. Here, design refers to the sizing phase, i.e. the choice of the optimal installed capacities for the system components, while operation determines their temporal dispatch. A realistic example could be the evolution of the European power system in 2050 under specific technical, economic, or environmental assumptions. .. _sec-economic-dispatch-problem: Economic dispatch problem ========================== If capacities are given and fixed, the problem becomes a special case of capacity expansion known as *economic dispatch* (ED). **Parameters** - :math:`P_{\text{PV}}` [MW]: installed PV capacity (fixed). - :math:`P_{\text{gas}}` [MW]: installed gas turbine capacity (fixed). - :math:`D(t)` [MW], :math:`t=1,\dots,T`: electricity demand. - :math:`G(t)` [MWh], :math:`\eta_{\text{PV}}`  : PV availability and conversion factor. - :math:`c_{\text{gas}}` [€ / MWh], :math:`\Delta t` [h]: gas marginal cost and time step. **Decision variables (operation)** .. math:: p_{\text{PV}}(t) [MW], \qquad p_{\text{gas}}(t) [MW] \qquad \forall t=1,\dots,T **Optimization problem** .. math:: \begin{aligned} \min_{\{p_{\text{PV}}(t),\, p_{\text{gas}}(t)\}} \quad & \sum_{t=1}^T c_{\text{gas}} \, p_{\text{gas}}(t)\, \Delta t \label{eq:ed_obj}\end{aligned} subject to .. math:: \begin{aligned} & p_{\text{PV}}(t) + p_{\text{gas}}(t) = D(t) \quad \forall t && \text{(demand balance)} \label{eq:ed_balance} \\ & 0 \le p_{\text{PV}}(t) \le \min\!\left\{ G(t)\,\eta_{\text{PV}},\; P_{\text{PV}} \right\} \quad \forall t && \text{(PV availability and capacity)} \label{eq:ed_pv_bounds} \\ & 0 \le p_{\text{gas}}(t) \le P_{\text{gas}} \quad \forall t && \text{(gas capacity limit)} \label{eq:ed_gas_bounds}\end{aligned} A realistic example could be the validation of a given model with historical data, where capacities are set (historical ones) and only operation is optimized. .. _sec-stochastic-optimization: Stochastic Optimization ======================= So far, we have assumed *deterministic* optimization: all input time series (demand, solar irradiance) and parameters (natural gas price) are perfectly known. However, in real life we often face uncertainty. Birthday party example. ----------------------- Suppose tomorrow is your birthday and you are going to have a party. You invited :math:`Y=20` people and everyone wants a pizza, but nobody has confirmed their presence yet. This means you do not know the actual number of guests :math:`y` who will show up. You must decide *today* how many pizzas :math:`x` to order, at a cost of 10 € each. If more guests arrive than pizzas ordered (:math:`y > x`), you will need to buy extra pizzas *last-minute* at 16 € each. There is no refund for leftovers. If :math:`y \le x` (over-ordering), you spend :math:`10x`, while :math:`10y` would have been enough. If :math:`y > x` (under-ordering), you spend :math:`10x` plus :math:`16(y-x)` for the extra pizzas. The challenge: :math:`x` must be chosen **today** — before knowing :math:`y`. Assume three equally likely scenarios: - 0 guest, nobody is coming - 5 guests, only your best friends are coming - 20 guests, everyone is coming Therefore, :math:`y \in \{0, 5, 20\}` guests. These scenarios have the same probability of occurring. **Scenario-by-scenario explanation and expected cost** - **Case A — \(0 <= x <= 5\)** - under-ordering can already happen at y=5: - Scenario y=0: no guests → you only pay the pre-ordered pizzas \(C(x,0)=10x\). - Scenario y=5: if x<5, you are short of \(5-x\) pizzas → last-minute at 16€ each \(C(x,5)=10x+16(5-x)\). If x=5, the extra term is 0. - Scenario y=20: you are short of \(20-x\) pizzas → last-minute at 16€ each \(C(x,20)=10x+16(20-x)\). **Expected cost** Each scenario has the same probability, so its contribution is weighted 1/3: .. math:: \mathbb{E}[C(x)] = \tfrac13\big(10x\big) + \tfrac13\big(10x+16(5-x)\big) + \tfrac13\big(10x+16(20-x)\big) = \frac{400 - 2x}{3} = 133.33 - \tfrac{2}{3}x. - **Case B — \(5 < x <= 20\)** - no shortage at y=5, only at y=20: - Scenario y=0: no guests → you only pay the pre-ordered pizzas \(C(x,0)=10x\). - Scenario y=5: enough pizzas \(x>5\) → no last-minute purchase \(C(x,5)=10x\). - Scenario y=20: you are short of \(20-x\) pizzas → last-minute at 16€ each \(C(x,20)=10x+16(20-x)=320-6x\). **Expected cost** .. math:: \mathbb{E}[C(x)] = \tfrac13\big(10x\big) + \tfrac13\big(10x\big) + \tfrac13\big(10x+16(20-x)\big) = \frac{320 + 14x}{3} = 106.67 + \tfrac{14}{3}x. **Minimizer.** Since :math:`\mathbb{E}[C(x)]` is decreasing on [0,5] and increasing on [5,20], the minimum is attained at the boundary :math:`x^\star=5`, with .. math:: \mathbb{E}[C(5)] = \frac{390}{3} = 130. **Interpretation:** Ordering 5 pizzas perfectly covers the medium scenario, avoids over-ordering in the low scenario, and limits the expensive last-minute purchases in the high scenario. Here, :math:`x` is a *here-and-now* decision taken under uncertainty, while the number of extra pizzas (if needed) is a *wait-and-see* decision made after the actual scenario is revealed. .. _sec-two-stage-formulation: Two-stage stochastic formulation. --------------------------------- Let :math:`\omega \in \Omega` be a scenario with probability :math:`p_\omega`. First-stage variables :math:`x` are chosen “here and now”, before knowing which scenario will occur. Second-stage variables :math:`y_\omega` are chosen “wait-and-see”, after the specific scenario :math:`\omega` is revealed. The two-stage stochastic problem can be written as: .. math:: \begin{aligned} \min_{x,\,\{y_\omega\}_{\omega\in\Omega}} \quad & f(x) + \sum_{\omega \in \Omega} p_\omega \, g(x, y_\omega, \omega) \label{eq:stoc_compact} \\ \text{s.t.} \quad & A_\omega x + B_\omega y_\omega \ge b_\omega, \quad \forall \omega \in \Omega, \\ & x \ge 0, \quad y_\omega \ge 0 \quad \forall \omega \in \Omega.\end{aligned} Here: - :math:`f(x)` represents the first-stage cost or contribution to the objective. - :math:`g(x, y_\omega, \omega)` represents the second-stage cost or contribution, depending on scenario :math:`\omega`. - The constraints must hold for every scenario :math:`\omega`. In the birthday party analogy: - :math:`x` = pizzas ordered today (*first stage*); - :math:`y_\omega` = guests in scenario :math:`\omega` (*second stage*). The same structure applies to energy systems: first-stage = investment decisions (capacities), second-stage = operational decisions (dispatch) under different scenarios of demand, renewable generation, or fuel prices. Capacity expansion under uncertainty (three gas price scenarios). ----------------------------------------------------------------- We consider the same system as in the deterministic case: - A photovoltaic (PV) plant with capacity :math:`P_{\text{PV}}` [MW]. - A gas generator with capacity :math:`P_{\text{gas}}` [MW]. - A fixed hourly electricity demand :math:`D(t)` [MW], :math:`t=1,\dots,T`. Now, however, the natural gas price is uncertain. We define three scenarios :math:`\omega \in \{1,2,3\}` with probabilities :math:`p_\omega`: .. math:: c_{\text{gas}}^{(1)}, \quad c_{\text{gas}}^{(2)}, \quad c_{\text{gas}}^{(3)} representing, for example, *low*, *medium*, and *high* price conditions. **Decision structure:** - **First-stage (here-and-now)**: capacities :math:`P_{\text{PV}}`, :math:`P_{\text{gas}}` (same for all scenarios). - **Second-stage (wait-and-see)**: dispatch profiles :math:`p_{\text{PV}}(t,\omega)`, :math:`p_{\text{gas}}(t,\omega)` (adapted to each scenario’s fuel cost). **Two-stage stochastic formulation:** .. math:: \begin{aligned} \min_{P_{\text{PV}},\, P_{\text{gas}}} \quad & C_{\text{PV}}^{\text{cap}} \cdot P_{\text{PV}} + C_{\text{gas}}^{\text{cap}} \cdot P_{\text{gas}} \nonumber \\ & + \sum_{\omega=1}^3 p_\omega \; Q(P_{\text{PV}},P_{\text{gas}},\omega) \label{eq:stoc_obj_energy}\end{aligned} where the second-stage operational cost for scenario :math:`\omega` is: .. math:: \begin{aligned} Q(P_{\text{PV}},P_{\text{gas}},\omega) \;=\; & \sum_{t=1}^T c_{\text{gas}}^{(\omega)} \cdot p_{\text{gas}}(t,\omega) \cdot \Delta t\end{aligned} subject to, for each scenario :math:`\omega`: .. math:: \begin{aligned} & p_{\text{PV}}(t,\omega) + p_{\text{gas}}(t,\omega) = D(t) && \forall t \quad \text{(demand balance)} \label{eq:stoc_bal} \\ & 0 \le p_{\text{PV}}(t,\omega) \le \min\{ G(t) \cdot \eta_{\text{PV}},\; P_{\text{PV}}\} && \forall t \quad \text{(PV limit)} \label{eq:stoc_pv_lim} \\ & 0 \le p_{\text{gas}}(t,\omega) \le P_{\text{gas}} && \forall t \quad \text{(gas capacity)} \label{eq:stoc_gas_lim}\end{aligned} **Interpretation:** - In the deterministic version, :math:`c_{\text{gas}}` is known, so the model optimizes capacities and dispatch for that single case. - In the stochastic version, the model chooses capacities that minimize investment cost plus the *expected* operational cost across all gas price scenarios. - The optimal solution is a compromise: it may not be optimal for any single scenario, but it is best *on average*, given the probabilities :math:`p_\omega`. **Note:** If multiple investment stages are allowed (e.g., building some capacity now and more in 10 years), the formulation extends to a *multi-stage* stochastic problem, where each stage has its own set of here-and-now decisions, followed by scenario-dependent operational decisions. .. _sec-sensitivity_analysis: Sensitivity Analysis ==================== **Sensitivity analysis** is performed *after* solving the optimization problem, by varying one or more parameters to see how the optimal solution changes. .. admonition:: Example Solve a deterministic CE model, then vary the fuel cost :math:`c_{\text{gas}}` from 50 to 100 €/MWh to see how the optimal capacities change. .. admonition:: Difference with stochastic optimization - **Sensitivity analysis**: changes are explored *post-optimization*. - **Stochastic optimization**: uncertainty is included *within* the optimization process. .. _sec-representative_days: Representative days. ==================== When optimizing over an entire year with hourly resolution (:math:`T=8760`), the problem size can become computationally heavy. A common approach is to select a reduced set of *representative days* that capture the main variability of demand and renewable generation profiles. Each representative day :math:`d \in \mathcal{D}` is assigned a *weight* :math:`w_d` indicating how many real days it represents in the year. This method is *deterministic*: there is only one scenario, but the time domain is reduced. The weights ensure that the reduced problem still approximates the annual cost and performance. The optimization problem becomes: .. math:: \begin{aligned} \min_{x,\,\{y_d\}_{d\in\mathcal{D}}} \quad & f(x) + \sum_{d \in \mathcal{D}} w_d \, g(x, y_d, d) \label{eq:repdays_obj} \\ \text{s.t.} \quad & A_d x + B_d y_d \ge b_d, \quad \forall d \in \mathcal{D}, \\ & x \ge 0, \quad y_d \ge 0 \quad \forall d \in \mathcal{D}.\end{aligned} The structure is similar to the stochastic formulation, but: - The :math:`w_d` are *weights*, not probabilities: they sum to the number of days in the year (e.g., 365), not to 1. - The problem is deterministic: all representative days are solved jointly as part of a single time series approximation. - There is no uncertainty: :math:`d` indexes clusters of days, not future scenarios. .. _sec-references: References =========== - :ref:`Theory of convex optimization ` – book with everything about convex optimization | S. P. Boyd, L. 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