docs/src/guides/problem.md

# Problem definition

The **Security-Constrained Unit Commitment Problem** (SCUC) is formulated in
UC.jl as a two-stage stochastic mixed-integer linear optimization problem that
aims to find the minimum-cost schedule for electricity generation while
satisfying various physical, operational and economic constraints. In its most
basic form, the problem is composed by:

- A set of generators, which produce power, at a given cost;
- A set of loads, which consume power;
- A transmission network, which delivers power from generators to the loads.

In addition to the basic components above, SCUC also include a wide variety of
additional components, such as _energy storage devices_, _reserves_ and _network
interfaces_, to name a few. On this page, we present a complete definition of
the problem, as modeled in UC.jl. Please note that different sources in the
literature may have significantly different definitions and assumptions.

!!! note

    UC.jl treats deterministic SCUC instances as a special case of the stochastic problem in which there is only one scenario, named `"s1"` by default. To access second-stage decisions, therefore, you must provide this scenario name as the value for `s`. For example, `model[:prod_above]["s1", g, t]`.

!!! warning

    The problem definition presented in this page is mathematically equivalent to the one solved by UC.jl. However, some constraints (ramping, piecewise-linear costs and start-up costs) have been simplified in this page for clarity. The set of constraints actually enforced by UC.jl better describes the convex hull of the problem and leads to better computational performance, but it is much more complex to describe. For further details, we refer to the package's source code and associated references.

## 1. General modeling assumptions

- **Time discretization:** SCUC is a multi-period problem, with decisions
  typically covering a 24-hour or 36-hour time window. UC.jl assumes that this
  time window is discretized into time steps of fixed length. The number of time
  steps, as well as the duration of each time step, are configurable. In the
  equations below, the set of time steps is denoted by $T=\{1,2,\ldots,|T|\}$.

- **Decision under uncertainty:** SCUC is a two-stage stochastic problem. In the
  first stage, we must decide the _commitment status_ of all thermal generators.
  In the second stage, we determine the remaining decision variables, such power
  output of all generators, the operation of energy storage devices and load
  shedding. Stochasticity is modeled through a discrete number of scenarios
  $s \in S$, each with given probability $p(S)$. The goal is to minimize the
  minimum expected cost.

## 2. Thermal generators

A _thermal generator_ is a power generation unit that converts thermal energy,
typically from the combustion of coal, natural gas or oil, into electrical
energy. Scheduling thermal generators is particularly complex due to their
operational characteristics, including minimum up and down times, ramping rates,
and start-up and shutdown limits.

### Important concepts

- **Commitment, power output and startup costs:** Thermal generators can either
  be online (on) or offline (off). When a thermal generator is on, it can
  produce between a minimum and a maximum amount of power; when it is off, it
  cannot produce any power. Switching a generator on incurs a startup cost,
  which depends on how long the unit has been offline. More precisely, each
  thermal generator $g$ has a number $K^{start}_g$ of startup categories (e.g.,
  cold, warm and hot). Each category $k$ has a corresponding startup cost
  $Z^{\text{start}}_{gk}$, and is available only if the unit has spent at most
  $M^{\text{delay}}_{gk}$ time steps offline.

- **Piecewise-linear production cost curve:** Besides startup costs, thermal
  generators also incur production costs based on their power output. The
  relationship between production cost and power output is not a linear, but a
  convex curve, which is simplified using a piecewise-linear approximation. For
  this purpose, each thermal generator $g$ has a number $K^{\text{cost}}_g$ of
  piecewise-linear segments and its power output $y^{\text{prod-above}}_{gts}$
  are broken down into
  $\sum_{k=1}^{K^{\text{cost}}_g} y^{\text{seg-prod}}_{gtks}$, so that
  production costs can be more easily calculated.

- **Ramping, minimum up/down:** Due to physical and operational limits, such as
  thermal inertia and mechanical stress, thermal generators cannot vary their
  power output too dramatically from one time period to the next. Similarly,
  thermal generators cannot switch on and off too frequently; after switching on
  or off, units must remain at that state for a minimum specified number of time
  steps.

- **Startup and shutdown limit:** A thermal generator cannot shut off if its
  output power level in the immediately preceding time step is very high (above
  a specified value); the unit must first ramp down, over potentially multiple
  time steps, and only then shut off. Similarly, the unit cannot produce a very
  large amount of power (above a specified limit) immediately after starting up;
  it must ramp up over potentially multiple time steps.

- **Initial status:** The optimization process finds optimal commitment status
  and power output level for all thermal generators starting at time period 1.
  Many constraints, however, require knowledge of previous time periods (0, -1,
  -2, ...) which are not part of the optimization model. For this reason, part
  of the input data is the initial power output $M^{\text{init-power}}_{g}$ of
  unit $g$ (that is, the output at time 0) and the initial status
  $M^{\text{init-status}}_{g}$ of unit g (how many time steps has it been
  online/offline at time time 0). If $M^{\text{init-status}}_{g}$ is positive,
  its magnitude indicates how many time periods has the unit been online; and if
  negative, how has it been offline.

- **Must-run:** Due to various factors, including reliability considerations,
  some units must remain operational regardless of whether it is economical for
  them to do so. Must-run constraints are used to enforce such requirements.

### Sets and constants

| Symbol                          | Unit   | Description                                                                                |
| :------------------------------ | :----- | :----------------------------------------------------------------------------------------- |
| $K^{cost}_g$                    |        | Number of piecewise linear segments in the production cost curve.                          |
| $K^{start}_g$                   |        | Number of startup categories (e.g. cold, warm, hot).                                       |
| $M^{\text{delay}}_{gk}$         |        | Delay for startup category $k$.                                                            |
| $M^{\text{init-power}}_{g}$     | MW     | Initial power output of unit $g$.                                                          |
| $M^{\text{init-status}}_{g}$    |        | Initial status of unit $g$.                                                                |
| $M^{\text{min-up}}_{g}$         |        | Minimum amount of time $g$ must stay on after switching on.                                |
| $M^{\text{must-run}}_{gt}$      | Binary | One if unit $g$ must be on at time $t$.                                                    |
| $M^{\text{pmax}}_{gt}$          | MW     | Maximum power output at time $t$.                                                          |
| $M^{\text{pmin}}_{gt}$          | MW     | Minimum power output at time $t$.                                                          |
| $M^{\text{ramp-down}}_{g}$      | MW     | Ramp down limit.                                                                           |
| $M^{\text{ramp-up}}_{g}$        | MW     | Ramp up limit.                                                                             |
| $M^{\text{seg-pmax}}_{gtks}$    | MW     | Maximum power output for piecewise-linear segment $k$ at time $t$ and scenario $s$.        |
| $M^{\text{shutdown-limit}}_{g}$ | MW     | Maximum power unit $g$ produces immediately before shutting down                           |
| $M^{\text{startup-limit}}_{g}$  | MW     | Maximum power unit $g$ produces immediately after starting up                              |
| $R_g$                           |        | Set of spinning reserves that may be served by $g$.                                        |
| $Z^{\text{pmin}}_{gt}$          | \$     | Cost to keep $g$ operational at time $t$ generating at minimum power.                      |
| $Z^{\text{pvar}}_{gtks}$        | \$/MW  | Cost for unit $g$ to produce 1 MW of power under piecewise-linear segment $k$ at time $t$. |
| $Z^{\text{start}}_{gk}$         | \$     | Cost to start unit $g$ at startup category $k$.                                            |
| $G^\text{therm}$                |        | Set of thermal generators.                                                                 |

### Decision variables

| Symbol                        | JuMP name           | Description                                                                                   | Unit   | Stage |
| :---------------------------- | :------------------ | :-------------------------------------------------------------------------------------------- | :----- | :---- |
| $x^{\text{is-on}}_{gt}$       | `is_on[g,t]`        | One if generator $g$ is on at time $t$.                                                       | Binary | 1     |
| $x^{\text{switch-on}}_{gt}$   | `switch_on[g,t]`    | One if generator $g$ switches on at time $t$.                                                 | Binary | 1     |
| $x^{\text{switch-off}}_{gt}$  | `switch_off[g,t]`   | One if generator $g$ switches off at time $t$.                                                | Binary | 1     |
| $x^{\text{start}}_{gtk}$      | `startup[g,t,s]`    | One if generator $g$ starts up at time $t$ under startup category $k$.                        | Binary | 1     |
| $y^{\text{prod-above}}_{gts}$ | `prod_above[s,g,t]` | Amount of power produced by $g$ at time $t$ in scenario $s$ above the minimum power.          | MW     | 2     |
| $y^{\text{seg-prod}}_{gtks}$  | `segprod[s,g,t,k]`  | Amount of power produced by $g$ at time $t$ in piecewise-linear segment $k$ and scenario $s$. | MW     | 2     |
| $y^{\text{res}}_{grts}$       | `reserve[s,r,g,t]`  | Amount of spinning reserve $r$ supplied by $g$ at time $t$ in scenario $s$.                   | MW     | 2     |

### Objective function terms

- Production costs:

```math
\sum_{g \in G^\text{therm}} \sum_{t \in T} x^{\text{is-on}}_{gt} Z^{\text{pmin}}_{gt}
+ \sum_{s \in S} p(s) \left[
    \sum_{g \in G^\text{therm}} \sum_{t \in T} \sum_{k=1}^{K^{cost}_g}
    y^{\text{seg-prod}}_{gtks} Z^{\text{pvar}}_{gtks}
\right]
```

- Start-up costs:

```math
\sum_{g \in G} \sum_{t \in T} \sum_{k=1}^{K^{start}_g} x^{\text{start}}_{gtk} Z^{\text{start}}_{gk}
```

### Constraints

- Some units must remain on, even if it is not economical for them to do so:

```math
x^{\text{is-on}}_{gt} \geq M^{\text{must-run}}_{gt}
```

- After switching on, unit must remain on for some amount of time
  (`eq_min_uptime[g,t]`):

```math
\sum_{i=max(1,t-M^{\text{min-up}}_{g}+1)}^t x^{\text{switch-on}}_{gi} \leq x^{\text{is-on}}_{gt}
```

- Same as above, but covering the initial time steps (`eq_min_uptime[g,0]`):

```math
\sum_{i=1}^{min(T,M^{\text{min-up}}_{g}-M^{\text{init-status}}_{g})} x^{\text{switch-off}}_{gi} = 0 \; \text{ if } \; M^{\text{init-status}}_{g} > 0
```

- After switching off, unit must remain offline for some amount of time
  (`eq_min_downtime[g,t]`):

```math
\sum_{i=max(1,t-M^{\text{min-down}}_{g}+1)}^t x^{\text{switch-off}}_{gi} \leq 1 - x^{\text{is-on}}_{gt}
```

- Same as above, but covering the initial time steps (`eq_min_downtime[g,0]`):

```math
\sum_{i=1}^{min(T,M^{\text{min-down}}_{g}+M^{\text{init-status}}_{g})} x^{\text{switch-on}}_{gi} = 0 \; \text{ if } \; M^{\text{init-status}}_{g} < 0
```

- If the unit switches on, it must choose exactly one startup category
  (`eq_startup_choose[g,t]`):

```math
x^{\text{switch-on}}_{gt} = \sum_{k=1}^{K^{start}_g} x^{\text{start}}_{gtk}
```

- If unit has not switched off in the last "delay" time periods, then startup
  category is forbidden (`eq_startup_restrict[g,t,s]`). The last startup
  category is always allowed. In the equation below, $L^{\text{start}}_{gtk}=1$
  if category should be allowed based on initial status.

```math
x^{\text{start}}_{gtk} \leq L^{\text{start}}_{gtk} + \sum_{i=min\left(1,t - M^{\text{delay}}_{g,k+1} + 1\right)}^{t - M^{\text{delay}}_{kg}} x^{\text{switch-off}}_{gi}
```

- Link the binary variables together (`eq_binary_link[g,t]`):

```math
\begin{align*}
& x^{\text{is-on}}_{gt} - x^{\text{is-on}}_{g,t-1} = x^{\text{switch-on}}_{gt} - x^{\text{switch-off}}_{gt} & \forall t > 1 \\
\end{align*}
```

- Cannot switch on and off at the same time (`eq_switch_on_off[g,t]`):

```math
x^{\text{switch-on}}_{gt} + x^{\text{switch-off}}_{gt} \leq 1
```

- If the unit is off, it cannot produce power or provide reserves. If it is on,
  it must to so within the specified production limits (`eq_prod_limit[s,g,t]`):

```math
y^{\text{prod-above}}_{gts} + \sum_{r \in R_g} y^{\text{res}}_{grts} \leq
(M^{\text{pmax}}_{gt} - M^{\text{pmin}}_{gt}) x^{\text{is-on}}_{gt}
```

- Break down the "production above" variable into smaller "segment production"
  variables, to simplify the objective function (`eq_prod_above_def[s,g,t]`):

```math
y^{\text{prod-above}}_{gts} = \sum_{k=1}^{K^{cost}_g} y^{\text{seg-prod}}_{gtks}
```

- Impose upper limit on segment production variables
  (`eq_segprod_limit[s,g,t,k]`):

```math
0 \leq y^{\text{seg-prod}}_{gtks} \leq M^{\text{seg-pmax}}_{gtks}
```

- Unit cannot increase its production too quickly (`eq_ramp_up[s,g,t]`):

```math
y^{\text{prod-above}}_{gts} + \sum_{r \in R_g} y^{\text{res}}_{grts} \leq
y^{\text{prod-above}}_{g,t-1,s} + M^{\text{ramp-up}}_{g}
```

- Same as above, for initial time (`eq_ramp_up[s,g,1]`):

```math
y^{\text{prod-above}}_{g,1,s} + \sum_{r \in R_g} y^{\text{res}}_{gr,1,s} \leq
\left(M^{\text{init-power}}_{g} - M^{\text{pmin}}_{gt}\right) + M^{\text{ramp-up}}_{g}
```

- Unit cannot decrease its production too quickly (`eq_ramp_down[s,g,t]`):

```math
y^{\text{prod-above}}_{gts} \geq
y^{\text{prod-above}}_{g,t-1,s} - M^{\text{ramp-down}}_{g}
```

- Same as above, for initial time (`eq_ramp_down[s,g,1]`):

```math
y^{\text{prod-above}}_{g,1,s} \geq
\left(M^{\text{init-power}}_{g} - M^{\text{pmin}}_{gt}\right) - M^{\text{ramp-down}}_{g}
```

- Unit cannot produce excessive amount of power immediately after starting up
  (`eq_startup_limit[s,g,t]`):

```math
y^{\text{prod-above}}_{gts} + \sum_{r \in R_g} y^{\text{res}}_{grts} \leq
(M^{\text{pmax}}_{gt} - M^{\text{pmin}}_{gt}) x^{\text{is-on}}_{gt} -
max\left\{0,M^{\text{pmax}}_{gt} - M^{\text{startup-limit}}_{g}\right\}
x^{\text{switch-on}}_{gt}
```

- Unit cannot shutoff it it's producing too much power
  (`eq_shutdown_limit[s,g,t]`):

```math
y^{\text{prod-above}}_{gts} \leq
(M^{\text{pmax}}_{gt} - M^{\text{pmin}}_{gt}) x^{\text{is-on}}_{gt} -
max\left\{0,M^{\text{pmax}}_{gt} - M^{\text{shutdown-limit}}_{g}\right\}
x^{\text{switch-off}}_{g,t+1}
```

## 3. Profiled generators

A _profiled generator_ is a simplified generator model that can be used to
represent renewable energy resources, including wind, solar and hydro. Unlike
thermal generators, which can be either on or off, profiled generators do not
have status variables; the only optimization decision is on their power output
level, which must remain between minimum and maximum time-varying amounts.
Production cost curves for profiled generators are linear, making them again
much simpler than thermal units.

### Constants

| Symbol                  | Unit  | Description                                        |
| :---------------------- | :---- | :------------------------------------------------- |
| $M^{\text{pmax}}_{sgt}$ | MW    | Maximum power output at time $t$ and scenario $s$. |
| $M^{\text{pmin}}_{sgt}$ | MW    | Minimum power output at time $t$ and scenario $s$. |
| $Z^{\text{pvar}}_{sgt}$ | \$/MW | Generation cost at time $t$ and scenario $s$.      |

### Decision variables

| Symbol                | JuMP name              | Unit | Description                                                   | Stage |
| :-------------------- | :--------------------- | :--- | :------------------------------------------------------------ | :---- |
| $y^\text{prod}_{sgt}$ | `prod_profiled[s,g,t]` | MW   | Amount of power produced by $g$ in time $t$ and scenario $s$. | 2     |

### Objective function terms

- Production cost:

```math
\sum_{s \in S} p(s) \left[
  \sum_{t \in T} y^\text{prod}_{sgt} Z^{\text{pvar}}_{sgt}
\right]
```

### Constraints

- Variable bounds:

```math
M^{\text{pmin}}_{sgt} \leq y^\text{prod}_{sgt} \leq M^{\text{pmax}}_{sgt}
```

## 4. Conventional loads

Loads represent the demand for electrical power by consumers and devices
connected to the system. This section describes _conventional_ (or inelastic)
loads, which are not sensitive to changes in electricity prices, and must always
be served. Each bus in the transmission network has exactly one load; multiple
loads in the same bus can be modelled by aggregating them. If there is not
enough production or transmission capacity to serve all loads, some load can be
curtailed, at a penalty.

### Constants

| Symbol                  | Unit  | Description                                                |
| :---------------------- | :---- | :--------------------------------------------------------- |
| $M^\text{load}_{sbt}$   | MW    | Conventional load on bus $b$ at time $s$ and scenario $s$. |
| $Z^\text{curtail}_{st}$ | \$/MW | Load curtailment penalty at time $t$ in scenario $s$.      |

### Decision variables

| Symbol                   | JuMP name        | Unit | Description                                                      | Stage |
| :----------------------- | :--------------- | :--- | :--------------------------------------------------------------- | :---- |
| $y^\text{curtail}_{sbt}$ | `curtail[s,b,t]` | MW   | Amount of load curtailed at bus $b$ in time $t$ and scenario $s$ | 2     |

### Objective function terms

- Load curtailment penalty:

```math
\sum_{s \in S} p(s) \left[
  \sum_{b \in B} \sum_{t \in T} y^\text{curtail}_{sbt} Z^\text{curtail}_{ts}
\right]
```

### Constraints

- Variable bounds:

```math
0 \leq y^\text{curtail}_{sbt} \leq M^\text{load}_{bts}
```

## 5. Price-sensitive loads

_Price-sensitive loads_ refer to components in the system which may increase or
reduce their power consumption according to energy prices. Unlike conventional
loads, described above, price-sensitive loads are only served if it is
economical to do so. More specifically, there are no constraints forcing these
loads to be served; instead, there is a term in the objective function rewarding
each MW served. Unlike conventional loads, there may be multiple price-sensitive
loads per bus.

!!! note

    Some unit commitment models allow price-sensitive loads to have a piecewise-linear convex revenue curves, similar to thermal generators. This can be achieved in UC.jl by adding multiple price-sensitive loads to the bus, one for each piecewise-linear segment.

### Sets and constants

| Symbol                       | Unit  | Description                                                      |
| :--------------------------- | :---- | :--------------------------------------------------------------- |
| $M^\text{psl-demand}_{spt}$  | MW    | Demand of price-sensitive load $p$ at time $t$ and scenario $s$. |
| $Z^\text{psl-revenue}_{spt}$ | \$/MW | Revenue from serving load $p$ at $t$ in scenario $s$.            |
| $\text{PSL}$                 |       | Set of price-sensitive loads.                                    |

### Decision variables

| Symbol               | JuMP name      | Unit | Description                                       | Stage |
| :------------------- | :------------- | :--- | :------------------------------------------------ | :---- |
| $y^\text{psl}_{spt}$ | `loads[s,p,t]` | MW   | Amount served to $p$ in time $t$ and scenario $s$ | 2     |

### Objective function terms

- Revenue from serving price-sensitive loads:

```math
  - \sum_{s \in S} p(s) \left[
    \sum_{p \in \text{PSL}} \sum_{t \in T} y^\text{psl}_{spt} Z^\text{psl-revenue}_{spt}
  \right]
```

### Constraints

- Variable bounds:

```math
0 \leq y^\text{psl}_{spt} \leq M^\text{psl-demand}_{spt}
```

## 6. Energy storage

_Energy storage_ units are able to store energy during periods of low demand,
then release energy back to the grid during periods of high demand. These
devices include _batteries_, _pumped hydroelectric storage_, _compressed air
energy storage_ and _flywheels_. They are becoming increasingly important in the
modern power grid, and can help to enhance grid reliability, efficiency and
integration of renewable energy resources.

### Concepts

- **Min/max energy level and charge rate:** Energy storage units can only store
  a limited amount of energy (in MWh). To maintain the operational safety and
  longevity of these devices, a minimum energy level may also be imposed. The
  rate (in MW) at which these units can charge and discharge is also limited,
  due to chemical, physical and operational considerations.

- **Operational costs:** Charging and discharging energy storage units may incur
  a cost/revenue. We assume that this cost/revenue is linear on the
  charge/discharte rate ($/MW).

- **Efficiency:** Charging an energy storage unit for one hour with an input of
  1 MW might not result in an increase of the energy level in the device by
  exactly 1 MWh, due to various inneficiencies in the charging process,
  including coversion losses and heat generation. For similar reasons,
  discharging a storage unit for one hour at 1 MW might reduce the energy level
  by more than 1 MWh. Furthermore, even when the unit is not charging or
  discharging, some energy level may be gradually lost over time, due to
  unwanted chemical reactions, thermal effects of mechanical losses.

- **Myopic effect:** Because the optimization process considers a fixed time
  window, there is an inherent bias towards exploiting energy storage units to
  their maximum within the window, completely ignoring their operation just
  beyond this horizon. For instance, without further constraints, the
  optimization algorithm will often ensure that all storage units are fully
  discharged at the end of the last time step, which may not be desirable. To
  mitigate this myopic effect, a minimum and maximum energy level may be imposed
  at the last time step.

- **Simultaneous charging and discharging:** Depending on charge and discharge
  costs/revenue, it may make sense mathematically to simultaneously charge and
  discharge the storage unit, thus keeping its energy level unchanged while
  potentially collecting revenue. Additional binary variables and constraints
  are required to prevent this incorrect model behavior.

### Sets and constants

| Symbol                                | Unit  | Description                                                                                           |
| :------------------------------------ | :---- | :---------------------------------------------------------------------------------------------------- |
| $\text{SU}$                           |       | Set of storage units                                                                                  |
| $Z^\text{charge}_{sut}$               | \$/MW | Linear charge cost/revenue for unit $u$ at time $t$ in scenario $s$.                                  |
| $Z^\text{discharge}_{sut}$            | \$/MW | Linear discharge cost/revenue for unit $u$ at time $t$ in scenario $s$.                               |
| $M^\text{discharge-max}_{sut}$        | \$/MW | Maximum discharge rate for unit $u$ at time $t$ in scenario $s$.                                      |
| $M^\text{discharge-min}_{sut}$        | \$/MW | Minimum discharge rate for unit $u$ at time $t$ in scenario $s$.                                      |
| $M^\text{charge-max}_{sut}$           | \$/MW | Maximum charge rate for unit $u$ at time $t$ in scenario $s$.                                         |
| $M^\text{charge-min}_{sut}$           | \$/MW | Minimum charge rate for unit $u$ at time $t$ in scenario $s$.                                         |
| $M^\text{max-end-level}_{su}$         | MWh   | Maximum storage level of unit $u$ at the last time step in scenario $s$                               |
| $M^\text{min-end-level}_{su}$         | MWh   | Minimum storage level of unit $u$ at the last time step in scenario $s$                               |
| $\gamma^\text{loss}_{s,u,t}$          |       | Self-discharge factor.                                                                                |
| $\gamma^\text{charge-eff}_{s,u,t}$    |       | Charging efficiency factor.                                                                           |
| $\gamma^\text{discharge-eff}_{s,u,t}$ |       | Discharging efficiency factor.                                                                        |
| $\gamma^\text{time-step}$             |       | Length of a time step, in hours. Should be 1.0 for hourly time steps, 0.5 for 30-min half steps, etc. |

### Decision variables

| Symbol                          | JuMP name               | Unit   | Description                                                  | Stage |
| :------------------------------ | :---------------------- | :----- | :----------------------------------------------------------- | :---- |
| $y^\text{level}_{sut}$          | `storage_level[s,u,t]`  | MWh    | Storage level of unit $u$ at time $t$ in scenario $s$.       | 2     |
| $y^\text{charge}_{sut}$         | `charge_rate[s,u,t]`    | MW     | Charge rate of unit $u$ at time $t$ in scenario $s$.         | 2     |
| $y^\text{discharge}_{sut}$      | `discharge_rate[s,u,t]` | MW     | Discharge rate of unit $u$ at time $t$ in scenario $s$.      | 2     |
| $x^\text{is-charging}_{sut}$    | `is_charging[s,u,t]`    | Binary | True if unit $u$ is charging at time $t$ in scenario $s$.    | 2     |
| $x^\text{is-discharging}_{sut}$ | `is_discharging[s,u,t]` | Binary | True if unit $u$ is discharging at time $t$ in scenario $s$. | 2     |

### Objective function terms

- Charge and discharge cost/revenue:

```math
\sum_{s \in S} p(s) \left[
  \sum_{u \in \text{SU}} \sum_{t \in T} \left(
    y^\text{charge}_{sut} Z^\text{charge}_{sut} +
    y^\text{discharge}_{sut} Z^\text{discharge}_{sut}
    \right)
\right]
```

### Constraints

- Prevent simultaneous charge and discharge
  (`eq_simultaneous_charge_and_discharge[s,u,t]`):

```math
x^\text{is-charging}_{sut} + x^\text{is-discharging}_{sut} \leq 1
```

- Limit charge/discharge rate (`eq_min_charge_rate[s,u,t]`,
  `eq_max_charge_rate[s,u,t]`, `eq_min_discharge_rate[s,u,t]` and
  `eq_max_discharge_rate[s,u,t]`):

```math
\begin{align*}
y^\text{charge}_{sut} \leq x^\text{is-charging}_{sut} M^\text{charge-max}_{sut} \\
y^\text{charge}_{sut} \geq x^\text{is-charging}_{sut} M^\text{charge-min}_{sut} \\
y^\text{discharge}_{sut} \leq x^\text{is-discharging}_{sut} M^\text{discharge-max}_{sut} \\
y^\text{discharge}_{sut} \geq x^\text{is-discharging}_{sut} M^\text{discharge-min}_{sut} \\
\end{align*}
```

- Calculate current storage level (`eq_storage_transition[s,u,t]`):

```math
y^\text{level}_{sut} =
(1 - \gamma^\text{loss}_{s,u,t}) y^\text{level}_{su,t-1} +
 \gamma^\text{time-step} \gamma^\text{charge-eff}_{s,u,t} y^\text{charge}_{sut} -
\frac{\gamma^\text{time-step}}{\gamma^\text{discharge-eff}_{s,u,t}} y^\text{charge}_{sut}
```

- Enforce storage level at last time step (`eq_ending_level[s,u]`):

```math
M^\text{min-end-level}_{su} \leq y^\text{level}_{sut} \leq M^\text{max-end-level}_{su}
```

## 7. Buses and transmission lines

So far, we have described generators, which produce power, loads, which consume
power, and storage units, which store energy for later use. Another important
element is the transmission network, which delivers the power produced by the
generators to the loads and storage units. Mathematically, the network is
represented as a graph $(B,L)$ where $B$ is the set of **buses** and $L$ is the
set of **transmission lines**. Each generator, load and storage unit is located
at a bus. The **net injection** at the bus is the sum of all power injected
minus withdrawn at the bus. To balance production and consumption, we must
enforce that the sum of all net injections over the entire network equal to
zero.

Besides the net balance equations, we must also enforce flow limits on the
transmission lines. Unlike flows in other optimization problems, power flows are
directly determined by net injections and transmission line parameters, and must
follow physical laws. UC.jl uses the DC linearization of AC power flow
equations. Under this linearization, the flow $f_l$ in transmission line $l$ is
given by $\sum_{b \in B} \delta_{bl} n_b$, where $\delta_{bl}$ is a constant
known as _injection shift factor_ (also commonly called _power transfer
distribution factor_), computed from the line parameters, and $n_b$ is the net
injection at bus $b$.

!!! warning

    To improve computational performance, power flow variables and constraints are generated on-the-fly, during `UnitCommitment.optimize!`; they are **not** added by `UnitCommitment.build_model`.

### Sets and constants

| Symbol                    | Unit  | Description                                                 |
| :------------------------ | :---- | :---------------------------------------------------------- |
| $M^\text{limit}_{slt}$    | MW    | Flow limit for line $l$ at time $t$ and scenario $s$.       |
| $Z^\text{overflow}_{slt}$ | \$/MW | Overflow penalty for line $l$ at time $t$ and scenario $s$. |
| $L$                       |       | Set of transmission lines.                                  |
| $B$                       |       | Set of buses.                                               |

### Decision variables

| Symbol                    | JuMP name              | Unit | Description                                                           | Stage |
| :------------------------ | :--------------------- | :--- | :-------------------------------------------------------------------- | :---- |
| $y^\text{flow}_{slt}$     | _(added on-the-fly)_   | MW   | Flow in line $l$ at time $t$ and scenario $s$.                        | 2     |
| $y^\text{inj}_{sbt}$      | `net_injection[s,b,t]` | MW   | Total net injection at bus $b$, time $t$ and scenario $s$.            | 2     |
| $y^\text{overflow}_{slt}$ | `overflow[s,l,t]`      | MW   | Amount of flow above limit for line $l$ at time $t$ and scenario $s$. | 2     |

### Objective function terms

- Penalty for exceeding line limits:

```math
  \sum_{s \in S} p(s) \left[
    \sum_{l \in L} \sum_{t \in T} y^\text{overflow}_{slt} Z^\text{overflow}_{slt}
  \right]
```

### Constraints

- Power produced equal power consumed (`eq_power_balance[s,t]`):

```math
\sum_{b \in B} \sum_{t \in T} y^\text{inj}_{sbt} = 0
```

- Definition of flow (_enforced on-the-fly_):

```math
y^\text{flow}_{slt} = \sum_{b \in B} \delta_{sbl} y^\text{inj}_{sbt}
```

- Flow limits (_enforced on-the-fly_):

```math
\begin{align*}
 y^\text{flow}_{slt} & \leq M^\text{limit}_{slt} + y^\text{overflow}_{slt} \\
-y^\text{flow}_{slt} & \leq M^\text{limit}_{slt} + y^\text{overflow}_{slt}
\end{align*}
```