ANL-CEEESA/UnitCommitment.jl
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@@ -416,142 +416,7 @@ loads per bus.
0 \leq y^\text{psl}_{spt} \leq M^\text{psl-demand}_{spt}
```
## 6. Buses and transmission lines
So far, we have described generators, which produce power, and loads, which
consume power. A third important element is the transmission network, which
delivers the power produced by the generators to the loads. 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 energy
storage device 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of all power injected minus withdrawn at the bus.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*}
```
## 7. Transmission interfaces
In some applications, such as energy exchange studies, it is important to
enforce flow limits not only on individual lines, but also on groups of
transmission lines. These groups are known as _interfaces_. More precisely, an
interface is composed by two sets of lines: the _inbound_ and the _outbound
lines_. The flow across the interface is defined as the sum of the flow in all
inbound lines minus the sum of the flow in all outbound lines. An upper and a
lower limit may be imposed on the flow across the interface, and a penalty is
imposed if the limit is exceeded.
### Sets and constants
| Symbol | Unit | Description |
| :-------------------------- | :---- | :----------------------------------------------------------------------------------------- |
| $L^\text{inbound}_{si}$ | | Set of inbound lines for interface $i$ in scenario $s$. |
| $L^\text{outbound}_{si}$ | | Set of outbound lines for interface $i$ in scenario $s$. |
| $M^\text{limit-down}_{sit}$ | MW | Lower flow limit for interface $i$ at time at time $t$ and scenario $s$ (negative number). |
| $M^\text{limit-up}_{sit}$ | MW | Upper flow limit for interface $i$ at time at time $t$ and scenario $s$ (positive number). |
| $Z^\text{overflow}_{sit}$ | \$/MW | Overflow penalty for interface $l$ at time $t$ and scenario $s$. |
| $\text{IF}$ | | Set of transmission interfaces. |
### Decision variables
| Symbol | JuMP name | Unit | Description | Stage |
| :-------------------------- | :-------------------------- | :--- | :--------------------------------------------------------------- | :---- |
| $y^\text{i-flow}_{sit}$ | `interface_flow[s,i,t]` | MW | Flow across interface $i$ at time $t$ and scenario $s$. | 2 |
| $y^\text{i-overflow}_{sit}$ | `interface_overflow[s,i,t]` | MW | Flow above limit for interface $i$ at time $t$ and scenario $s$. | 2 |
### Objective function terms
- Penalty for exceeding interface limits:
```math
\sum_{s \in S} p(s) \left[
\sum_{i \in \text{IF}} \sum_{t \in T} y^\text{i-overflow}_{sit} Z^\text{overflow}_{sit}
\right]
```
### Constraints
- Definition of interface flow (`eq_if_flow`):
```math
y^\text{i-flow}_{sit} = \sum_{b \in B} y^\text{inj}_{sbt} \left[
\sum_{l \in L^\text{outbound}_{si}} \delta_{sbl} -
\sum_{l \in L^\text{inbound}_{si}} \delta_{sbl}
\right]
```
- Interface flow limits (`eq_if_limit_up` and `eq_if_limit_up`)
```math
\begin{align*}
y^\text{i-flow}_{sit} & \leq M^\text{limit-up}_{sit} + y^\text{i-overflow}_{sit} \\
-y^\text{i-flow}_{sit} & \leq -M^\text{limit-down}_{sit} + y^\text{i-overflow}_{sit}
\end{align*}
```
## 7. Energy storage
## 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
@@ -618,67 +483,198 @@ integration of renewable energy resources.
| 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 $s$. |
| $y^\text{charge}_{sut}$ | `charge_rate[s,u,t]` | MW | Charge rate of unit $u$ at time $t$ in scenario $s$. |
| $y^\text{discharge}_{sut}$ | `discharge_rate[s,u,t]` | MW | Discharge rate of unit $u$ at time $t$ in scenario $s$. |
| $x^\text{is-charging}_{sut}$ | `is_charging[s,u,t]` | Binary | True if unit $u$ is charging at time $t$ in scenario $s$. |
| $x^\text{is-discharging}_{sut}$ | `is_discharging[s,u,t]` | Binary | True if unit $u$ is discharging at time $t$ in scenario $s$. |
| $y^\text{i-flow}_{sit}$ | `interface_flow[s,i,t]` | MW | Flow across interface
$i$ at time $t$ and scenario $s$. | 2 | | $y^\text{i-overflow}_{sit}$ |
`interface_overflow[s,i,t]` | MW | Flow above limit for interface $i$ at time
$t$ and scenario $s$. | 2 |
| $y^\text{level}_{sut}$ | `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]`):
$$
x^\text{is-charging}_{sut} + x^\text{is-discharging}_{sut} \leq 1
$$
```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]`):
$$
```math
M^\text{min-end-level}_{su} \leq y^\text{level}_{sut} \leq M^\text{max-end-level}_{su}
$$
```
## 8. Contingencies
## 7. Buses and transmission lines
## 9. Reserves
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*}
```
## 8. Transmission interfaces
In some applications, such as energy exchange studies, it is important to
enforce flow limits not only on individual lines, but also on groups of
transmission lines. These groups are known as _interfaces_. More precisely, an
interface is composed by two sets of lines: the _inbound_ and the _outbound
lines_. The flow across the interface is defined as the sum of the flow in all
inbound lines minus the sum of the flow in all outbound lines. An upper and a
lower limit may be imposed on the flow across the interface, and a penalty is
imposed if the limit is exceeded.
### Sets and constants
| Symbol | Unit | Description |
| :-------------------------- | :---- | :----------------------------------------------------------------------------------------- |
| $L^\text{inbound}_{si}$ | | Set of inbound lines for interface $i$ in scenario $s$. |
| $L^\text{outbound}_{si}$ | | Set of outbound lines for interface $i$ in scenario $s$. |
| $M^\text{limit-down}_{sit}$ | MW | Lower flow limit for interface $i$ at time at time $t$ and scenario $s$ (negative number). |
| $M^\text{limit-up}_{sit}$ | MW | Upper flow limit for interface $i$ at time at time $t$ and scenario $s$ (positive number). |
| $Z^\text{overflow}_{sit}$ | \$/MW | Overflow penalty for interface $l$ at time $t$ and scenario $s$. |
| $\text{IF}$ | | Set of transmission interfaces. |
### Decision variables
| Symbol | JuMP name | Unit | Description | Stage |
| :-------------------------- | :-------------------------- | :--- | :--------------------------------------------------------------- | :---- |
| $y^\text{i-flow}_{sit}$ | `interface_flow[s,i,t]` | MW | Flow across interface $i$ at time $t$ and scenario $s$. | 2 |
| $y^\text{i-overflow}_{sit}$ | `interface_overflow[s,i,t]` | MW | Flow above limit for interface $i$ at time $t$ and scenario $s$. | 2 |
### Objective function terms
- Penalty for exceeding interface limits:
```math
\sum_{s \in S} p(s) \left[
\sum_{i \in \text{IF}} \sum_{t \in T} y^\text{i-overflow}_{sit} Z^\text{overflow}_{sit}
\right]
```
### Constraints
- Definition of interface flow (`eq_if_flow`):
```math
y^\text{i-flow}_{sit} = \sum_{b \in B} y^\text{inj}_{sbt} \left[
\sum_{l \in L^\text{outbound}_{si}} \delta_{sbl} -
\sum_{l \in L^\text{inbound}_{si}} \delta_{sbl}
\right]
```
- Interface flow limits (`eq_if_limit_up` and `eq_if_limit_up`)
```math
\begin{align*}
y^\text{i-flow}_{sit} & \leq M^\text{limit-up}_{sit} + y^\text{i-overflow}_{sit} \\
-y^\text{i-flow}_{sit} & \leq -M^\text{limit-down}_{sit} + y^\text{i-overflow}_{sit}
\end{align*}
```
## 9. Contingencies
## 10. Reserves