Security Constrained Unit Commitment (SCUC) — Full Model Documentation

This document describes the mathematical optimization model implemented in the provided C++ code (“SCUC”).

Scope note

• The model includes thermal unit commitment, piecewise-linear production costs, startup categories, reserves (with optional shortfall), profiled generators, price-sensitive loads, conventional load curtailment, and a DC network with PTDF base-case limits and LODF N-1 limits (with soft overflow penalties).

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1) Indices, Sets, and Time Convention

All indices are 0-based, consistent with the code.

Sets

• Time steps:
  $\mathcal{T}={0,1,\dots,T-1}$

• Buses:
  $\mathcal{B}={0,1,\dots,|\mathcal{B}|-1}$

• Transmission lines:
  $\mathcal{L}={0,1,\dots,|\mathcal{L}|-1}$

• Thermal generators:
  $\mathcal{G}={0,1,\dots,|\mathcal{G}|-1}$

• Profiled generators:
  $\mathcal{P}={0,1,\dots,|\mathcal{P}|-1}$

• Price-sensitive loads (PSL):
  $\mathcal{D}={0,1,\dots,|\mathcal{D}|-1}$

• Reserve requirement products (each is a spinning requirement time series):
  $\mathcal{R}={0,1,\dots,|\mathcal{R}|-1}$

• Reserve-eligible thermal generators:
  Let $\mathcal{G}^{res}\subseteq \mathcal{G}$ be the subset of thermal gens that can provide reserve.
  Code builds a mapping: map_res: $\mathcal{G}^{res}\to {0,1,\dots,|\mathcal{G}^{res}|-1}$ and reserve variables are indexed by $i\in{0,\dots,|\mathcal{G}^{res}|-1}$.

• Relevant N-1 contingency pairs:
  The code builds a list of monitored/outage pairs: $\mathcal{Q}\subseteq {(k,\ell)\in\mathcal{L}\times\mathcal{L}: k\neq \ell}$ including $(k,\ell)$ if the contingency list contains outage line $k$ and $|\text{LODF}_{\ell,k}|>\varepsilon$.
  Each pair is indexed by $q\in{0,\dots,|\mathcal{Q}|-1}$ with:

  • outage line $k(q)$
  • monitored line $\ell(q)$

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2) Parameters (Inputs)

2.1 Thermal generator parameters (for each $g\in\mathcal{G}$)

• Min/max power: $P^{min}_g,; P^{max}_g$

• Ramp limits: $RU_g,; RD_g$

• Startup/shutdown power limits (used for tightening): $P^{SUlim}_g,; P^{SDlim}_g$

• Minimum up/down times: $U_g,; D_g \in \mathbb{Z}_{\ge 0}$

• No-load cost: $C^{NL}_g$

• Initial status (integer):

  • init_status_g $> 0$: unit initially ON for $|init_status_g|$ periods
  • init_status_g $< 0$: unit initially OFF for $|init_status_g|$ periods

• Initial power: $P^{init}_g$

• Must-run flag and optional fixed commitment status:

  • must-run: must_run_g $\in{0,1}$
  • optional fixed commitment sequence fix$_{g,t}\in{0,1}$ for some times $t$

Piecewise-linear production segments (for each thermal $g$)

Let $\mathcal{S}_g={0,1,\dots,S_g-1}$ be the segment set.

• Segment length $\ell_{g,s}$ and slope $c_{g,s}$ for $s\in\mathcal{S}_g$

Startup categories / stages (for each thermal $g$)

Let $\mathcal{K}_g={0,1,\dots,K_g-1}$ be the startup stage set.

• Stage $k$ has: $Del_{g,k}\in\mathbb{Z}{\ge 0}, \quad C^{SU}{g,k}\ge 0$

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2.2 Profiled generator parameters (for each $p\in\mathcal{P}$)

• Time-varying bounds: $P^{min}{p,t},; P^{max}{p,t}$
• Linear generation cost: $C^{prof}_p$

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2.3 Price-sensitive load parameters (for each $d\in\mathcal{D}$)

• Demand cap and revenue: $D^{max}{d,t},\quad Rev{d,t}$

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2.4 Reserve requirement parameters (for each $r\in\mathcal{R}$)

• Required amount: $Req_{r,t}$
• Shortfall penalty (enables shortfall only if positive): $Pen^{res}_r$

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2.5 Conventional (inelastic) load and curtailment penalty

• Nodal demand: $Load_{t,b}$
• Curtailment penalty: $Pen^{curt}$

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2.6 Network parameters (DC approximation)

• PTDF matrix: $PTDF_{\ell,b}$
• LODF matrix: $LODF_{\ell,k}$
• Line limits and penalties:
  • Base-case (normal) limit: $Lim^{N}_{\ell}$
  • Contingency (emergency) limit: $Lim^{E}_{\ell}$
  • Overflow penalty: $Pen^{flow}_{\ell}$

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3) Decision Variables

3.1 Thermal UC variables (for each $g\in\mathcal{G}, t\in\mathcal{T}$)

• Commitment: $u_{g,t}\in{0,1}$
• Shutdown: $w_{g,t}\in{0,1}$
• Power: $p_{g,t}\ge 0$

Startup stage binaries

• $v_{g,k,t}\in{0,1}$ for $k\in\mathcal{K}_g$

Define the (implicit) switch-on indicator: $$y_{g,t}\triangleq \sum_{k\in\mathcal{K}g} v{g,k,t}$$

Segment production

• $p^{seg}_{g,s,t}\ge 0$ for $s\in\mathcal{S}_g$

Above-min production (not explicit variable in code): $$p^{above}{g,t}\triangleq \sum{s\in\mathcal{S}g} p^{seg}{g,s,t}$$

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3.2 Reserves

• Generator reserve: $r_{i,t}\ge 0$ for $i\in{0,\dots,|\mathcal{G}^{res}|-1}$
• Reserve shortfall (only if $Pen^{res}r>0$): $sh{r,t}\ge 0$

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3.3 Profiled generation

• $p^{prof}_{p,t}\ge 0$

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3.4 Price-sensitive load served

• $s^{psl}_{d,t}\ge 0$

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3.5 Conventional load curtailment

• $curt_{b,t}\ge 0$ with bounds: $0 \le curt_{b,t} \le Load_{t,b}$

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3.6 Network overflow variables (soft limits)

• Base overflow: $ov^{base}_{\ell,t}\ge 0$
• Contingency overflow: $ov^{cont}_{q,t}\ge 0$

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4) Objective Function

Minimize total cost: $$\min; \Big( C^{thermal} + C^{startup} + C^{prof} - Rev^{psl} + Pen^{res_short} + Pen^{curt} + Pen^{flow} \Big)$$

4.1 Thermal no-load + energy

$$C^{thermal} = \sum_{g}\sum_{t} C^{NL}_g,u_{g,t} + \sum_{g}\sum_{s}\sum_{t} c_{g,s},p^{seg}_{g,s,t}$$

4.2 Startup costs

$$C^{startup} = \sum_{g}\sum_{k}\sum_{t} C^{SU}_{g,k},v_{g,k,t}$$

4.3 Profiled generation cost

$$C^{prof}=\sum_{p}\sum_{t} C^{prof}_p,p^{prof}_{p,t}$$

4.4 PSL revenue (subtracted)

$$Rev^{psl}=\sum_{d}\sum_{t} Rev_{d,t},s^{psl}_{d,t}$$

4.5 Reserve shortfall penalty (only if $Pen^{res}_r>0$)

$$Pen^{res_short} = \sum_{r:Pen^{res}_r>0}\sum_{t} Pen^{res}_r,sh_{r,t}$$

4.6 Curtailment penalty

$$Pen^{curt}=\sum_{b}\sum_{t} Pen^{curt},curt_{b,t}$$

4.7 Line overflow penalties

$$Pen^{flow} = \sum_{\ell}\sum_{t} Pen^{flow}_\ell,ov^{base}_{\ell,t} + \sum_{q}\sum_{t} Pen^{flow}_{\ell(q)},ov^{cont}_{q,t}$$

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5) Constraints

5.1 System power balance

For each $t\in\mathcal{T}$: $$\sum_{g} p_{g,t} + \sum_{p} p^{prof}{p,t} + \sum{b} curt_{b,t} - \sum_{d} s^{psl}{d,t} = \sum{b} Load_{t,b}$$

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5.2 Reserve requirements

For each reserve product $r\in\mathcal{R}$ and time $t\in\mathcal{T}$:

• If $Pen^{res}r>0$: $$\sum{i} r_{i,t} + sh_{r,t} \ge Req_{r,t}$$

• If $Pen^{res}r\le 0$: $$\sum{i} r_{i,t} \ge Req_{r,t}$$

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6) Thermal Generator Constraints

Fix a generator $g\in\mathcal{G}$.

6.1 Initial boundary fixes (remaining min up/down)

Let $h_g=|init_status_g|$, and initOn_g $=\mathbf{1}[init_status_g>0]$.

If initOn_g $=1$, define $remUp_g=\max(0,U_g-h_g)$ and enforce: $$u_{g,t}=1 \quad \forall t=0,\dots,\min(T-1,remUp_g-1)$$

If initOn_g $=0$, define $remDn_g=\max(0,D_g-h_g)$ and enforce: $$u_{g,t}=0 \quad \forall t=0,\dots,\min(T-1,remDn_g-1)$$

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6.2 Initial transition at $t=0$

$$u_{g,0}-initOn_g = y_{g,0}-w_{g,0}$$

6.2a Optional tightening at $t=0$ (enforced in code)

$$w_{g,0} \le initOn_g$$ $$y_{g,0} \le 1-initOn_g$$

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6.3 Power definition

$$p_{g,t}=P^{min}_g,u_{g,t}+\sum_{s\in\mathcal{S}_g}p^{seg}_{g,s,t}$$

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6.4 Segment bounds

$$0\le p^{seg}_{g,s,t}\le \ell_{g,s},u_{g,t}$$

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6.5 Commitment logic for $t\ge 1$

For $t=1,\dots,T-1$: $$u_{g,t}-u_{g,t-1}=y_{g,t}-w_{g,t}$$

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6.6 Start/shut gating (as in code)

For all $t$:

• Startup implies on: $$y_{g,t} \le u_{g,t}$$
• For $t\ge 1$, startup only if was previously off: $$y_{g,t} \le 1-u_{g,t-1}$$
• For $t\ge 1$, shutdown only if was previously on: $$w_{g,t} \le u_{g,t-1}$$
• For $t\ge 1$, shutdown implies off now: $$w_{g,t}+u_{g,t}\le 1$$

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6.7 One action per period

$$y_{g,t}+w_{g,t}\le 1 \quad \forall t\in\mathcal{T}$$

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6.8 Minimum up/down time (cumulative)

Minimum up-time: $$\sum_{\tau=\max(0,t-U_g+1)}^{t} y_{g,\tau} \le u_{g,t} \quad \forall t\in\mathcal{T}$$

Minimum down-time (equivalent to code’s form): $$\sum_{\tau=\max(0,t-D_g+1)}^{t} w_{g,\tau} \le 1-u_{g,t} \quad \forall t\in\mathcal{T}$$

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6.9 Ramping (implemented on above-min)

Define: $$p^{above}{g,t}=\sum{s\in\mathcal{S}g} p^{seg}{g,s,t}.$$ If $g$ is reserve-eligible, define $r_{g,t}=r_{\text{map_res}(g),t}$; otherwise $r_{g,t}=0$.

Ramp-up ($t\ge 1$): $$p^{above}{g,t}+r{g,t}-p^{above}_{g,t-1}\le RU_g$$

Ramp-down ($t\ge 1$): $$p^{above}{g,t-1}-p^{above}{g,t}\le RD_g$$

Initial ramps ($t=0$): Let initOn_g $=\mathbf{1}[init_status_g>0]$ and $$p^{init,above}_g= \begin{cases} \max(0, P^{init}g-P^{min}g) & \text{if } initOn_g=1 \ 0 & \text{if } initOn_g=0 \end{cases}$$ Then: $$p^{above}{g,0}+r{g,0}\le p^{init,above}g+RU_g$$ $$p^{above}{g,0}\le RD_g - p^{init,above}_g$$

6.10 Startup/shutdown tightening

Let: $$Cap^{above}_g=P^{max}_g-P^{min}_g$$ $$A_g=\max(0,P^{max}_g-P^{SUlim}_g),\quad B_g=\max(0,P^{max}_g-P^{SDlim}_g)$$

Startup limit (all $t$): $$p^{above}{g,t}+r{g,t}\le Cap^{above}g,u{g,t}-A_g,y_{g,t}$$

Shutdown limit (for $t=0,\dots,T-2$): $$p^{above}{g,t}\le Cap^{above}g,u{g,t}-B_g,w{g,t+1}$$

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6.11 Must-run / fixed commitment

If must-run: $$u_{g,t}=1 \quad \forall t$$ If fixed commitment fix${g,t}$ is provided: $$u{g,t}=fix_{g,t} \quad \text{for those } t$$

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6.12 Reserve headroom (reserve-eligible only)

For $g\in\mathcal{G}^{res}$: $$p_{g,t}+r_{\text{map_res}(g),t}\le P^{max}g,u{g,t}$$

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7) Startup Category Logic

This section describes the exact logic enforced by the code for choosing startup categories.

Assume stages are indexed so that: $$Del_{g,0}\le Del_{g,1}\le \dots \le Del_{g,K_g-1}$$

Let initOn_g $=\mathbf{1}[init_status_g>0]$ and if initOn_g $=0$ define initial offline duration: $$initOffDur_g = -init_status_g \quad (\text{else } initOffDur_g=0).$$

The code adds constraints only for categories $k=0,\dots,K_g-2$ (the last category is not restricted by this routine).

For each time $t\in\mathcal{T}$ and each $k\in{0,\dots,K_g-2}$, define:

• An initial-status allowance term $L^{init}_{g,k,t}\in{0,1}$:

  • If the unit is initially OFF, define offdur $= initOffDur_g + t$.
  • Then: $$L^{init}{g,k,t} = \begin{cases} 1 & \text{if } offdur \in [Del{g,k},, Del_{g,k+1}-1] \ 0 & \text{otherwise} \end{cases}$$
  • If initially ON, then $L^{init}_{g,k,t}=0$.

• A shutdown window: $$lb = \max(0,; t-Del_{g,k+1}+1), \qquad ub = t-Del_{g,k}.$$

Then the code enforces: $$v_{g,k,t} - \sum_{i=lb}^{ub} w_{g,i} \le L^{init}_{g,k,t}, \quad \text{whenever } ub\ge 0 \text{ and } lb\le ub.$$

If the window is empty (i.e., $ub<0$ or $lb>ub$), the constraint reduces to: $$v_{g,k,t} \le L^{init}_{g,k,t}.$$

Interpretation (code-consistent):

• Selecting startup category $k$ at time $t$ is permitted if either:
  • the initial offline duration implies category $k$ is feasible at time $t$ (via $L^{init}=1$), or
  • there has been at least one shutdown event $w_{g,i}=1$ in the window $i\in[lb,ub]$, i.e., a shutdown occurred at a time consistent with being offline for a duration in $[Del_{g,k},,Del_{g,k+1}-1]$.

The coldest category $K_g-1$ is not restricted by this routine and is therefore always feasible (subject to the other startup constraints like $y_{g,t}\le u_{g,t}$ and $y_{g,t}\le 1-u_{g,t-1}$).

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8) Profiled Generator Constraints

For each $p\in\mathcal{P}$ and $t\in\mathcal{T}$: $$P^{min}{p,t}\le p^{prof}{p,t}\le P^{max}_{p,t}$$

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9) Price-Sensitive Load Constraints

For each $d\in\mathcal{D}$ and $t\in\mathcal{T}$: $$0\le s^{psl}{d,t}\le D^{max}{d,t}$$

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10) Curtailment Bounds

For each $b\in\mathcal{B}$ and $t\in\mathcal{T}$: $$0\le curt_{b,t}\le Load_{t,b}$$

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11) Network Model (PTDF + LODF, soft)

The code does not explicitly create net-injection or flow variables; it builds line constraints directly by expanding injections into generator/load variables using PTDF rows.

11.1 Net injections (conceptual)

For bus $b$ and time $t$: $$inj_{b,t}= \sum_{g\in\mathcal{G}(b)}p_{g,t} + \sum_{p\in\mathcal{P}(b)}p^{prof}{p,t} - \sum{d\in\mathcal{D}(b)}s^{psl}{d,t} - Load{t,b} + curt_{b,t}$$

11.2 Base flow (conceptual)

$$flow^{base}_{\ell,t}=\sum_{b} PTDF_{\ell,b},inj_{b,t}$$

11.3 Base soft limits

For each line $\ell$ and time $t$: $$flow^{base}{\ell,t}\le Lim^{N}{\ell}+ov^{base}{\ell,t}$$ $$-flow^{base}{\ell,t}\le Lim^{N}{\ell}+ov^{base}{\ell,t}$$

11.4 Contingency flow for pair $q=(k(q),\ell(q))$

Let $k=k(q)$ and $\ell=\ell(q)$. Then: $$flow^{cont}{q,t} = \sum{b}\Big(PTDF_{\ell,b}+LODF_{\ell,k},PTDF_{k,b}\Big),inj_{b,t}$$

11.5 Contingency soft limits

For each pair $q$ and time $t$: $$flow^{cont}{q,t}\le Lim^{E}{\ell(q)}+ov^{cont}{q,t}$$ $$-flow^{cont}{q,t}\le Lim^{E}{\ell(q)}+ov^{cont}{q,t}$$

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12) Summary

This SCUC model chooses:

• thermal unit on/off schedules and dispatch,
• startup categories and costs,
• reserve provision (and optional shortfall),
• profiled generation within availability bounds,
• price-sensitive load served if profitable,
• load curtailment only if necessary (penalized),
• while enforcing base-case and N-1 network limits using PTDF/LODF with soft overflow penalties.