Overview

The production cost model (PCM) simulates chronological operations given fixed infrastructure. PCM currently defaults to full-hourly resolution; representative-day reduction is planned for a future update.

Model class by configuration:

  • unit_commitment = 0: LP (no UC binaries)
  • unit_commitment = 1: MILP (integer UC)
  • unit_commitment = 2: LP relaxation of UC

If unit_commitment = 1 and write_shadow_prices = 1, HOPE solves MILP first, then fixes discrete variables and re-solves LP to recover dual/LMP outputs.

Active Mode Switches

  • network_model:
    • 0: no network constraints (copper plate)
    • 1: zonal transport
    • 2: nodal DCOPF angle-based
    • 3: nodal DCOPF PTDF-based
  • operation_reserve_mode:
    • 0: off
    • 1: REG + SPIN
    • 2: REG + SPIN + NSPIN
  • clean_energy_policy (0/1)
  • carbon_policy (0/1/2)
  • flexible_demand (0/1)

Problem Formulation

Objective

\[\min \; C^{startup} + C^{op}_{gen} + C^{op}_{sto} + C^{DR} + C^{LS} + C^{RPS\_pen} + C^{CO2\_pen}\]

Expanded form used in code:

\[\begin{aligned} \min \Gamma =\; &\sum_{t\in T}N_t\sum_{g\in G}\sum_{h\in H_t} VCG_g\,p_{g,h} + \sum_{t\in T}N_t\sum_{s\in S}\sum_{h\in H_t} VCS_s\,(c_{s,h}+dc_{s,h}) \\ &+ \sum_{t\in T}N_t\sum_{i\in I}\sum_{h\in H_t} VOLL\,p^{LS}_{i,h} \\ &+ \mathbb{1}_{UC}\sum_{t\in T}N_t\sum_{g\in G^{UC}}\sum_{h\in H_t} STC_g\,P^{max}_g\,su_{g,h} \\ &+ \mathbb{1}_{FD}\sum_{t\in T}N_t\sum_{r\in R}\sum_{h\in H_t} DRC_r\,(dr^{DF}_{r,h}+dr^{PB}_{r,h}) \\ &+ \mathbb{1}_{RPS}\,PT^{rps}\sum_{w\in W} pt^{rps}_w + \mathbb{1}_{CO2}\,PT^{emis}\sum_{w\in W} em^{emis}_w \end{aligned}\]

where indicators $\mathbb{1}_{UC}$, $\mathbb{1}_{FD}$, $\mathbb{1}_{RPS}$, $\mathbb{1}_{CO2}\in\{0,1\}$ are controlled by unit_commitment, flexible_demand, clean_energy_policy, and carbon_policy. In current full-hourly PCM runs, T={1}, H_1=H, and N_1=1.

Constraint Blocks

Constraint IDs in code comments use the same labels below (for example, PCM-C1.2 in src/PCM.jl).

1. [PCM-C1] Power balance and network by network_model

[PCM-C1.0] network_model = 0 (copper plate)

One system balance per hour:

\[\sum_g p_{g,h} + \sum_s (dc_{s,h} - c_{s,h}) + \sum_i NI_{i,h} = \sum_i Load_{i,h} + \sum_i DR^{opt}_{i,h} - \sum_i p^{LS}_{i,h}\]

No transmission constraints are enforced.

[PCM-C1.1] network_model = 1 (zonal transport)

Zonal balance:

\[\sum_{g \in G_i} p_{g,h} + \sum_{s \in S_i}(dc_{s,h}-c_{s,h}) - \sum_{l \in LS_i} f_{l,h} + \sum_{l \in LR_i} f_{l,h} + NI_{i,h} = Load_{i,h} + DR^{opt}_{i,h} - p^{LS}_{i,h}\]

When transmission_loss = 1, HOPE adds endpoint-allocated line losses to the zonal balance:

\[\mathrm{ZoneLineLoss}_{i,h} = \frac{1}{2}\sum_{l \in LS_i \cup LR_i} loss_{l,h}, \qquad loss_{l,h} = \rho_l |f_{l,h}|\]

Corridor flow bounds:

\[-F^{max}_l \le f_{l,h} \le F^{max}_l\]

[PCM-C1.2] network_model = 2 (nodal DCOPF, angle-based)

Nodal balance per bus n:

\[\sum_{g \in G_n} p_{g,h} + \sum_{s \in S_n}(dc_{s,h}-c_{s,h}) - \sum_{l \in LS_n} f_{l,h} + \sum_{l \in LR_n} f_{l,h} + NI^{actual}_{n,h} = Load_{n,h}\]

When transmission_loss = 1, HOPE adds endpoint-allocated line losses to each nodal balance:

\[\mathrm{NodeLineLoss}_{n,h} = \frac{1}{2}\sum_{l \in LS_n \cup LR_n} loss_{l,h}, \qquad loss_{l,h} = \rho_l |f_{l,h}|\]

Nodal load comes from load_timeseries_nodal. Nodal interchange uses direct bus-level input from ni_timeseries_nodal when provided; otherwise HOPE falls back to allocating system NI from zone-level terms using bus load shares. In the ISO-NE 250-bus example, that nodal NI file is built from official interface chronology, scaled to the synthetic case NI magnitude, and blended with a small load-share balancing tail so the nodal case remains solvable without smearing NI uniformly across the system.

When both ni_timeseries_nodal_target and ni_timeseries_nodal_cap are provided, HOPE instead treats nodal NI as a decision variable bounded between the target and cap profiles. Deviation from the target profile is penalized in the objective:

\[NI^{actual}_{n,h} - NI^{target}_{n,h} = dev^{+}_{n,h} - dev^{-}_{n,h}\]

\[\mathrm{NIDeviationPenalty} = PT^{NI}_{dev}\sum_{t\in T}N_t\sum_{h\in H_t}\sum_{n\in N} \left(dev^{+}_{n,h}+dev^{-}_{n,h}\right)\]

DC line physics:

\[f_{l,h} = B_l(\theta_{from(l),h} - \theta_{to(l),h})\]

Reference angle and optional bounds:

\[\theta_{ref,h}=0,\quad -\theta^{max} \le \theta_{n,h} \le \theta^{max}\]

Optional per-line angle-difference limits (if enabled in data):

\[-\Delta\theta^{max}_l \le \theta_{from(l),h} - \theta_{to(l),h} \le \Delta\theta^{max}_l\]

[PCM-C1.3] network_model = 3 (nodal DCOPF, PTDF-based)

Nodal injection definition:

\[inj_{n,h} = \sum_{g \in G_n} p_{g,h} + \sum_{s \in S_n}(dc_{s,h}-c_{s,h}) + NI^{actual}_{n,h} - Load_{n,h}\]

Injection balance:

\[\sum_n inj_{n,h}=0\]

PTDF flow mapping:

\[f_{l,h} = \sum_n PTDF_{l,n}\,inj_{n,h}\]

PTDF mode in the current HOPE release is lossless. Keep transmission_loss = 0 when network_model = 3.

Line flow bounds in PTDF mode:

\[-F^{eff}_l \le f_{l,h} \le F^{eff}_l\]

F^{eff}_l is used for PTDF mode and equals thermal limit by default; it can be tightened by angle-difference limits via:

\[F^{eff}_l = \min\left(F^{max}_l,\; |B_l|\Delta\theta^{max}_l\right)\]

2. [PCM-C2] Operating reserve

Reserve variables:

  • Thermal generators: $r^{REG\uparrow}_{G,g,h}$, $r^{REG\downarrow}_{G,g,h}$, $r^{SPIN}_{G,g,h}$, $r^{NSPIN}_{G,g,h}$
  • Storage: $r^{REG\uparrow}_{S,s,h}$, $r^{REG\downarrow}_{S,s,h}$, $r^{SPIN}_{S,s,h}$, $r^{NSPIN}_{S,s,h}$

System requirements by mode:

  • Mode 1: $REG^\uparrow$, $REG^\downarrow$, $SPIN$ active; $NSPIN$ fixed to zero
  • Mode 2: $REG^\uparrow$, $REG^\downarrow$, $SPIN$, $NSPIN$ all active
  • Mode 0: all reserve variables fixed to zero

Thermal eligibility:

  • Reserve requirements are supplied by thermal units ($G^{F}$) and storage.
  • Non-thermal generators are forced to zero reserve provision.

Headroom/downward room and reserve capability limits are enforced with UC-aware variants for units in $G^{UC}$.

Ramp-response limits link reserve products to response windows:

\[r \le RampRate \cdot P^{max} \cdot \Delta\]

with product-specific windows $\Delta \in \{\Delta^{REG}, \Delta^{SPIN}, \Delta^{NSPIN}\}$.

3. [PCM-C3] Generator and UC blocks

Base dispatch limits use energy plus upward reserve terms.

If UC is enabled:

  • commitment state $o_{g,h}$
  • startup/shutdown $su_{g,h},\;sd_{g,h}$
  • minimum-run variable $pmin_{g,h}$
  • transition, min up/down, and UC-adjusted ramp constraints.

4. [PCM-C4] Storage blocks

  • Charge and discharge are co-limited with downward/upward reserve, respectively.
  • SOC dynamics:

\[soc_{s,h} = soc_{s,h-1} + \eta^{ch}_s c_{s,h} - dc_{s,h}/\eta^{dis}_s\]

  • Cyclic yearly SOC closure.
  • Current code enforces both:
    • soc[s,1] = soc[s,H[end]]
    • soc[s,H[end]] = 0.5 * SECAP[s]
  • Reserve deliverability from SOC over response windows:

\[r_{S,s,h}\cdot \Delta \le soc_{s,h}\]

5. [PCM-C5] RPS and carbon policies

RPS uses $pwe_{g,w,w^\prime}$ (REC exports from state $w$ to $w^\prime$), with:

  • state renewable generation accounting $pw_{g,w}$
  • REC export/import feasibility
  • state RPS balance with slack $pt^{rps}_w$.

Carbon policy options:

  • carbon_policy = 1: state annual emissions cap with slack
  • carbon_policy = 2: state allowance cap and allowance-emission balance with slack.
  • carbon_policy = 0: no carbon-policy constraints (no carbon slack variable/constraints are added).

Code expression for annual emissions accounting:

\[StateCarbonEmission_w = \sum_{t\in T}N_t\sum_{i\in I_w}\sum_{g\in G_i\cap G^F}\sum_{h\in H_t} EF_g\,p_{g,h}\]

6. [PCM-C6] Flexible demand

Current code uses the backlog load-shifting formulation over DR resources r \in R:

\[b_{r,h} = b_{r,h-1} + dr^{DF}_{r,h} - \eta^{DR}_r\,dr^{PB}_{r,h}\]

Boundary conditions per period:

\[b_{r,h_0(t)} = 0,\quad b_{r,h_{end}(t)} = 0\]

Bounds:

\[dr^{DF}_{r,h} \le DR^{DF,max}_{r,h},\quad dr^{PB}_{r,h} \le DR^{PB,max}_{r,h},\quad b_{r,h} \le \tau^{DR}_r\cdot DR^{DF,peak}_r\]

DR^{opt}_{i,h} enters power balance as net load shift per zone:

\[DR^{opt}_{i,h} = \sum_{r\in R_i}\left(dr^{PB}_{r,h} - dr^{DF}_{r,h}\right)\]

LMP and Congestion Outputs

When duals are available, PCM writes:

  • zonal/nodal prices
  • nodal price decomposition (energy, congestion, loss)
  • line shadow prices and congestion rent
  • optional summary analytics in output/Analysis/Summary_*.csv when summary_table = 1.

For unit_commitment = 1 MILP runs, set write_shadow_prices = 1 to trigger fixed-LP re-solve for dual recovery.

Price Sign Convention

HOPE exports PCM prices as the marginal objective change from a +1 MW increase in load. That is the economic quantity users typically mean by LMP.

Important implementation note:

  • The raw sign of dual(constraint) is not stable across algebraically equivalent equality constraints.
  • In particular, multiplying an equality row by -1 can flip the raw dual sign without changing the economics.

So HOPE now applies a formulation-specific sign rule when writing prices:

  • network_model = 0, 1, or 2:
    • power balance is written in supply == load form
    • exported price = dual(balance_constraint)
  • network_model = 3:
    • the reported nodal price comes from the PTDF injection-definition row inj == supply - load
    • exported price = -dual(PTDFInjDef_con)

This rule is now regression-tested against two independent checks:

  • shadow_price
  • a direct +1 MW load perturbation test

The regression is implemented in test/test-lmp-sign-regression.jl and currently covers:

  • the ISO-NE 250-bus nodal angle case
  • the RTS24 nodal PTDF case

So the sign convention is now tied to the formulation family, not to a specific case's historical behavior.