Overview

The generation and transmission expansion planning (GTEP) model co-optimizes investment and operations under policy constraints.

The model jointly decides:

  • investment/retirement decisions for generators, transmission, and storage;
  • hourly operations (generation, transmission flows, storage charging/discharging, and optional demand response);
  • policy compliance under RPS/REC trading and carbon-policy modes.

Temporal structure supports either full-year hourly chronology or representative-day compression with weighted periods. Planning reserve adequacy can be enforced at system or zonal level, and operating reserve is modeled as SPIN in the current GTEP implementation.

Model class by configuration:

  • inv_dcs_bin = 1: MILP (binary build/retire decisions)
  • inv_dcs_bin = 0: relaxed LP

If a MILP is solved and write_shadow_prices = 1, HOPE performs a fixed-LP re-solve to recover dual values (for example, zonal power-balance prices).

Active Mode Switches

  • planning_reserve_mode:
    • 0: off
    • 1: system RA
    • 2: zonal RA
  • operation_reserve_mode:
    • 0: off
    • 1: SPIN reserve only
  • clean_energy_policy:
    • 0: RPS off
    • 1: RPS on
  • carbon_policy:
    • 0: off
    • 1: state emissions cap with slack penalty
    • 2: state allowance balance with slack penalty
  • flexible_demand:
    • 0: DR off
    • 1: DR backlog formulation on

Problem Formulation

Objective

Minimize total annual system cost:

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

Expanded form:

\[\begin{aligned} \min \Gamma =\; &\sum_{g\in G^+} \tilde I_g\,x_g + \sum_{l\in L^+} \tilde I_l\,y_l + \sum_{s\in S^+} \tilde I_s\,z_s \\ &+ \sum_{g\in G,t\in T} VCG_g\,N_t \sum_{h\in H_t} p_{g,h} + \sum_{s\in S,t\in T} VCS_s\,N_t \sum_{h\in H_t}(c_{s,h}+dc_{s,h}) \\ &+ \sum_{i\in I,t\in T} VOLL\,N_t\sum_{h\in H_t} p^{LS}_{i,h} \\ &+ \mathbb{1}_{FD}\sum_{t\in T}N_t\sum_{i\in I}\sum_{r\in R_i}\sum_{h\in H_t} DRC\,(dr^{DF}_{r,h}+dr^{PB}_{r,h}) \\ &+ \mathbb{1}_{RPS}\,PT^{rps}\sum_{w\in W} pt^{rps}_w + \mathbb{1}_{CO2}\,PT^{emis}\sum_{w\in W} em^{emis}_w \end{aligned}\]

Decision vector:

\[\Gamma = \{x_g,\;x_g^{RET},\;y_l,\;z_s,\;f_{l,h},\;p_{g,h},\;p^{LS}_{i,h},\;c_{s,h},\;dc_{s,h},\;soc_{s,h}, \;pt_w^{rps},\;pw_{g,w},\;pwe_{g,w,w^\prime},\;em_w^{emis},\;a_g,\;b_{r,h},\;dr_{r,h}^{DF},\;dr_{r,h}^{PB}, \;r^{SPIN}_{g,h},\;r^{SPIN}_{s,h}\}\]

Notes:

  • \[DR^{opt}_{i,h}\]

    is an auxiliary net-shift term in zonal power balance: $DR^{opt}_{i,h}=\sum_{r\in R_i}(dr^{PB}_{r,h}-dr^{DF}_{r,h})$.

  • \[a_g\]

    is active only when carbon_policy = 2.

  • \[em_w^{emis}\]

    is active only when $carbon\_policy \in \{1,2\}$.
  • Indicators $\mathbb{1}_{FD}$, $\mathbb{1}_{RPS}$, $\mathbb{1}_{CO2}\in\{0,1\}$ are controlled by flexible_demand, clean_energy_policy, and carbon_policy.

Constraint Blocks

Constraint IDs in code comments use the same labels below (for example, GTEP-C5 in src/GTEP.jl).

1. [GTEP-C1] Investment budgets

\[\sum_{g\in G^{+}} INV_g\,P^{max}_g\,x_g \le IBG\]

\[\sum_{l\in L^{+}} \tilde I_l\,y_l \le IBL\]

\[\sum_{s\in S^{+}} INV_s\,SCAP_s\,z_s \le IBS\]

2. [GTEP-C2] Power balance

For each zone and modeled hour:

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

If flexible_demand = 0, $DR^{opt}_{i,h}=0$.

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

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

3. [GTEP-C3] Transmission limits

For existing lines $l\in L^{E}$:

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

For candidate lines $l\in L^{+}$:

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

4. [GTEP-C4] Generator operating limits (with SPIN headroom)

Energy and SPIN are co-limited by available capacity.

For existing non-retirement units $g\in G^{E}\setminus G^{RET}$:

\[P^{min}_g \le p_{g,h} + r^{SPIN}_{g,h} \le P^{max}_g \cdot AF_{g,h}\]

For retirement-eligible existing units $g\in G^{RET}$:

\[P^{min}_g(1-x^{RET}_g) \le p_{g,h} + r^{SPIN}_{g,h} \le P^{max}_g AF_{g,h}(1-x^{RET}_g)\]

For candidate units $g\in G^{+}$:

\[P^{min}_g x_g \le p_{g,h} + r^{SPIN}_{g,h} \le P^{max}_g AF_{g,h} x_g\]

Must-run generators are enforced at available maximum output (with candidate/retirement status applied consistently in code).

5. [GTEP-C5] Storage operation

Power/energy limits and SOC dynamics:

\[dc_{s,h} + r^{SPIN}_{s,h} \le SD_s \cdot SCAP_s\]

\[0 \le c_{s,h} \le SC_s\cdot SCAP_s,\quad 0 \le dc_{s,h} \le SD_s\cdot SCAP_s,\quad 0 \le soc_{s,h} \le SECAP_s\]

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

SPIN deliverability from stored energy:

\[r^{SPIN}_{s,h} \cdot \Delta^{SPIN} \le soc_{s,h}\]

Storage chronology depends on time structure:

  • Full-year mode (endogenous_rep_day = 0 and external_rep_day = 0): cyclic SOC wrap from hour 8760 to hour 1.
  • Representative-day mode (endogenous_rep_day = 1 or external_rep_day = 1):
    • Short-duration storage ($S^{SD}$) uses start/end anchor $\alpha^{anchor}_{s,t}$ (alpha_storage_anchor[s,t] in code).
    • Long-duration storage ($S^{LD}$) links SOC across representative periods.
    • With Feature 6 (link_storage_rep_days = 1 in HOPE_rep_day_settings.yml), endogenous representative-day mode uses predecessor weights derived from the real day-to-representative mapping rather than a simple fixed t-1 ordering.

6. [GTEP-C6] Planning reserve adequacy

Activated by planning_reserve_mode:

System RA mode (planning_reserve_mode = 1):

\[\sum_{g\in G^{E}} CC_gP^{max}_g +\sum_{g\in G^{+}} CC_gP^{max}_g x_g +\sum_{s\in S^{E}} CC_sSCAP_s +\sum_{s\in S^{+}} CC_sSCAP_s z_s +DR^{RA}_{sys} \ge (1+PRM)\,PK_{sys}\]

Zonal RA mode (planning_reserve_mode = 2), for each zone $i$:

\[\sum_{g\in G_i\cap G^{E}} CC_gP^{max}_g +\sum_{g\in G_i\cap G^{+}} CC_gP^{max}_g x_g +\sum_{s\in S_i\cap S^{E}} CC_sSCAP_s +\sum_{s\in S_i\cap S^{+}} CC_sSCAP_s z_s +DR^{RA}_{i} \ge (1+PRM_i)\,PK_i\]

RA includes DR capacity credit terms when flexible_demand = 1.

7. [GTEP-C7] Operating reserve (SPIN only)

Activated by operation_reserve_mode = 1:

\[\sum_{g\in G} r^{SPIN}_{g,h} + \sum_{s\in S} r^{SPIN}_{s,h} \ge \rho^{SPIN} \cdot Load_h\]

If reserve mode is off, reserve variables are constrained to zero.

8. [GTEP-C8] RPS with REC trading

RPS uses $pwe_{g,w,w^\prime}$:

  • Meaning: annual renewable credits generated by unit $g$ in state $w$ and exported from $w$ to $w^\prime$.
  • Export and import feasibility constraints are enforced via $WER_w$ and $WIR_w$.
  • State RPS balance includes in-state renewable accounting $pw_{g,w}$, net REC imports/exports, and slack $pt^{rps}_w$.

When clean_energy_policy = 1, the code enforces:

\[G_w := \bigcup_{i\in I_w} G_i\]

[GTEP-C8.1] State-level renewable accounting:

\[pw_{g,w}=\sum_{t\in T}N_t\sum_{h\in H_t} p_{g,h}, \quad g\in G^{RPS}\cap G_w\]

[GTEP-C8.2] REC export feasibility:

\[pw_{g,w}\ge \sum_{w^\prime\in WER_w} pwe_{g,w,w^\prime}\]

[GTEP-C8.3] REC import feasibility:

\[pw_{g,w^\prime}\ge pwe_{g,w^\prime,w}, \quad \forall w^\prime\in WIR_w\]

[GTEP-C8.4] State RPS balance with REC trading and slack:

\[\begin{aligned} &\sum_{g\in G^{RPS}\cap G_w} pw_{g,w} +\sum_{w^\prime\in WIR_w}\sum_{g\in G^{RPS}\cap G_{w^\prime}} pwe_{g,w^\prime,w} -\sum_{w^\prime\in WER_w}\sum_{g\in G^{RPS}\cap G_w} pwe_{g,w,w^\prime} + pt^{rps}_w \\ &\ge \sum_{t\in T}N_t\sum_{h\in H_t}\sum_{i\in I_w}\sum_{d\in D_i} P_{h,d}PK_d\,RPS_w \end{aligned}\]

If clean_energy_policy = 0, the model enforces:

\[pt^{rps}_w=0,\quad \forall w\in W\]

9. [GTEP-C9A/C9B/C9O] Carbon policy

Formulation by flag:

  • carbon_policy = 1:

\[\sum_{t\in T}N_t\sum_{h\in H_t}\sum_{g\in G_w\cap G^{F}} EF_g\,p_{g,h} \le ELMT_w + em^{emis}_w,\quad \forall w\in W\]

  • carbon_policy = 2:

Cap-and-trade formulation (state allowance cap + allowance-emission balance):

\[\sum_{g\in G_w\cap G^{F}} a_g \le ALW_w,\quad \forall w\in W\]

\[\sum_{t\in T}N_t\sum_{h\in H_t}\sum_{g\in G_w\cap G^{F}} EF_g\,p_{g,h} \le \sum_{g\in G_w\cap G^{F}} a_g + em^{emis}_w,\quad \forall w\in W\]

If carbon_policy = 0, no carbon-policy constraints are imposed.

10. [GTEP-C10] Flexible demand (backlog form)

If enabled, DR uses deferred $dr^{DF}_{r,h}$, payback $dr^{PB}_{r,h}$, and backlog $b_{r,h}$ with:

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

for non-first hours in each modeled period, with start/end closure:

\[b_{r,h^{start}_t}=0,\qquad b_{r,h^{end}_t}=0\]

and bounds:

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

11. Variable domains

Binary (or relaxed to $[0,1]$ when inv_dcs_bin = 0):

\[x_g,\;x_g^{RET},\;y_l,\;z_s \in \{0,1\}\]

Nonnegative variables:

\[a_g,\;b_{r,h},\;p_{g,h},\;p^{LS}_{i,h},\;c_{s,h},\;dc_{s,h},\;soc_{s,h},\;pt_w^{rps}, \;pw_{g,w},\;pwe_{g,w,w^\prime},\;em_w^{emis},\;dr^{DF}_{r,h},\;dr^{PB}_{r,h},\; r^{SPIN}_{g,h},\;r^{SPIN}_{s,h}\ge 0\]

Output and Dual Notes

  • power_price.csv in GTEP requires duals of PB_con.
  • For MILP (inv_dcs_bin = 1), enable write_shadow_prices = 1 to trigger fixed-LP dual recovery before output writing.