Overview
The production cost model (PCM) is for system operation analysis under various policies and energy transition scenarios. This model is a Linear Programming Problem.
The objective of this model is to minimize the system's (given by the planning model) total cost, including variable operation costs and penalties for non-compliance with policies. The constraints of this model are power balance, transmission transfer limit, generator operation constraints, storage operation constraints, resource adequacy requirements, and policy constraints (i.e., renewable portfolio standards (RPS) and carbon emission limitations). The continuous decision variables of the PCM model are hourly operations of resources for a target year.
Problem Formulation
Objective function
(1) Minimize total system cost:
\[\begin{aligned} \min_{\Gamma} \quad &\sum_{g \in G, t \in T}VCG_{g} \times N_{t} \times \sum_{h \in H}p_{g,h} + \\ &\sum_{s \in S, t \in T} VCS \times \sum_{h \in H} (c_{s,h} + dc_{s,h}) + \\ &\sum_{d \in D, t \in T} VOLL_{d} \times \sum_{h \in H} p_{d,h}^{LS} + \\ &\sum_{g \in G^{F}, t \in T} CP_{g} \times \sum_{h \in H} p_{g,h} + \\ &\sum_{w \in W, h \in H} PT^{rps} \times pt_{w,h}^{rps} + \\ &\sum_{t \in T} \sum_{w \in W, h \in H} PT^{emis} \times em_{w,h}^{emis} \end{aligned}\]
\[\Gamma = \Bigl\{ a_{g,t}, b_{g,t}, f_{l,h}, p_{g,h}, p_{d,h}^{LS}, c_{s,h}, dc_{s,h}, soc_{s,h}, pt_{h}^{rps}, em^{emis}_{h}, r_{g,h}^{G}, r_{g,h}^{S} \Bigr\}\]
Constraints
(2) Power balance:
\[\sum_{g \in G_{i}} P_{g,h} + \sum_{s \in S_{i}} (dc_{s,h} - c_{s,h}) - \sum_{l \in LS_{i}} f_{l,h} \\ = \sum_{d \in D_{i}} (P_{d,h} - P_{d,h}^{LS}); \forall i \in I, h \in H\]
(3) Transmission:
\[- F_{l}^{max} \le f_{l,h} \le F_{l}^{max}; \forall l \in L, h \in H\]
(4) Operation:
\[P_{g}^{min} \le p_{g,h} + r_{g,h}^{G} \le (1 - FOR_{g}) \times P_{g}^{max}; \forall g \in G\]
(5) Spinning reserve limit:
\[r_{g,h}^{G} \le RM_{g}^{SPIN} \times (1 - FOR_{g}) \times P_{g}^{max}; \forall g \in G^{F}\]
(6) Ramp limits - 1:
\[(p_{g,h} + r_{g,h}^{G}) - p_{g, h-1} \le RU_{g} \times (1 - FOR_{g}) \times P_{g}^{max}; \forall g \in G^{F}, h \in H\]
(7) Ramp limits - 2:
\[(p_{g,h} + r_{g,h}^{G}) - p_{g, h-1} \ge -RU_{g} \times (1 - FOR_{g}) \times P_{g}^{max}; \forall g \in G^{F}, h \in H\]
(8) Load shedding limit:
\[0 \le p_{d,h}^{LS} \le P_{d}; \forall d \in D\]
(9) Renewables generation availability:
\[p_{g,h} \le AFRE_{g,h,i} \times P_{g}^{max}; \forall h \in H, g \in G_{PV} \cup G^{W}), i \in I\]
(10) Storage charging rate limit:
\[\frac{c_{s,h}}{SC_{s}} \le SCAP_{s}; \forall h \in H\]
(11) Storage discharging rate limit:
\[\frac{dc_{s,h}}{SD_{s}} \le SCAP_{s}; \forall h \in H\]
(12) Storage operation limit - 1:
\[0 \le soc_{s,h} \le SECAP_{s}; \forall h \in H, s \in S\]
(13) Storage operation limit - 2:
\[dc_{s,h} + r_{s,h}^{S} \le SD_{s} \times SCAP_{s}; \forall h \in H\]
(14) Storage operation limit - 3:
\[soc_{s,h} = soc_{s,h-1} + \epsilon_{ch} \times c_{s,t,h} - \frac{dc_{s,t,h}}{\epsilon_{dis}}; \forall h \in H\]
(15) RPS policy - State renewable credits export limitation:
\[pw_{g,w} \ge \sum_{w' \in WER_{w}} pwi_{g,w,w'}; \forall g \in (\bigcup_{i \in I_{w'}} G_{i}) \cap (G^{RPS}), w \in W\]
(16) RPS policy - State renewable credits import limitation:
\[pw_{g,w'} \ge pwi_{g,w,w'}; \forall g \in (\bigcup_{i \in I_{w'}} G_{i}) \cap (G^{RPS}), w \in W, w' \in WIR_{w}\]
(17) RPS policy - Renewable credits trading meets state RPS requirements:
\[\begin{aligned} \sum_{g \in (\bigcup_{i \in I_{w'}} G_{i}) \cap (G^{RPS}), w' \in WIR_{w}} pwi_{g,w,w'} - \sum_{g \in (\bigcup_{i \in I_{w'}} G_{i}) \cap (G^{RPS}), w' \in WER_{w}} pwi_{g,w',w} + \sum_{w \in W, h \in H} pt_{w,h}^{rps} \\ \ge \sum_{i \in I_{w},h \in H} \sum_{d \in D_{i}} p_{d,h} \times RPS_{w};\\ w \in W \end{aligned}\]
(18) Cap & Trade - State carbon allowance cap:
\[\sum_{g \in (\bigcup_{i \in I_{w}} G_{i}) \cap G^{F}} a+{g,t} - \sum_{t \in T} N_{t} em_{w,h}^{emis} \le ALW_{t,w}; w \in W\]
(19) Cap & Trade - Balance between allowances and emissions:
\[\sum_{h \in H} EF_{g} \times p_{g,h} = a_{g,t} + b_{g,t-1} = b_{g,t}; g \in (\bigcup_{i \in I_{w}} G_{i}) \cap G_{F}, w \in W, t \in T\]
(20) Cap & Trade - No cross-year banking:
\[b_{g,1} = b_{g,end} = 0; g \in G_{F}\]
(21) Nonnegative variable:
\[a_{g,t}, b_{g,t}, p_{g,h}, p_{d,h}^{LS}, c_{s,h}, soc_{s,h}, pt^{rps}, pw_{g,w}, pwi_{g,w,w'}, em^{emis} \\ \ge 0\]