Update objective_function.md

- Added missing "+" after 7th sum
- Fixed typo from "Omega^pv" to "Omega^wind" in 12th sum
- Fixed net installed energy storage capacity symbol in description of 2nd sum from "Delta^total" to "Delta^total,energy"
- Fixed net installed charging power capacity symbol in description of 3rd sum from "Delta^total" to "Delta^total,charge"
- Removed extra word "over" in description of 3rd sum
- Removed extra "to" in description of 10th sum

docs/src/Model_Concept_Overview/objective_function.md

@@ -10,14 +10,14 @@ The objective function of GenX minimizes total annual electricity system costs o
     &\sum_{y \in \mathcal{O}} \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST,energy}_{y,z} \times    \Omega^{energy}_{y,z}) + (\pi^{FOM,energy}_{y,z} \times  \Delta^{total,energy}_{y,z})\right) +  \\
     &\sum_{y \in \mathcal{O}^{asym}}  \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST,charge}_{y,z} \times    \Omega^{charge}_{y,z}) + (\pi^{FOM,charge}_{y,z} \times  \Delta^{total,charge}_{y,z})\right) +  \\
     & \sum_{y \in \mathcal{G}} \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}} \left( \omega_{t}\times(\pi^{VOM}_{y,z} + \pi^{FUEL}_{y,z})\times \Theta_{y,z,t}\right) + \sum_{y \in \mathcal{O \cup DF} } \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}} \left( \omega_{t}\times\pi^{VOM,charge}_{y,z} \times \Pi_{y,z,t}\right) + \\
-    &\sum_{s \in \mathcal{S}} \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}} \left(\omega_{t} \times n_{s}^{slope} \times \Lambda_{s,z,t}\right) + \sum_{t \in \mathcal{T}} \left(\omega_{t} \times \pi^{unmet}_{rsv} \times r^{unmet}_{t}\right)  \\
+    &\sum_{s \in \mathcal{S}} \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}} \left(\omega_{t} \times n_{s}^{slope} \times \Lambda_{s,z,t}\right) + \sum_{t \in \mathcal{T}} \left(\omega_{t} \times \pi^{unmet}_{rsv} \times r^{unmet}_{t}\right) + \\
     &\sum_{y \in \mathcal{H}} \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}}\left(\omega_{t} \times \pi^{START}_{y,z} \times \chi_{s,z,t}\right) +  \\
     & \sum_{l \in \mathcal{L}}\left(\pi^{TCAP}_{l} \times \bigtriangleup\varphi^{max}_{l}\right) + \\
     &\sum_{y \in \mathcal{VS}^{inv}} \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST, inv}_{y,z} \times \Omega^{inv}_{y,z})
     + (\pi^{FOM, inv}_{y,z} \times  \Delta^{total,inv}_{y,z})\right) + \\
     &\sum_{y \in \mathcal{VS}^{pv}} \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST, pv}_{y,z} \times \Omega^{pv}_{y,z})
     + (\pi^{FOM, pv}_{y,z} \times  \Delta^{total,pv}_{y,z})\right) + \\
-    &\sum_{y \in \mathcal{VS}^{wind}} \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST, wind}_{y,z} \times \Omega^{pv}_{y,z})
+    &\sum_{y \in \mathcal{VS}^{wind}} \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST, wind}_{y,z} \times \Omega^{wind}_{y,z})
     + (\pi^{FOM, wind}_{y,z} \times  \Delta^{total,wind}_{y,z})\right) + \\
     &\sum_{y \in \mathcal{VS}^{asym,dc,dis}} \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST,dc,dis}_{y,z} \times \Omega^{dc,dis}_{y,z})
     + (\pi^{FOM,dc,dis}_{y,z} \times  \Delta^{total,dc,dis}_{y,z})\right) + \\
@@ -40,10 +40,10 @@ The objective function of GenX minimizes total annual electricity system costs o
 The first summation represents the fixed costs of generation/discharge over all zones and technologies, which reflects the sum of the annualized capital cost, $\pi^{INVEST}_{y,z}$, times the total new capacity added (if any), plus the Fixed O&M cost, $\pi^{FOM}_{y,z}$, times the net installed generation capacity, $\overline{\Omega}^{size}_{y,z} \times \Delta^{total}_{y,z}$ (e.g., existing capacity less retirements plus additions).
 
 The second summation corresponds to the fixed cost of installed energy storage capacity and is summed over only the storage resources ($y \in \mathcal{O} \cup \mathcal{VS}^{stor}$).
-This term includes the sum of the annualized energy capital cost, $\pi^{INVEST,energy}_{y,z}$, times the total new energy capacity added (if any), plus the Fixed O&M cost, $\pi^{FOM, energy}_{y,z}$, times the net installed energy storage capacity, $\Delta^{total}_{y,z}$ (e.g., existing capacity less retirements plus additions).
+This term includes the sum of the annualized energy capital cost, $\pi^{INVEST,energy}_{y,z}$, times the total new energy capacity added (if any), plus the Fixed O&M cost, $\pi^{FOM, energy}_{y,z}$, times the net installed energy storage capacity, $\Delta^{total,energy}_{y,z}$ (e.g., existing capacity less retirements plus additions).
 
-The third summation corresponds to the fixed cost of installed charging power capacity and is summed over only over storage resources with independent/asymmetric charge and discharge power components ($y \in \mathcal{O}^{asym}$).
-This term includes the sum of the annualized charging power capital cost, $\pi^{INVEST,charge}_{y,z}$, times the total new charging power capacity added (if any), plus the Fixed O&M cost, $\pi^{FOM, energy}_{y,z}$, times the net installed charging power capacity, $\Delta^{total}_{y,z}$ (e.g., existing capacity less retirements plus additions).
+The third summation corresponds to the fixed cost of installed charging power capacity and is summed over only storage resources with independent/asymmetric charge and discharge power components ($y \in \mathcal{O}^{asym}$).
+This term includes the sum of the annualized charging power capital cost, $\pi^{INVEST,charge}_{y,z}$, times the total new charging power capacity added (if any), plus the Fixed O&M cost, $\pi^{FOM, energy}_{y,z}$, times the net installed charging power capacity, $\Delta^{total,charge}_{y,z}$ (e.g., existing capacity less retirements plus additions).
 
 The fourth and fifth summations correspond to the operational cost across all zones, technologies, and time steps.
 The fourth summation represents the sum of fuel cost, $\pi^{FUEL}_{y,z}$ (if any), plus variable O&M cost, $\pi^{VOM}_{y,z}$ times the energy generation/discharge by generation or storage resources (or demand satisfied via flexible demand resources, $y\in\mathcal{DF}$) in time step $t$, $\Theta_{y,z,t}$, and the weight of each time step $t$, $\omega_t$. The fifth summation represents the variable charging O&M cost, $\pi^{VOM,charge}_{y,z}$ times the energy withdrawn for charging by storage resources (or demand deferred by flexible demand resources) in time step $t$ , $\Pi_{y,z,t}$ and the annual weight of time step $t$,$\omega_t$.
@@ -59,7 +59,7 @@ The ninth term corresponds to the transmission reinforcement or construction cos
 Transmission reinforcement costs are equal to the sum across all lines of the product between the transmission reinforcement/construction cost, $\pi^{TCAP}_{l}$, times the additional transmission capacity variable, $\bigtriangleup\varphi^{max}_{l}$. Note that fixed O&M and replacement capital costs (depreciation) for existing transmission capacity is treated as a sunk cost and not included explicitly in the GenX objective function.
 
 The tenth term onwards specifically relates to the breakdown investment, fixed O&M, and variable O&M costs associated with each configurable component of a co-located VRE and storage resource.
-The tenth term represents to the fixed cost of installed inverter capacity and is summed over only the co-located resources with an inverter component ($y \in \mathcal{VS}^{inv}$).
+The tenth term represents the fixed cost of installed inverter capacity and is summed over only the co-located resources with an inverter component ($y \in \mathcal{VS}^{inv}$).
 This term includes the sum of the annualized inverter capital cost, $\pi^{INVEST,inv}_{y,z}$, times the total new inverter capacity added (if any), plus the Fixed O&M cost, $\pi^{FOM, inv}_{y,z}$, times the net installed inverter capacity, $\Delta^{total,inv}_{y,z}$ (e.g., existing capacity less retirements plus additions).
 The eleventh term represents the fixed cost of installed solar PV capacity and is summed over only the co-located resources with a solar PV component ($y \in \mathcal{VS}^{pv}$).
 This term includes the sum of the annualized solar PV capital cost, $\pi^{INVEST,pv}_{y,z}$, times the total new solar PV capacity added (if any), plus the Fixed O&M cost, $\pi^{FOM, pv}_{y,z}$, times the net installed solar PV capacity, $\Delta^{total,pv}_{y,z}$ (e.g., existing capacity less retirements plus additions).
@@ -94,4 +94,4 @@ Note however that each of these components are considered jointly and the optimi
 While the objective function is formulated as a cost minimization problem, it is also equivalent to a social welfare maximization problem, with the bulk of demand treated as inelastic and always served, and the utility of consumption for price-elastic consumers represented as a segment-wise approximation, as per the cost of unserved demand summation above.
 
 # Objective Scaling
-Sometimes the model will be built into an ill form if some objective terms are quite large or small. To alleviate this problem, we could add a scaling factor to scale the objective function during solving while leaving all other expressions untouched. The default ```ObjScale``` is set to 1 which has no effect on objective. If you want to scale the objective, you can set the ```ObjScale``` to an appropriate value in the ```genx_settings.yml```. The objective function will be multiplied by the ```ObjScale``` value during the solving process.
+Sometimes the model will be built into an ill form if some objective terms are quite large or small. To alleviate this problem, we could add a scaling factor to scale the objective function during solving while leaving all other expressions untouched. The default ```ObjScale``` is set to 1 which has no effect on objective. If you want to scale the objective, you can set the ```ObjScale``` to an appropriate value in the ```genx_settings.yml```. The objective function will be multiplied by the ```ObjScale``` value during the solving process.