Update app.py

app.py

@@ -253,14 +253,14 @@ In our renewable energy model, the **objective function** is to minimize the tot
 st.latex(r"""
 \text{Minimize } \quad \sum_{r, g} \text{Cost}_{g} \times \text{Capacity}_{r, g} + \text{Battery Cost} \times \text{Battery Capacity}
 """)
-st.markdown("""
+st.latex("""
 where:
-- \( r \) represents the region (in this case, a single region),
-- \( g \) represents the generation technology (solar, onshore wind, offshore wind, river),
-- \( \text{Cost}_{g} \) is the per-MW cost of technology \( g \),
-- \( \text{Capacity}_{r, g} \) is the installed capacity of technology \( g \) in region \( r \),
-- \( \text{Battery Cost} \) represents the cost per MWh of battery storage,
-- \( \text{Battery Capacity} \) is the total installed battery capacity.
+- $r$ represents the region (in this case, a single region),
+- $g$ represents the generation technology (solar, onshore wind, offshore wind, river),
+- $\text{Cost}_{g}$ is the per-MW cost of technology $g$,
+- $\text{Capacity}_{r, g}$ is the installed capacity of technology $g$ in region $r$,
+- $\text{Battery Cost}$ represents the cost per MWh of battery storage,
+- $\text{Battery Capacity}$ is the total installed battery capacity.
 """)
 
 st.markdown("""
@@ -272,23 +272,23 @@ MGA addresses this need by generating **alternative solutions** that are close t
 """)
 
 st.write("## How MGA Works: Adding a Cost Constraint")
-st.markdown("""
-To generate alternatives, MGA introduces a **cost tolerance** parameter \\( \\epsilon \\), which represents the acceptable increase in total cost relative to the optimal solution. The cost constraint for alternative solutions is expressed as:
+st.latex("""
+To generate alternatives, MGA introduces a **cost tolerance** parameter $\epsilon$, which represents the acceptable increase in total cost relative to the optimal solution. The cost constraint for alternative solutions is expressed as:
 """)
 st.latex(r"""
 \text{Total Cost} \leq (1 + \epsilon) \times \text{Optimal Cost}
 """)
-st.markdown("""
+st.latex("""
 where:
-- \\( \\epsilon \\) is the cost deviation percentage (e.g., if \\( \\epsilon = 0.05 \\), then the solution can be up to 5% more expensive than the optimal cost),
-- \\( \\text{Optimal Cost} \\) is the minimum cost obtained from the initial optimization.
+- $\epsilon$ is the cost deviation percentage (e.g., if $\epsilon = 0.05$ , then the solution can be up to 5% more expensive than the optimal cost),
+- $\text{Optimal Cost}$ is the minimum cost obtained from the initial optimization.
 This constraint allows for flexibility in cost, enabling the exploration of solutions that are **near-optimal** but differ in terms of installed capacities for each technology.
 """)
 
 st.markdown("""
 ### MGA Process in This Application
-1. **Initial Optimization**: First, we solve for the optimal solution to obtain the minimal total cost, referred to as \\( \\text{Optimal Cost} \\).
-2. **Setting the Cost Threshold**: We introduce a range of \\( \\epsilon \\) values (0%, 5%, 10%, etc.) to explore how alternative solutions differ as we allow for higher costs.
+1. **Initial Optimization**: First, we solve for the optimal solution to obtain the minimal total cost, referred to as $\text{Optimal Cost}$ .
+2. **Setting the Cost Threshold**: We introduce a range of $\epsilon$ values (0%, 5%, 10%, etc.) to explore how alternative solutions differ as we allow for higher costs.
 3. **Minimizing and Maximizing Capacities**: For each selected technology (e.g., solar, wind, hydro), we attempt to:
    - **Minimize the installed capacity** within the allowed cost threshold, identifying configurations with the lowest feasible capacity for that technology.
    - **Maximize the installed capacity** under the same conditions, exploring configurations with higher reliance on that technology.