Update app.py

app.py

@@ -253,7 +253,7 @@ 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.latex("""
+st.markdown("""
 where:
 - $r$ represents the region (in this case, a single region),
 - $g$ represents the generation technology (solar, onshore wind, offshore wind, river),
@@ -268,39 +268,43 @@ st.markdown("""
 Typically, optimization models produce a **single optimal solution** that minimizes the cost under a given set of constraints. However, in many real-world applications, there are **multiple near-optimal solutions** that achieve similar costs but vary in other characteristics. This diversity is valuable because:
 - **Flexibility**: Different solutions might be preferable depending on policy objectives, geographic constraints, or social preferences.
 - **Robustness**: Exploring near-optimal solutions reveals which elements (e.g., specific technologies or infrastructure investments) are consistently essential, regardless of slight variations in cost.
+
 MGA addresses this need by generating **alternative solutions** that are close to the optimal cost but differ in technological composition.
 """)
 
 st.write("## How MGA Works: Adding a Cost Constraint")
-st.latex("""
+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(r"""
 \text{Total Cost} \leq (1 + \epsilon) \times \text{Optimal Cost}
 """)
-st.latex("""
+st.markdown("""
 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),
+- $\epsilon$ 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 $\text{Optimal Cost}$ .
+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.
+
 These steps generate a set of **alternative solutions** that are close in cost but vary significantly in their reliance on each technology, revealing **flexibility** and **trade-offs** in the renewable energy system configuration.
 """)
 
 st.write("## Interpreting the Cost Threshold (\\( \\epsilon \\))")
 st.markdown("""
 The cost threshold parameter \\( \\epsilon \\) is crucial in MGA, as it determines the range within which we consider solutions to be "near-optimal." For example:
-- **\\( \\epsilon = 0 \\%**: Only the exact optimal solution is considered.
-- **\\( \\epsilon = 5 \\%**: Solutions within 5% of the optimal cost are considered acceptable, allowing for slightly more flexibility in technology choice.
-- **\\( \\epsilon = 10 \\%**: Solutions within 10% of the optimal cost are allowed, providing even greater flexibility.
+- **\\( \\epsilon = 0\\%**: Only the exact optimal solution is considered.
+- **\\( \\epsilon = 5\\%**: Solutions within 5% of the optimal cost are considered acceptable, allowing for slightly more flexibility in technology choice.
+- **\\( \\epsilon = 10\\%**: Solutions within 10% of the optimal cost are allowed, providing even greater flexibility.
+
 By exploring a range of \\( \\epsilon \\) values, we can see how the system configuration changes as we relax the cost constraint, offering a broader view of feasible solutions.
 """)
 
@@ -308,9 +312,11 @@ st.markdown("""
 ## Visualization of Results
 - **Cost Breakdown**: The total cost of each solution, broken down by technology, helps us see the contribution of each technology to the total cost.
 - **Capacity Ranges**: For each technology, we plot the minimum and maximum capacities across different \\( \\epsilon \\) values, showing the flexibility in system design as cost thresholds change.
+
 This visualization provides insights into:
 - Which technologies are essential (appear consistently in solutions across all \\( \\epsilon \\) values),
 - Which technologies offer flexibility (capacities vary widely as \\( \\epsilon \\) increases),
 - The cost impact of relying more or less on specific technologies.
+
 Through MGA, we can make more **informed decisions** about the renewable energy mix and identify robust, flexible strategies that align with broader goals beyond cost minimization.
 """)