Update app.py

app.py

@@ -281,7 +281,7 @@ st.latex(r"""
 """)
 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),
 - ${Optimal\, Cost}$ 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.
@@ -289,7 +289,7 @@ This constraint allows for flexibility in cost, enabling the exploration of solu
 
 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 ${Optimal\, Cost}$.
 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.
@@ -298,24 +298,24 @@ st.markdown("""
 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 ($psilon$ )")
+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% $**: Solutions within 10% of the optimal cost are allowed, providing even greater flexibility.
+The cost threshold parameter ($\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.
 
-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.
+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.
 """)
 
 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.
+- **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),
+- 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.