Objective Function

This page provides concise explanations of how objective functions work specifically in the Real-Time Control (RTC) context.

In the RTC context, the objective function typically focuses on:

1. Operational cost minimization: Minimizing immediate electricity costs based on current prices and forecasted prices for a short horizon.

   \[f_{\text{cost}}(\mathbf{x}) = \sum_{t=1}^{T} c_{\text{grid}}(t) \cdot P_{\text{grid,import}}(t) - c_{\text{feed-in}}(t) \cdot P_{\text{grid,export}}(t)\]

   where \(T\) is typically a short horizon (e.g., 1-24 hours).

2. Comfort satisfaction: Maintaining user comfort within acceptable ranges for the immediate future.

   \[f_{\text{comfort}}(\mathbf{x}) = \sum_{t=1}^{T} w_{\text{comfort}} \cdot (T_{\text{indoor}}(t) - T_{\text{setpoint}}(t))^2\]

3. Renewable energy utilization: Maximizing self-consumption of available renewable generation.

   \[f_{\text{renewable}}(\mathbf{x}) = \sum_{t=1}^{T} w_{\text{renewable}} \cdot P_{\text{curtailed}}(t)\]

The RTC objective function is characterized by:

• Short time horizon: Focus on immediate or near-term optimization (typically hours rather than days)

• Computational efficiency: Formulated for fast solving to enable real-time decision-making

• Reactive approach: Emphasis on responding to current conditions rather than long-term planning

• Simplified formulation: Less complex than forecast-based controllers to enable faster solution times