Power Mismatch

Power mismatch (also called power residual) is the difference between the specified (scheduled) power injection at a bus and the power injection calculated from the current voltage estimates during an iterative power flow solution. Driving all mismatches to zero (within tolerance) is the fundamental objective of the power flow algorithm.

Key Aspects of Power Mismatch:

• Active and Reactive Components: At each PQ (load) bus, both an active power mismatch ΔP and a reactive power mismatch ΔQ are computed. At PV (generator) buses, only the active power mismatch ΔP is relevant because the voltage magnitude is held fixed and reactive power is free to adjust. The slack bus has no mismatches because both P and Q are free variables.
• Mathematical Definition: The active power mismatch at bus i is ΔP_i = P_specified,i - P_calculated,i, where P_calculated depends on the voltages and admittances of bus i and all its neighbors through the power flow equations. The reactive mismatch ΔQ_i follows the same pattern.
• Convergence Indicator: The maximum mismatch across all buses is the primary convergence criterion. When max(|ΔP|, |ΔQ|) falls below the tolerance (typically 10⁻⁴ to 10⁻² p.u.), the solution is declared converged. Monitoring how mismatches decrease across iterations also reveals the convergence rate and health of the solution process.
• Diagnostic Value: Persistently large mismatches at specific buses can point to data errors (incorrect load values, missing branches), unrealistic operating conditions (overloaded system), or proximity to voltage collapse. The location and magnitude of the largest mismatches guide engineers in troubleshooting non-converging cases.
• Role in Newton-Raphson: In Newton-Raphson, the mismatch vector [ΔP; ΔQ] forms the right-hand side of the linear correction equation solved at each iteration. The Jacobian then maps these mismatches into the voltage corrections needed to reduce them.