Lagrange Multipliers as a Metric for Early-Stage Shipboard Power System Design

Integrated power systems are essential in the operation of electric ships and must function under a diverse range of conditions. In addition to routine operation, ships should be prepared to respond to challenging events that require significant amounts of power. Demanding scenarios, coupled with complex and interconnected power systems, present challenging issues. The power and energy ratings of equipment are crucial factors in the operational success of the ship; however, to meet economic constraints, balance between performance and cost must be upheld. Details regarding equipment models and specifications are typically unavailable during early-stage design. Power system modeling and simulation can be used to test many designs at low resolution with minimal investment. Optimization of power flow offers insight into the potential performance and quality of a power system’s design. Multi-period optimization solves all time steps simultaneously, indicating the upper bounds on performance by allowing for the optimal control of energy storage and other state variables. In the following study, linear programming-based techniques are used to model a notional electric shipboard power system and optimize power flow following a multi-period approach. Lagrange multipliers are extracted from the optimal solution and applied as a metric to quantify the marginal change in performance per change in rating of the equipment. The Lagrange multipliers are used to create performance road maps to visualize trade-offs between ratings and performance, producing a tool to aid in early-stage ship design. The notion of performance road maps is extended from a single scenario to the aggregation of multiple scenarios using average Lagrange multipliers and Lagrange multipliers associated with the scenario with minimum performance.