Optimal flow for general multi-carrier energy systems,including load flow equations

Optimization is an important tool for the operation of an energy system. Multi-carrier energy systems (MESs) have recently become more important. Load ow (LF) equations are used within optimization to determine if physical network limits are violated. The way these LF equations are included in the optimal ow (OF) problem, influences the solvability of the OF problem and the convergence of the optimization algorithms. This paper considers two ways to include the LF equations within the OF problem for general MESs. In the first formulation, optimization is over the combined control and system-state variables, with the LF equations included explicitly as equality constraints. In the second formulation, optimization is over the control variables only. The system-state variables are solved from the LF equations in a separate subsystem, given the control variables. Hence, the LF equations are included only implicitly in the second formulation. The two formulations are compared theoretically. The effect of the two formulations on the solvability of the OF problem is illustrated by optimizing two MESs. Both formulation I and formulation II result in a solvable OF problem. For the two example MESs, the optimization algorithms require significantly fewer iterations with formulation II than with formulation I. For formulation II, the direct and the adjoint approach can be used to determine the required derivatives within the optimization algorithms. Scaling is needed to solve the OF problem for MESs. Both matrix scaling and per unit scaling can be used, but they are not equivalent.