To efficiently operate hybrid electric vehicles and, in general, electromechanical systems powered by two energy sources, it is necessary to determine the instantaneous power split between the sources in order to minimize the energy consumption of the whole system in the long time. This task is carried out by a high level controller, usually called "supervisory controller" whose strategy has to be defined.

Many real-time supervisory control algorithms for HEVs are based in the so called "Equivalent Consumption Minimization
Strategy"(ECMS). This strategy consists of multiplying the power supplied by the electrical storage system by a weighting
factor in order to turn it into an equivalent fuel power able to be added to that supplied by the internal combustion engine. The interest of these approach raises from the fact that, by means of this equivalence, the problem of minimizing consumption that is intrinsically a global problem, may be turned into an instantaneous minimization of this weighted sum of powers, so allowing its use in real time. The supervisory control problem then resumes to the problem of determining this "equivalent consumption" factor.

However, this weighting factor varies strongly according to the features of the velocity cycle required to the vehicle. A great effort in current research is devoted to the determination of this parameter for different driving scenarios. In previous work we have formulated the supervisory control problem in HEVs as an optimal control problem with state and control constraints. The solution was searched by means of solving the optimality conditions given by the Pontryagin Maximum Principle. In this approach, it is introduced a variable named the adjoint state which is found to be a nondimensional scaling of the equivalent consumption factor. Hence, by solving the optimality conditions and obtaining the evolution in time of the adjoint state, it is possible to obtain the evolution of the equivalent consumption factor.

The numerical solution of the optimal control problems with constraints is not straightforward because, generally, the optimality conditions are differential-algebraic equations. Sometimes, the algebraic equations may be solved independently and, after suitable replacements, the problem may be turned into an ordinary differential equations boundary value problem. In addition, in this particular problem, the control and state constraints may introduce discontinuities in the solution and in the RHS of the differential equations. Moreover these discontinuities occur at unknown times, since they depend on the solution itself.

PASVA4 is a software tool able to solve non linear boundary value problems with discontinuities in the RHS and in the solution, even in unknown locations, multipoint boundary values and unknown algebraic parameters. This tool has been successfully used for problems that include the above difficulties that appear in geophysics, electronics, mechanics, etc. In this work we present the use of this tool to solve the above problem.