An equation-oriented (EO) mathematical model for the optimization of utility systems has been developed. The model includes an accurate physical description of water/steam and air process streams. These together with EO models for the required unit operations were used to optimize two utility systems: a small model containing two steam turbines and a larger model of a combined heat and power plant. The applicability of the model was established using a novel SQP (Sequential Quadratic Programming) method which employs a 'tolerance tube' approach (Zoppke-Donaldson, A Tolerance-Tube Approach to Sequential Quadratic Programming with Applications, PhD thesis, 1995) to decide whether to accept a step from the QP subproblem. Comparison was then made with a further new penalty-free optimization method named filterSQP (Nonlinear Programming without a Penalty Function, 1997), which was used to solve the model containing two steam turbines. The tuning of both the model and the use of filterSQP were further explored with numerical experiments on the effect of scaling: comparison of dense and sparse versions of filterSQP; and different initial sizes for the trust region. Also, the correctness is demonstrated of the 'shadow price' given by the SQP code for the steam generation temperature. The results show that the sparse filterSQP solver works very efficiently and provides a basis for modelling and optimization of larger, industrial size problems which are reported in a companion paper.