This paper presents the design and realization of a linear Model Predictive Controller (MPC) and state estimator for a multi-zone heat pump in the Modelica modeling language, in order to validate closed-loop performance prior to experimental testing. The vapor compression system uses a variable speed compressor and a set of expansion valves for control, and it is required to regulate zone temperatures to set-points without offset. Constraints are imposed on all control inputs and also the values of both measured and unmeasured system outputs. Because experimental testing is both expensive and time-consuming, we have developed a tool chain for software-in-the-loop validation that uses a Modelica model for the plant, integrated with a software representation of the MPC that is realized in a combination of Modelica and C that is suitable for real-time use. We show the results of closedloop tests of the controller with a nonlinear system model, which provide a partial validation of the controller and tool chain.