The creation of low-order dynamic models for complex industrial systems is complicated by disturbances and limited sensor accuracy. This work presents a system identification procedure that uses machine learning methods and process knowledge to robustly identify a low-order closed-loop model of a municipal solid waste (MSW) grate incineration plant. These types of plants are known for their strong disturbances coming from fuel composition variations. Using Bayesian Optimization, the algorithm both ranks and selects inputs from the available sensor data and chooses the model structure from a broad grey-box model class. This results in accurate low-order models that respect the known physics of the process. Multiple flue gas composition measurements are used as inputs to provide information on the fuel composition. The method is applied and validated using data of an industrial MSW incineration plant and compared against four established methods, of which the resulting models either show unphysical dynamic behaviour or have lower performance than the proposed method. Also on a numerical benchmark, the proposed method outperforms the alternative methods. The obtained MSW incinerator models give excellent predictions and confidence intervals for the steam capacity and intermediate quantities such as supply air flow and flue gas temperature. The identified continuous-time models are fully given, and their step-response dynamics are discussed. The models can be used to develop model-based coordinated unit control schemes for grate incineration plants. The presented method shows great potential for low-order grey-box identification of systems with partial knowledge of the model structure.