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## MPC for large-scale HVAC systems

All the solutions discussed above assume a centralized control architecture. Their appli-cation in the case of large-scale systems may face various challenges due to the various factors like the size of the system, the internal couplings, the communication network diﬃculties, etc. Generally, this is addressed by means of the decomposition of large-scale systems into the subsystems and the coordination between controllers applied to each subsystem (see Figure 1.2). Several architectures and decomposition methods are proposed in various articles in the literature to address the above issues. Siljack [Sil91] has summarized various decomposition methods. He has introduced the overlapping and non-overlapping decomposition methods based on the sharing of the variables between the subsystems. Some widely used methods of partitioning large-scale systems into the sub-systems are based on bipartite graph theory, decomposition and the inclusion principle [Sca09a] [Lun92]. Other interesting approaches such as relative gain array matrix [Hag97] and Grammians [KFM03] use input-output mapping. Sometimes, due to the structural properties of large-scale systems, it is straightforward to derive subsystem models by a system identification procedure, instead of partitioning the centralized large-scale model.

The coordination between the controllers of subsystems is closely related to the degree of interaction between the subsystems [VR06]. If the interaction between subsystems is negligible, each subsystem is controlled independently without any coordination between subsystem controllers. This is named as decentralized model predictive control [Lun92]. If the interaction between the subsystems is strong, then the coordination between the subsystem controllers improves overall performance of the system. These coordination strategies may diﬀer depending on the several ways of information exchange between controllers mainly classified as noncooperative and cooperative type [RM12]. In noncoop-erative architecture, the subsystem controller optimizes locally the MPC problem using information of other subsystems and it reaches to Nash equilibrium. On the other hand, in the cooperative architecture, the subsystem controller optimizes the global objective and it achieves a Pareto optimal solution [VR06]. The coordination can be in a hierarchical architecture [Sca09b], where master level optimization problem comprises the shared vari-ables or constraints. The optimal solutions are sent to lower level i.e. the subsystem level controllers as coordination parameters. In some articles, DMPC problem is viewed as the partitioning of the Centralized Model Predictive (CMPC) Problem. This is motivated by some decomposition methods of large-scale convex optimization problems e.g. Dantzig Wolfe decomposition, Benders decomposition [MBDB10], primal and dual decomposition techniques [MN13] [PAL14].

When dealing with the DMPC problem, the decomposition of large-scale systems into subsystems and the coordination between the subsystem controllers are addressed independently. In this work, we propose a novel approach of addressing the system de-composition and controller coordination issues in two distinct stages. In first stage of this proposed method, the optimality conditions of a large-scale optimization problem are formulated and then decomposed to obtain the subsystems. Further, in second stage, the idea of coordination among the controllers is presented using an optimality condition decomposition approach [Con06]. We implement this distributed control scheme on a given multizone building without compromising the main objectives of energy eﬃcient operations.

**Distributed moving horizon estimation**

In the final part of this thesis, we concentrate on the estimation techniques in the context of large-scale buildings. This focus on the estimation is motivated due to some obvious advantages like the fault detection and isolation techniques, the possibilities of replacing the measurements with an estimation in case of faulty sensors and the minimization of the number of sensors to save the capital cost of the installation, etc.

The Kalman filter is still viewed as the best available strategy for state space estima-tion. Despite the current developments in the Kalman filter estimation methodologies, other state estimation techniques are also investigated by the researchers. We can find diﬀerent type of observer designs [JR99] and moving horizon estimations (MHE) [Jor04]. MHE is becoming popular as it is essentially formulated as an optimization problem that facilitates the inclusion of some physical constraints. This optimization problem over hori-zon N is solved in a receding horizon manner (similarly to MPC) allowing to estimate the states minimizing the errors introduced by the disturbances and noises. To minimize the uncertainty in the initial states, an extra term is introduced in the objective function. This procedure to solve this problem is repeated at each time instant using a sliding window of N values, hence it is termed as moving horizon estimation. The idea of MHE is extended for nonlinear systems in [ZLB08] [ABBZ11]. Also, the extension for MHE considering systems with bounded disturbances is presented in [JRH+16]. Nevertheless, the size of the optimization problem in MHE for large-scale systems may increase exponentially as the size of the system increases so the application of MHE for large-scale systems is an emerging topic. Diﬀerent schemes of partitioning centralized MHE problem for large-scale systems are proposed in [FFTS12] while [FFTS09] suggests the implementation of MHE for each sensor fulfilling convergence properties. [SM16] proposes a sensitivity based par-tition technique with a detailed discussion on convergence and stability. However, much work still has to be done in the regard of Distributed MHE (DMHE).

**VAV type HVAC system**

In large scale non-residential and commercial buildings, the HVAC system must meet the varying needs of diﬀerent spaces since diﬀerent zones of the building may have diﬀerent heating and cooling needs. In that respect, VAV systems were developed to be more energy-eﬃcient and to meet the varying heating and cooling needs of diﬀerent building zones. All the VAVs receive the supply airflow from a central air handling unit (AHU). Then, VAVs control the supply airflow into the zones by adjusting the damper position to maintain the thermal comfort. Examples of equipments are shown in Figure 2.2. We explain the working principle of a VAV type HVAC system with its mathematical model based on the general thermal behavior given in previous section.

### FCU type HVAC system

FCUs are widely used in various types of building topologies due to their simplicity. Most favored traits of the FCUs units are ease of installation and operation, lower noise levels and versatility in type of mounting (floor or ceiling) etc… Numerous forms of FCU units are available in the market such as the example shown in Figure 2.4. A typical FCU unit comprises components as a heating coil, supply fan, filters, mixer and noise attenuation and their forms can vary depending on their internal arrangement inside the FCU unit. We consider a blow type FCU unit where, as the name suggests, the supply fan is placed before the heating coil (See Figure 2.5). To derive a mathematical model, it is necessary to understand the working principle of the FCU system which is explained next.

Let us refer to the general building topology based on FCU system is shown in Figure 2.5. Each zone is equipped with a temperature sensor, a FCU to supply airflow and a return air plenum. The return air plenum recirculates the fraction of the return air to the FCU. The mixer combines the return air flow from the plenum with the outside air. The supply fan maintains constant supply air flow through the heating coil. The heating coil is a water-to-air heat exchanger which maintains the supply air at required temperature by manipulating hot water flow from a boiler or heat pump. This temperature control is achieved with embedded PID controllers. Thus, the supply air of constant flow is circulated into the zone at a suitable temperature. Zone temperature is controlled by modulating supply air temperature directly with the heating coil. Supply air flow is viewed as the control signal for achieving the zone temperature. The supply air temperature is controlled through the heating coil by controlling hot water flow, depending on the temperature of space to which FCU serves.

#### Maintenance-aware Economic Model Predictive Control

Various interpretations of the economic MPC are available as discussed in the literature survey. In this work, we formulate the desired objectives considering the benchmark HVAC systems provided in Chapter 1. The proposed MPC details are explained consid-ering the VAV type HVAC building system. Note that, referring to the mechanism of FCU type HVAC configuration given in Chapter 1, the cost function formulation can be eﬀortlessly extended for this type of HVAC system.

There has been a great evolution in the formulation of MPC in the literature. We consider the discrete-time state space model (2.9) which reads as: y(k) = Cx(k)x(k + 1) = Ax(k) + Bu(k) + Gd(k) (3.1).

where x(k) ∈ Rnx are the states representing the zone temperatures, u(k) ∈ Rnu are the inputs of the system denoting the supply airflow, nx and nu are number of states and number of inputs respectively. d(k) ∈ Rnd are the disturbances consisting the weather temperature and heat flux due to occupants and k is the discrete time. A, B and G are system dynamic matrices with appropriate dimensions.

**Table of contents :**

**Chapter 1 Introduction**

1.1 Literature review

1.2 Contributions

1.3 Outline

**Chapter 2 Building Frameworks and Mathematical Modeling**

2.1 Thermal Zone Model

2.2 VAV type HVAC system

2.3 FCU type HVAC system

2.4 Concluding Remarks

**Chapter 3 Centralized control**

3.1 Centralized Control – Fault Free Case

3.2 Centralized Control – Faulty case

**Chapter 4 Distributed Control and Estimation**

4.1 Optimality Conditions Decomposition based Distributed Model Predictive Control

4.2 Sensitivity Based Distributed Model Predictive Control

4.3 Distributed Estimation

**Chapter 5 Conclusions and Future Perspectives**

**Bibliography**