Extremum seeking controllers have been investigated for multivariable data-driven energy optimization in heat pumps. In particular, proportional–integral extremum seeking control (PI-ESC) has demonstrated potential for significant acceleration compared to other ESC variants for nonlinear closed-loop control systems. A barrier to PI-ESC’s utilization in self-optimizing control is the fact that the PI-ESC algorithm is fragile. That is, unless the PI-ESC gains (e.g., controller gains, estimator gains) are carefully tuned, small perturbations to these gains can render the closed-loop unstable. Since arbitrary combinations of PI-ESC gains can result in instabilities, we propose a failure-robust Bayesian optimization (FRBO) algorithm that computes PI-ESC gains that ensure the closed-loop system can be driven rapidly to the optimum, while identifying and avoiding regions in the space of PI-ESC gains that are likely to result in instabilities (i.e., failures). The FRBO-tuned PI-ESC is shown to result in rapid closed-loop convergence to optimal values both on benchmark examples and a production-level model of an air conditioning system.

This paper proposes an extremum-seeking controller (ESC) design for a class of discrete-time nonlinear control systems subject to input constraints or quantized inputs. The proposed method implements a proportional-integral ESC design along with a discrete-time anti-windup mechanism. The anti-windup enforces input saturation while preserving the input dither signal. The technique incorporates a mechanism for adjusting the amplitude of the extremum seeking control dither signal. This mechanism ensures that any violation of constraints due to the dither signal is removed while maintaining the probing signal active. An amplitude update routine is also proposed. The amplitude update is coupled with a saturation bias estimation algorithm that correctly accounts for the inherent bias associated with systems operated at or near saturation conditions. The amplitude update is designed to remove the dither signal when the system approaches the optimum. It also ensures that a lower bound of the amplitude is enforced to guarantee that excitation conditions are maintained.

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 closed-loop tests of the controller with a nonlinear system model, which provide a partial validation of the controller and tool chain.

Multi-split variable refrigerant flow (VRF) air-conditioning systems, composed of a single outdoor heat exchanger and multiple indoor heat exchangers, are becoming increasingly popular because of the ease with which they can be installed, their high energy savings potential, and the opportunity to realize individualized comfort control. Because these systems exhibit strong coupling effects between different indoor units, coordinated control of compressor speed, fan speeds, and valve openings is required. A literature review indicates there is a dearth of comprehensive studies that explore the complex multivariable dynamic behavior of air-source VRF systems with commercially prevalent architectures. This paper describes a first-principles model of such an air-conditioner with four indoor units that is validated against experimental data. This validated simulation model is then used to study the dynamic system response subject to various sets of conditions, including variations in the compressor speed, fan speeds, and valve openings. The insights gained from such a system analysis will be beneficial to future energy performance analysis and model-based controls design. IEA Heat Pump Conference This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Copyright c © Mitsubishi Electric Research Laboratories, Inc., 2017 201 Broadway, Cambridge, Massachusetts 02139 Dynamic Characteristics of an R410a Multi-split Variable Refrigerant Flow Air-conditioning System Hongtao Qiao1, Christopher R. Laughman, Daniel J. Burns, Scott A. Bortoff Mitsubishi Electric Research Laboratories Cambridge, MA 02139, USA

We derive a power-optimizing output feedback controller for a multi-zone heat pump that (1) regulates individual zone temperatures, rejecting unknown heat load disturbances, (2) regulates condenser subcooling and (3) the compressor discharge temperature, and (4) minimizes electrical power consumption at steady-state operating conditions. The design is a cascade of a linear inner-loop and a nonlinear outer-loop. The inner-loop is designed for robust disturbance rejection using H ∞ loop-shaping methods. The outer-loop uses a model of compressor and fan power consumption and a gradient descent feedback to drive the system to its power-minimizing equilibrium for constant values of references and disturbances. The controller uses only temperature measurements for feedback; refrigerant pressure sensors, which are not present in many products for cost reasons, are not required. A proof of exponential stability is provided and preliminary experimental tests demonstrate satisfactory transient responses for a commercial multi-zone heat pump.

While the previous chapter presented modeling and control strategies for vapor compression systems in general, in this chapter, a model predictive controller is designed for a multi-zone vapor compression system. Controller requirements representing desired performance of production-scale equipment are provided and include baseline requirements common in control literature (constraint enforcement, reference tracking, disturbance rejection) and also extended requirements necessary for commercial application (selectively deactivating zones, implementable on embedded processors with limitedComputation memory/computation, compatibility with demand response events.). A controller architecture is presented based on model predictive control to meet the requirements. Experiments are presented validating constraint enforcement and automatic deactivation of zones.

This work considers real-time continuous curvature (CC) path planning for car-like robots. It is motivated by the fact that Reeds-Shepp's (RS) based path planning remains unmatched in terms of computation efficiency and reliability when compared with various CC path planning results. Similar to [1], this paper post-processes RS paths to enforce the CC property, while ensuring CC paths contained in a neighborhood of the RS paths to maintain obstacle clearance. Targeting to alleviate concerns about reliability and computational efficiency, we exploit the geometric insights casted by μ-tangency conditions [2] to post-process RS paths. Specifically, distinctive postprocessing scheme is devised offline for each type of discontinuous curvature junctions. The proposed schemes, though suboptimal, are straightforward, and result in CC path planning with guaranteed completeness at the negligible increase of computation. Effectiveness of proposed schemes and resultant algorithms is validated by numerical simulations.

An observed behavior of refrigerant mass distribution in multi-path heat exchangers is exploited for control purposes. In this paper, we describe the following empirical property exploited for control: as the inlet valve position is decreased, refrigerant mass flow rate entering the heat exchanger is reduced, and for some flow rates, refrigerant is shown to preferentially flow in some paths more than others, causing maldistribution. This uneven refrigerant distribution is repeatable, reduces the capacity in a continuous manner and can be exploited with feedback controllers to regulate the per-zone cooling. A controller is designed to provide stability and robustness to per-zone conditions and setpoints for this controller that relate per-path superheat temperature to overall evaporator capacity is created in such a way as to be robust to changes in local zone temperatures and the overall system evaporating temperature. This strategy provides zone decoupling and ultimately creates a virtual control input for a model predictive controller. Experiments demonstrate the effectiveness of this approach on a two-zone air conditioner in laboratory tests.

Multi-split variable refrigerant flow (VRF) air-conditioning systems, composed of a single outdoor heat exchanger and multiple indoor heat exchangers, are becoming increasingly popular because of the ease with which they can be installed, their high energy savings potential, and the opportunity to realize individualized comfort control. Because these systems exhibit strong coupling effects between different indoor units, coordinated control of compressor speed, fan speeds, and valve openings is required. A literature review indicates there is a dearth of comprehensive studies that explore the complex multivariable dynamic behavior of air-source VRF systems with commercially prevalent architectures. This paper describes a first-principles model of such an air-conditioner with four indoor units that is validated against experimental data. This validated simulation model is then used to study the dynamic system response subject to various sets of conditions, including variations in the compressor speed, fan speeds, and valve openings. The insights gained from such a system analysis will be beneficial to future energy performance analysis and model-based controls design. © 2017 Stichting HPC 2017. Selection and/or peer-review under responsibility of the organizers of the 12th IEA Heat Pump Conference 2017.

This paper proposes a proportional-integral extremum-seeking control technique for a class of discrete-time nonlinear dynamical systems with unknown dynamics. The technique is a generalization of existing time-varying extremum-seeking control techniques that provides fast transient performance of the closed-loop system to the optimum equilibrium of a measured objective function. The main contribution of the proposed technique is the addition of a proportional action that can be used to minimize the impact of a time-scale separation on the transient performance of the extremum-seeking control system. The integral action fulfills the role of standard ESC techniques to identify optimal equilibrium conditions. The effectiveness of the proposed approach is demonstrated using a simulation example.

One promising path for reducing the power consumption and improving the performance of building HVAC systems involves the coordinated design and operation of individual subsystems. This objective requires careful consideration of the dynamic interactions between the subsystems serving the occupied spaces. We propose a physical model-based approach for evaluating, optimizing, and designing control algorithms for integrated collections of subsystems to achieve high overall system performance. This approach is used in this paper to optimize the setpoints of and design dynamic controllers for an overall HVAC system consisting of a dedicated outdoor air system (DOAS) and a variable refrigerant flow (VRF) system serving a large occupied space. By optimizing the system inputs, we demonstrate that the energy consumption can be reduced by 15% as compared to the system with non-optimized inputs, and that nonintuitive system dynamics can be managed through systematic control analysis and design.

Multi-evaporator vapor compression systems (ME-VCS) provide cooling to multiple zones. The thermodynamic conditions in these zones are independent: the heat loads often differ, and the occupants of these spaces often have different desired room temperatures. Therefore, in order to regulate each zone to its desired setpoint temperature, the amount of thermal energy removed by each evaporator must be controlled independently. However, a common evaporating pressure introduces coupling between all the evaporators that makes this objective difficult—the valve and piping arrangement imposes the constraint that all evaporators operate at the same temperature. In order to control the per-zone cooling, a common control strategy employed in the literature is to duty cycle the evaporator (alternate between a fully-open and fully-closed valve). However, duty cycling causes periodic disturbances to not only the local zone, but also to many critical machine temperatures and pressures, and these disturbances are often not transient but instead persist indefinitely. Fluctuations induced by the periodic disturbances can degrade the ability of the machine to regulate zone temperatures with zero steady state error, cause excessively high or low temperatures during peaks of the period, and couple into most machine signals of interest in ways that are difficult to describe with low order dynamical models. An observed behavior of refrigerant mass distribution in multi-path heat exchangers is exploited for control purposes. Multi-path heat exchangers are characterized by an inlet header pipe that splits refrigerant flow to two or more parallel paths through the heat exchanger and collects those paths into a common outlet header pipe. In the paper, we describe the following empirical phenomenon exploited for control: as the inlet valve is decreased, refrigerant mass flow rate entering the heat exchanger is reduced, and at some critical flow rate, refrigerant is shown to preferentially flow in some paths more than others, causing maldistribution. This uneven refrigerant distribution is repeatable, reduces the capacity in a continuous manner and can be exploited with feedback controllers to regulate the per-zone cooling. A feedback controller is designed to provide stability and robustness to per-zone conditions and setpoints for this controller that relate per-path superheat temperature to overall evaporator capacity is created in such a way as to be robust to changes in local zone temperatures and the overall system evaporating temperature. This strategy provides zone decoupling and ultimately creates a virtual control input for a supervisory controller such as a model predictive controller. Experiments demonstrate the effectiveness of this approach on a two-zone air conditioner in laboratory tests.

In this paper, we optimize vapor compression system power consumption through the application of a newly-developed proportional-integral extremum seeking controller (PI-ESC) that converges at the same timescale as the process. This method modifies the control law to include terms proportional to the estimated gradient, but this modification of the control law requires a more sophisticated gradient estimator in order to avoid bias. We develop a PI-ESC for which this bias is eliminated. PI-ESC is applied to the problem of compressor discharge temperature setpoint selection for a vapor compression system where setpoints are automatically determined so that power consumption is minimized. The vapor compression system operates with a regulating feedback controller configured to drive the compressor discharge temperature to setpoints selected by the PI-ESC, and we use a physics-based simulation model to demonstrate that power consumption is minimized dramatically faster than by traditional perturbation-based methods.

This paper considers the control of a multievaporator vapor compression system (ME-VCS) where individual evaporators are permitted to turn ON or OFF. We present a model predictive controller (MPC) that can be easily reconfigured for different ON/OFF configurations of the system. In this approach, only the cost function of the constrained finite-time optimal control problem is updated depending on the system configuration. Exploiting the structure of the system dynamics, the cost function is modified by zeroing elements of the state, input, and terminal cost matrices. The advantage of this approach is that cost matrices for each configuration of the ME-VCS do not need to be stored or computed online. This reduces the effort required to tune and calibrate the controller and the amount of memory required to store the controller parameters in a microprocessor. The reconfigurable MPC is compared with a conventional approach in which individual model predictive controllers are independently designed for each ON/OFF configuration. The simulations show that the reconfigurable MPC method provides a similar closedloop performance in terms of reference tracking and constraint satisfaction to the set of individual model predictive controllers. Further, we show that our controller requires substantially less memory than the alternative approaches. Experiments on a residential two-zone vapor compression system further validate the reconfigurable MPC method.

While traditional perturbation-based extremum seeking controllers (ESC) for vapor compression systems have proven effective at optimizing power without requiring a process model, the algorithm’s requirement for multiple distinct timescales has limited the applicability of this method to laboratory tests where boundary conditions can be carefully controlled, or simulation studies with unrealistic convergence times. In this paper, we optimize power consumption through the application of a newly-developed proportional–integral extremum seeking controller (PI-ESC) that converges at the same timescale as the process. This method uses an improved gradient estimation routine previously developed by the authors but also modifies the control law part of the algorithm to include terms proportional to the estimated gradient. PI-ESC is applied to the problem of compressor discharge temperature selection for a vapor compression system so that power consumption is minimized. We test the performance of this method using a custom-developed model of a vapor compression system written in the Modelica object-oriented modeling language. We compare the convergence times of PI-ESC to our previously developed time-varying ESC method and the conventional perturbation-based ESC method. For the conditions tested, PI-ESC is shown to converge to the optimum in about 15 minutes, whereas TV-ESC converges in 45 minutes and perturbation ESC requires more than 7,000 minutes due to its ine_cient estimation of the gradient. Because of the improved convergence properties of PI-ESC, self-optimization algorithms for HVAC equipment can be deployed into situations where previous methods have failed.

In many formulations of controller architectures for variable-speed vapor compression machines, evaporator superheat temperature is commonly selected as a regulated variable due to its correlation with cycle efficiency. Further, the superheat temperature setpoint is conveniently taken as a constant value over the wide range of operating conditions. However, direct measurement of superheat is not always available, and estimates of superheat have limited robustness. Therefore identifying alternate signals in the control of vapor compression machines that correlate to efficiency is desired. In this paper, we consider a model-free extremum seeking algorithm that adjusts compressor discharge temperature setpoints in order to optimize energy efficiency. While perturbation-based extremum seeking methods have been known for some time, they suffer from slow convergence rates-a problem emphasized in application by the long time constants associated with thermal systems. Our method uses a new algorithm (time-varying extremum seeking), which has dramatically faster and more reliable convergence properties. In particular, we regulate the compressor discharge temperature using setpoints selected from a model-free time-varying extremum seeking algorithm. We show that the relationship between compressor discharge temperature and power consumption is convex (a requirement for this class of realtime optimization), and use time-varying extremum seeking to drive these setpoints to values that minimize power. The results are compared to the traditional perturbation-based extremum seeking approach. Experiments are performed demonstrating discharge temperature optimization from 72°C to 62°C for a particular set of experimental conditions where the power consumption is decreased from 525 W to 450 W, resulting in an increase in observed coefficient of performance (COP) of 14%.

In recent years, a number of extremum seeking algorithms have been proposed. While each approach aims to estimate the gradient of a performance metric in realtime and steer inputs to values that optimize the metric, the way in which each method accomplishes this goal can have practical implications that depend on the application. In this paper, we compare the performance of traditional perturbation-based extremum seeking to time-varying extremum seeking in the context of optimizing the energy efficiency of a vapor compression system. In order to benchmark these algorithms, we simulate their performance using a moving-boundary model of a vapor compression machine that has been tuned and calibrated to data gathered from a multi-split style room air conditioner operating in cooling mode. We show that while perturbation-based extremum seeking appears simplest to tune, some challenging minima are not obtained. Also, we find that time-varying extremum seeking converges faster and more reliably than the other method tested.