Optimal Refrigeration Control For Soda Vending Machines

Energy, Controls, & Applications Lab Energy Systems and Control: May 2015 DeWitt & RoeschkeOptimal Refrigeration Control for Soda VendingMachinesZoltan DeWitt and Matthew RoeschkeUniversity of California, BerkeleyAbstractThe prevalent soda vending machine industry in the US could yield reductions in energy consumption by addressing operational use.A study by the National Renewable Energy Laboratory estimates that each of the 4.6 million vending machines in the US consumesbetween 7 and 13kWh per day.[1] Currently, soda vending machines keep their products at a consistent temperature regardless of the timeof day. Although no formal soda vending machine usage patterns have been observed, we hypothesize that usage patterns primarily followtime of day with high utilization during midday and afternoon and low utilization during the night and morning. However, soda isgenerally non-perishable and does not need to be refrigerated during periods of low to no soda demand. In this report, we construct athermodynamic, state space refrigerator model and integrate a hypothetical soda demand schedule in order to optimize the operation of asoda vending machine that minimizes energy and carbon impact while maximizing the delivery of the appropriately chilled soda.I.Introductionefrigeration, and space conditioning in general, occupies a reasonably large portion of the total energyusage in the United States. The U.S. Department ofEnergy estimates that refrigeration accounts for approximately 7% of total commercial building energy usage. Thebygone era of cheap and plentiful electricity provided littleincentive to push for more efficient refrigerators in both thehome and commercial installations. Gradually, the energyconsumption per refrigerator unit increased, outpacing therate at which the physical size of each refrigerator unit wasgrowing (Figure 1). Regulations at both the state and federallevel were enacted which finally required steady reductionsin the energy usage of these appliances; refrigerator energy consumption began to decline dramatically afterwards.Clearly, without any incentive to increase efficiency, littletechnological improvements were made in the refrigeratorsector.The commercial, soda vending machine sector faces aneconomic obstacle that hinders the incentives for increasedenergy efficiency. Most vending machines are owned by avending or beverage company which contracts with building managers to have a machine placed on their premises.This arrangement sets an economic disconnect between theowner of the machine (the vending company) and the payerof the electrical bill (the building manager). The vendingcompany is not incentivized to improve the energy efficiency of their equipment since they do not pay for theenergy consumption. Also, no Energy Star rating is currently established for soda vending machines, althoughthere is some movement to establish one.[3]RRefrigerated devices have gained significant interest fordynamic demand management in the power utility sectoras these devices are viewed as a flexible, energy storageresource. Refrigerated systems can help stabilize powerdemand fluctuations in the grid by advancing or retarding their cooling cycles while still staying within a desiredtemperature band. Large thermal ballasts inside the refrigerated areas help to keep the temperature more stable duringperiods when it may be desirable to turn off the compressorfor grid-related reasons.[1]While these special "ancillary" services for reliabilitymanagement are of interest for all thermostatically controlled loads, soda vending machines are of unique interestbecause soda has a much wider, acceptable temperaturerange. While most commercial refrigeration units mustkeep perishables below 40 F, soda has no storage temperature restriction except to serve the product acceptably coldthe moment it is sold. Currently, vending machines operateto keep soda cold at all times in case someone wants to purchase one. The energy consumption of vending machinescan be significantly reduced by regulating the compressorbased on a thermal model of the vending machine and sodademand throughout the day.II.1.Technical DescriptionTestbed and Data AcquisitionThe testbed for this project consists of a mini-fridge, anArduino microcontroller, four temperature sensors, and onecurrent sensor connected to the fridge compressor (Figure2). The temperature sensors measure the main refrigerator1

Energy, Controls, & Applications Lab Energy Systems and Control: May 2015 DeWitt & RoeschkeFigure 1: Refrigerator Energy Use Over Time [3]compartment temperature, soda bottle temperature, waterbottle temperature, and ambient room temperature. Themicrocontroller also controls the actuation of the refrigerator compressor, receiving commands to maintain a setpointtemperature within the bounds of a specified deadbandwidth. In order to better simulate a well-mixed environment such as in a commercial vending machine, a fan wasadded inside the fridge cavity.The microcontroller performs two main functions: temperature data logging and deadband control. The temperature readings from each sensor are logged at one-minuteintervals and stored on a memory card for later analysis. Inorder to adapt this model to a commercial machine, a separate temperature data set would need to be acquired froma test unit and analyzed. However, this setup can serve asa proof of concept for these methods. Only in this test machine is there a need for more than one sensor; a commercialunit would only monitor the fridge temperature.Deadband control is also performed by the microcontroller, keeping the fridge temperature within a certainbounds. The target setpoint is programmable on an hourlybasis for a 24-hour period. This setpoint schedule wouldeventually be used in a commercial unit, possibly receiving2daily values from a remote server. During our test phase,this schedule was adjusted several times to collect a rangeof data for more accurate results.2.NomenclatureCs Thermal Capacitance of SodaC f Thermal Capacitance of Refrigerator AirRs Thermal Resistance of Soda ContainerR f Thermal Resistance of Refrigerator WallQc Compressor Heat PowerTs Temperature of SodaT f Temperature of Refrigerator AirTo Temperature of Ambient Airs Compressor State (1 On, 0 Off)e Rate Schedule for Electric Powerc Carbon Intensity of Electric Powerλ Cost Function Weighting FactorP Power Consumption of Compressor

Energy, Controls, & Applications Lab Energy Systems and Control: May 2015 DeWitt & RoeschkeFigure 2: Data Acquisition Equipment3.ModelingThe modeling objective is to understand how the soda temperature behaves given the temperature of the refrigeratorwhich is influenced by ambient temperature and refrigerator compressor. The temperature dynamics of the soda andrefrigerator is governed by the heat transfer between thesoda, refrigerator air, ambient air outside the refrigerator,and heat removed by the compressor. Mathematically, therefrigerator and soda temperature evolve according to thefollowing equations:dTs1 ( Ts (t) T f (t))dtRsdT f1Cf ( To (t) T f (t))dtRfCs (1)11x x2Rs Cs 1 Rs Cs11ẋ2 u1 x2Rf CfRf Cfẋ1 (3)(4)11Qcx1 x2 u2Rs C fRs C fCfWhere x1 and x2 are the soda and referigerator statesrespectively and u1 and u2 are the ambient temperature andcompressor state inputs respectively.With the following parameter assignments, equations3 and 4 can be arranged in the following matrix form inpreparation for identification:(2)1( Ts (t) T f (t)) Qc s(t)RsThe states, Ts and T f , are to be estimated given theuncontrollable input, To , and controllable input, s. The unknown parameters of this model are Cs , C f , Rs , R f , Qc andassumed to be independent.p0 1Rs Cs ẋ1ẋ2p1 1Rs C fp0p1p2 0p21Rf Cfp3 x1 x20 u1 x2 p3u2 QcCf (5)or4.Parameter Estimation and ResultsThe target states evolve according to the following equations:z(t) θ T φ(6)From equation 6, the normalized recursive gradient update law is applied to identify parameters.3

Energy, Controls, & Applications Lab Energy Systems and Control: May 2015 DeWitt & Roeschkeθ̂ (t) Γ φ(t)e T (t)(7)θ̂ (0) θ̂0e(t) z(t) θ̂ T φ(t)m2 ( t )m2 ( t ) 1 φ T ( t ) φ ( t )(9)State Estimation and ResultsAlthough our test bed is capable of measuring soda temperature, vending machines do not typically measure thisstate. In order to emulate this limitation, the soda temperature is estimated using our thermodynamic state spacesystem with the identified parameters, process noise w(t),and sensor noise n(t): ẋ1ẋ2 p 0p1 p0 p1 p2 x2 0 x1x2 00u1 wp2 p3u2ym (t) Cx (t) n(t)(13)The noise terms are assumed to be Gaussian around azero mean with covariances W and N for processor andsensor noise respectively. N is additionally assumed to bepositive definite.The states of our linear, thermodynamic system are estimated using the Kalman filter algorithm:x̂ A x̂ (t) Bu(t) L(t)(ym C x̂ )(14)x̂ (0) x̂0L ( t ) Σ ( t ) C ( t ) N 1 , t 0(15)T(16)Σ̇(t) Σ(t) A AΣ(t) WT Σ(t)C NΣ (0) Σ0 1CΣ(t)(17)Equation 15 is the observer gain of the system, and equation 16 is the Riccati differential equation that solves forΣ ( t ).The Kalman Filter algorithm was implemented inPython, and the soda temperature state was estimated overrefrigerator temperature, ambient temperature, and currentmeasurements taken over a span of 4 days. These measurements are different than the data used for parameteridentification. Soda temperature was also measured butwas not used as feedback in the Kalman Filter algorithm.The soda temperature measurements are used to evaluatethe estimation error as seen in Figure 5.6.Model DiscretizationIn preparation for the optimization program, equation 12,which is continuous in the time domain, is discretized usingthe exponentiation formulation. [2] 1 or4(12)(8)Where the update gain, Γ, is a non-negative matrix of thesame size as θ, e(t) is the normalized prediction error, andm2 (t) is the normalization signal. The Hadamard productis denoted by , which is an element-wise multiplication oftwo matrices of the same size. The update gain matrix isadjusted to have appropriate gain for each correspondingparameter estimate in the θ̂ matrix.Soda temperature, refrigerator temperature, ambienttemperature, and current (which was used to determinecompressor state) was measured from the test bed at oneminute intervals for one week. Two different controlschemes were tested during the week as seen in the oneday examples in Figure 3. The first control scheme was astandard, refrigerator temperature control scheme based ona fixed set point temperature and dead-band. The secondcontrol scheme involved deactivating the compressor for anextended period of time then implementing rapid coolingto simulate a potential, overnight vending machine controlstrategy.The recursive gradient update law was implementedin Python, and the parameter values converged quickly tosteady state values as seen in Table 15.ẋ (t) Ax (t) Bu(t) w(t)x1x2"(10) n(11)eA0B0#! t Ad0BdI Where t is 1 minute, the desired timestep of thediscrete-time equations, and Ad and Bd are the discretizedmatrices of A and B respectively. Using the parametersidentified in Table 1, equation 12 is discretized as follows:

Energy, Controls, & Applications Lab Energy Systems and Control: May 2015 DeWitt & RoeschkeFigure 3: Compressor and Ambient Temperature Inputs (Left: Custom Control, Right: Normal Control)Table 1: Parameter Estimatesp0p1p2p3 9.8 10 35.4 10 22.1 10 2 1.6 10 1Figure 4: Parameter Estimation with Gradient Decent5