Dynamics of Heating and Cooling Loads: Models

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1 Dynamics of Heating and Cooling Loads: Models An Overview of Modelling of Thermostatically Controlled Load Presentation at Smart Grid Journal Club Xinbo Geng Texas A&M University April Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

2 Overview 1 Motivation 2 Models Start Point Deterministic Differential Equation Stochastic Difference Equation Markov Chain Matrix Equation Hybrid Partial Differential Equation 3 Relationships Among Models & Summary Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

3 Motivation Motivation Thermostatically Controlled Load (TCL): air conditioners (ACs), heat pumps, water heaters, and refrigerators, etc. In this presentation, TCL ACs In 2013, 40% of total U.S. energy consumption was consumed in residential and commercial buildings. Residential Thermostatically Controlled Loads (TCLs) such as represent about 20% of the total electricity consumption in the United States. Features of TCL load: installed storage like Energy Storage (batteries) inertia change the temperature set point, the effects will NOT be shown immediately controllable direct/indirect control Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

4 Motivation Need For Models... Models from the Department of Civil Engineering: objective: analyze the indoor environment too many parameters, too complicated too accurate Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

5 Motivation Wanted: Models Need For Models concise (less parameters) output (power) state variables (indoor temperature) and control Need For Models of Aggregated TCLs 1 TCL: 5kW 1 Generator: 1MW only aggregated TCLs matter models for a large population of TCLs Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

6 Models 1 Motivation 2 Models Start Point Deterministic Differential Equation Stochastic Difference Equation Markov Chain Matrix Equation Hybrid Partial Differential Equation 3 Relationships Among Models & Summary Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

7 Control Strategies in Common Dead-band Control (example: heating load) Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

8 Start Point Start Point Newton s Heating/Cooling Law dt dt = k(t T a) (1) reference any high school textbooks t time T temperature of an object T a temperature of its surroundings k proportionality constant. k 1/t. 1/k is a time constant. Verified by numerous experiments. Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

9 Deterministic Differential Equation Model#1: Deterministic Differential Equation dt dt = 1 τ (T T a WT g ) (2) reference Ihara, S., & Schweppe, F. C. (1981). Physically Based Modeling of Cold Load Pickup. IEEE Transactions on Power Apparatus and Systems t time T spatial average temperature of the interior of the house T a T g τ effective thermal time constant ambient temperature temperature gain of heater/cooler, proportional with the AC Power W binary variable. W = 0 AC is OFF; W = 1 AC is ON. Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

10 Stochastic Difference Equation Model#2: Stochastic Difference Equation T [(n+1) t] = e t τ T [n t] (1 e t τ )(Ta W [n t]t g )+V [n t] (3) reference: Mortensen, R. E., & Haggerty, K. P. (1988). A stochastic computer model for heating and cooling loads. IEEE Transactions on Power Systems t time T spatial average temperature of the interior of the house V ( ) error, sequence of independent Gaussian random variables of mean zero and variance τ effective thermal time constant ambient temperature T a T g temperature gain of heater/cooler W binary variable (indicating ON/OFF). n number of the sampling instant Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

11 Markov Chain Matrix Equation Model#3: Markov Chain Matrix Equation Reference: Mortensen, R. E., & Haggerty, K. P. (1988). A stochastic computer model for heating and cooling loads. IEEE Transactions on Power Systems t ON the time of cooling the room from T s + to T s t OFF the time of heating the room from T s to T s + Assumption The t ON satisfies a negative binomial distribution: P(t ON = n) = Cm 1 n 1 (1 p) n m if n m (4) P(t ON = n) = 0 if n < m (5) Estimation Run simulation using Model#2 and get a distribution of t ON, and find the best parameter p ON, m ON to fit the distribution Estimation similarly, we can estimate p OFF, m OFF. Markov Chain then we can formulate a Markov Chain to generate t ON and t OFF Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

12 Markov Chain Matrix Equation Model#3: Markov Chain Matrix Equation Criticism: Estimation good estimation for a long period, but could be very in accurate in a short time. Power States of TCLs Failed to reveal the relationship between P(t) and other variables. Not Very Helpful we cannot formulate control strategies based on this. An Alternative Approach Quantizing the indoor temperature T into discrete level (of Model #2), and use this as the states of a Markov Chain. Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

13 Markov Chain Matrix Equation A very special case: p i,i+1 = 1. Reference: Lu, N., & Chassin, D. P. (2004). A state-queueing model of thermostatically controlled appliances. IEEE Transactions on Power Systems Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19 Model#3.1: A State Queue Model

14 Hybrid Partial Differential Equation Model of Aggregated TCLs 2 Possible Approaches: Track Individual TCL Method have numerous copies of Model#1 or Model#2, add the power together Pros Accurate, (maybe) deterministic Cons privacy issue, (not necessary) to be exactly accurate Model the Distribution Overall behavior ignore many details Estimation Good Estimation when Large Population (Large Number s Law). Control partial differential equation control Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

15 Hybrid Partial Differential Equation Model#4: Hybrid Partial Differential Equation (CPFE Model) Reference: Malhame, R., & Chong, C.-Y. (1985). Electric load model synthesis by diffusion approximation of a high-order hybrid-state stochastic system. IEEE Transactions on Automatic Control Estimation Model TCLs as particles. Estimate by a diffusion process. Fokker-Planck Equation brief summary: describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces. check (forward) Chapman-Kolmogorov Equation and Fokker-Planck Equation for more details. 2 Process at the same time 1 process of indoor temperature (continuous) 2 process of state of TCLs (discrete/binary) Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

16 Hybrid Partial Differential Equation Model#4: Hybrid Partial Differential Equation (Details) Coupled Fokker - Planck Equations Model (CFPE Model) f 1 t f 0 t = T [f 1 τ (T (t) T a + T g )] + σ2 2 = T [f 0 τ (T (t) T a)] + σ2 2 2 f 1 T 2 (6) 2 f 0 T 2 (7) f 1 (T, t)dt = P(T < x(t) T + dt AC is ON at time t) (8) f 0 (T, t)dt = P(T < x(t) T + dt AC is OFF at time t) (9) And 6 more equations about boundary conditions: Absorbing Boundaries, Conditions at Infinity, Continuity Conditions, Probability Conservation Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

17 Hybrid Partial Differential Equation Model#4: Criticism not solvable Analytical closed form solutions to the Fokker-Planck partial differential equation are difficult to obtain except in particular cases. discretize to solve these equations numerically, it is necessary first to discretize them both in time and space. the transition to an implementable computer program. drawbacks of discretize whatever virtue the sophisticated continuous-state continuous-time mathematical formalism may possess, some of it is lost in the process of discretizing Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

18 Relationships Among Models & Summary Relationships Among Models & Summary Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

19 Relationships Among Models & Summary Any Questions? Dynamics of Heating and Cooling Loads: Models An Overview of Modelling of Thermostatically Controlled Load Presentation at Smart Grid Journal Club Xinbo Geng Texas A&M University April Xinbo Geng (TAMU) Dynamics of Heating and Cooling Loads: Models April / 19

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