Abstract
Due to time-varying utility prices, peak demand charges, and variable-efficiency equipment, optimal operation of heating ventilation, and air conditioning systems in campuses or large buildings is nontrivial. Given forecasts of ambient conditions and utility prices, system energy requirements can be reduced by optimizing heating/cooling load within buildings and then choosing the best combination of large chillers, boilers, etc., to meet that load while accounting for switching constraints and equipment performance. With the presence of energy storage, utility costs can be further reduced by temporally shifting production, which adds an additional layer of complexity. Furthermore, due to changes in market and weather conditions, it is necessary to revise a given schedule regularly as updated information is received, which means the problem must be tractable in real time (e.g., solvable within 15 min). In this paper, we present a mixed-integer linear programming model for this problem along with reformulations, decomposition approaches, and approximation strategies to improve tractability. Simulations are presented to illustrate the effectiveness of these methods. By removing symmetry from identical equipment, decomposing the problem into subproblems, and approximating longer-timescale behavior, large instances can be solved in real time to within 1% of the true optimal solution.
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Acknowledgements
Funding, equipment models, and sample data provided by Johnson Controls, Inc. Additional funding provided by the National Science Foundation (Grant #CTS-1603768).
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Appendices
Appendix A: Notation
1.1 A.1 Sets
- \(i \in \mathbf{I}\) :
-
Temperature zones
- \(j \in \mathbf{J}\) :
-
Type of generators
- \(k \in \mathbf{K}\) :
-
Resources/utilities
- \(t \in \mathbf{T}\) :
-
Time periods
- \(m \in \mathbf{M}\) :
-
Interpolation regions for equipment models
- \(n \in \mathbf{N}\) :
-
Interpolation points for equipment models
- \(\mathbf{M}_{j} \subseteq \mathbf{M}\) :
-
Interpolation regions for equipment j
- \(\mathbf{N}_{jm} \subseteq \mathbf{N}\) :
-
Interpolation points contained in region m for equipment j
1.2 A.2 Parameters
The following parameters are given based on available forecasts or desired system performance.
- \(\phi _{kt}\):
Forecast secondary resource demand
- \(\rho _{kt}\):
Forecast resource prices
- \(\rho ^{\text {max}}_{k}\):
Demand charge
- \(\psi _{kt}\):
Exogenous demand included in peak calculation
- \(\beta _{kt}\):
Penalty for unmet secondary demand
- \(\varTheta ^-_{it}, \varTheta ^+_{it}\):
Lower and upper temperature comfort bounds for zones
- \(\varPi ^{\text {max}}_{k}\):
Previous maximum resource demand
- \(\mu _j\):
Number of units of each type available for use
- \(\zeta _{jknt}\):
Interpolation points for piecewise-linear models
- \(\sigma _k\):
Fractional retention of stored resource
- \(\delta ^+_j, \delta ^-_j\):
Minimum on/off dwell time for generators
- \(\varPi _k\):
Upper bound for resource purchase
- \(\varUpsilon _k\):
Bound for single-period storage charge/discharge
- \(\varSigma _k\):
Storage capacity
- \(\alpha _{ii'}, \omega _{ii'k}\):
Dynamic coefficients in airside model
- \(\theta _{it}\):
Time-varying temperature disturbance for zones
- \(\chi ^+_{it}, \chi ^-_{it}\):
Penalties for comfort violation in zones
- \(\varGamma _{ik}\):
Upper bound on resource flow to zones
- \(\xi _{jnt}\):
Part-load coefficients for piecewise-linear models
- \(\lambda _{ikt}\):
Dual multiplier for resource supply to airside
1.3 A.3 Decision variables
The following decision variables are primarily for the waterside subsystem.
- \(U_{jt} \in \{0,\cdots ,\mu _j\}\):
Integer number of generators on during a given time
- \(U^+_{jt}, U^-_{jt}\):
Number of generators switched on/off at the beginning of a time period
- \(Q_{jkt}\):
Production of resources in equipment
- \(P_{kt} \in {[}0,\varPi _k{]}\):
Amount of purchased resources
- \(P^{\text {max}}_{k} \in [\varPi ^{\text {max}}_{k},\infty )\):
Maximum single-period purchase of resource k
- \(Y_{kt} \in {[}-\varUpsilon _k,\varUpsilon _k{]}\):
Charge (\(<0\)) or discharge (\(>0\)) of stored resources
- \(S_{kt} \in {[}0,\varSigma _k{]}\):
Stored inventory at the end of a time period
- \(B_{kt} \in {[}0,\phi _{kt}{]}\):
Unmet secondary demand
- \(V_{jmt} \in \{0,\cdots ,\mu _j\}\):
Number of units operating in each interpolation region
- \(Z_{jmnt} \in {[}0,\mu _j{]}\):
Weighting of interpolation points in each region
- \(X_{lt}\):
States of the airside dynamic model
- \(T_{it}\):
Zone temperature of zone at end of a time period
- \(G_{ikt} \in {[}0,\varGamma _{ik}{]}\):
Primary demand of resources by zones
- \(T^+_{it}, T^-_{it}\):
Positive and negative comfort violation for zones
Appendix B: Formulation comparison
The following table shows performance results for different formulations and decomposition strategies applied to various instance. The first four columns give the number of temperature zones I, generators J, resources K, and time points T for the instance. The remaining four columns give the performance the following four formulations:
The original Symmetric formulation
The Symmetry-Free reformulation from Sect. 3.1
The Airside/Waterside decomposition based on Lagrangian relaxation
The Hierarchical decomposition using aggregate system curves.
Each entry shows the objective function and estimated optimality gap, along with the solution time required to obtain those values. Note that in each row, the objective function is normalized to the best value at 0% with other objectives given as a percentage increase over that value. Gaps are also given as a percentage of the best solution. For problems without an airside model, the two decomposition strategies are unnecessary (and thus listed as N/A for those rows).
I | J | K | T | Symmetric | Symmetry-free | Airside/waterside | Hierarchical |
|---|---|---|---|---|---|---|---|
0 | 10 | 11 | 48 | 0.04% (0.71% Gap) | 0.00% (0.00% Gap) | N/A | N/A |
10.01 min | 0.40 min | ||||||
0 | 10 | 11 | 120 | 0.21% (1.02% Gap) | 0.00% (0.01% Gap) | N/A | N/A |
10.01 min | 10.00 min | ||||||
0 | 10 | 11 | 168 | 0.39% (1.13% Gap) | 0.00% (0.03% Gap) | N/A | N/A |
10.01 min | 10.00 min | ||||||
0 | 20 | 11 | 48 | 0.03% (1.00% Gap) | 0.00% (0.00% Gap) | N/A | N/A |
33.59 min | 3.40 min | ||||||
0 | 20 | 11 | 120 | 0.53% (1.73% Gap) | 0.00% (0.18% Gap) | N/A | N/A |
10.01 min | 10.00 min | ||||||
0 | 20 | 11 | 168 | 1.19% (2.48% Gap) | 0.00% (0.23% Gap) | N/A | N/A |
10.02 min | 10.00 min | ||||||
0 | 30 | 11 | 48 | 0.41% (1.26% Gap) | 0.00% (0.00% Gap) | N/A | N/A |
10.01 min | 0.82 min | ||||||
0 | 30 | 11 | 120 | No Solution 0.00% | (0.06% Gap) | N/A | N/A |
10.01 min | 10.00 min | ||||||
0 | 30 | 11 | 168 | No Solution 0.00% | (0.06% Gap) | N/A | N/A |
10.01 min | 10.01 min | ||||||
0 | 50 | 11 | 48 | 0.09% (1.62% Gap) | 0.00% (0.00% Gap) | N/A | N/A |
10.01 min | 3.30 min | ||||||
0 | 50 | 11 | 120 | No Solution 0.00% | (0.12% Gap) | N/A | N/A |
10.02 min | 10.00 min | ||||||
0 | 50 | 11 | 168 | No Solution 0.00% | (0.18% Gap) | N/A | N/A |
10.02 min | 10.00 min | ||||||
0 | 100 | 11 | 48 | 0.27% (2.40% Gap) | 0.00% (0.00% Gap) | N/A | N/A |
10.01 min | 3.18 min | ||||||
0 | 100 | 11 | 120 | No Solution 0.00% | (0.01% Gap) | N/A | N/A |
10.02 min | 10.00 min | ||||||
0 | 100 | 11 | 168 | No Solution 0.00% | (0.02% Gap) | N/A | N/A |
10.03 min | 10.00 min |
I | J | K | T | Symmetric | Symmetry-free | Airside/waterside | Hierarchical |
|---|---|---|---|---|---|---|---|
4 | 10 | 4 | 48 | 0.09% (1.01% Gap) | 0.01% (0.07% Gap) | 0.00% (0.53% Gap) | 0.00% (1.28% Gap) |
10.00 min | 10.00 min | 5.25 min | 5.28 min | ||||
4 | 10 | 4 | 120 | 0.06% (0.76% Gap) | 0.02% (0.13% Gap) | 0.00% (0.22% Gap) | 0.01% (2.46% Gap) |
10.00 min | 10.00 min | 8.18 min | 5.31 min | ||||
4 | 10 | 4 | 168 | 0.32% (1.06% Gap) | 0.03% (0.17% Gap) | 0.00% (0.27% Gap) | 0.03% (2.70% Gap) |
10.01 min | 10.00 min | 10.00 min | 6.61 min | ||||
20 | 10 | 4 | 48 | 0.26% (1.21% Gap) | 0.00% (0.02% Gap) | 0.00% (0.49% Gap) | 0.00% (1.74% Gap) |
10.00 min | 10.00 min | 4.61 min | 4.05 min | ||||
20 | 10 | 4 | 120 | 0.95% (1.69% Gap) | 0.02% (0.26% Gap) | 0.00% (0.25% Gap) | 0.01% (3.06% Gap) |
10.00 min | 10.00 min | 7.86 min | 5.39 min | ||||
20 | 10 | 4 | 168 | 0.88% (1.64% Gap) | 0.15% (0.50% Gap) | 0.00% (0.28% Gap) | 0.05% (5.13% Gap) |
10.01 min | 10.01 min | 10.02 min | 6.14 min | ||||
50 | 10 | 4 | 48 | 0.31% (1.27% Gap) | 0.06% (0.20% Gap) | 0.00% (0.50% Gap) | 0.00% (1.75% Gap) |
10.00 min | 10.00 min | 3.74 min | 5.16 min | ||||
50 | 10 | 4 | 120 | No Solution | 1.77% (0.00% Gap) | 0.00% (0.25% Gap) | 0.00% (1.53% Gap) |
10.01 min | 6.67 min | 7.80 min | 10.12 min | ||||
50 | 10 | 4 | 168 | No Solution | 1.00% (1.44% Gap) | 0.00% (0.27% Gap) | 0.12% (8.33% Gap) |
10.13 min | 10.03 min | 10.08 min | 8.08 min | ||||
100 | 10 | 4 | 48 | 0.38% (1.35% Gap) | 0.34% (1.02% Gap) | 0.00% (0.50% Gap) | 0.00% (1.75% Gap) |
20.01 min | 20.03 min | 10.39 min | 5.33 min | ||||
100 | 10 | 4 | 120 | No Solution | 1.08% (1.71% Gap) | 0.00% (0.24% Gap) | 0.01% (1.50% Gap) |
20.04 min | 20.03 min | 12.59 min | 14.89 min | ||||
100 | 10 | 4 | 168 | 29.25% (29.99% Gap) | No Solution | 0.02% (0.28% Gap) | 0.00% (2.46% Gap) |
20.05 min | 20.04 min | 15.94 min | 14.54 min | ||||
250 | 10 | 4 | 48 | No Solution | No Solution | 0.00% (0.51% Gap) | 0.00% (1.75% Gap) |
30.06 min | 30.18 min | 15.92 min | 16.52 min | ||||
250 | 10 | 4 | 120 | No Solution | No Solution | 0.01% (0.24% Gap) | 0.00% (2.23% Gap) |
30.14 min | 30.14 min | 18.54 min | 19.53 min | ||||
250 | 10 | 4 | 168 | No Solution | No Solution | 0.00% (0.27% Gap) | 0.05% (2.30% Gap) |
30.24 min | 30.27 min | 21.17 min | 34.51 min |
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Risbeck, M.J., Maravelias, C.T., Rawlings, J.B. et al. Mixed-integer optimization methods for online scheduling in large-scale HVAC systems. Optim Lett 14, 889–924 (2020). https://doi.org/10.1007/s11590-018-01383-9
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DOI: https://doi.org/10.1007/s11590-018-01383-9