A multistage stochastic energy model with endogenous probabilities and a rolling horizon

Energy and Buildings 135 (2017) 338–349

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A multistage stochastic energy model with endogenous probabilities and a rolling horizon Billy R. Champion a,∗ , Steven A. Gabriel b a b

Civil Systems Program, Department of Civil & Environmental Engineering, University of Maryland, College Park, MD 20742, USA Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA

a r t i c l e

i n f o

Article history: Received 1 September 2016 Received in revised form 20 November 2016 Accepted 28 November 2016 Available online 30 November 2016 Keywords: Energy Conservation Optimization Multistage Stochastic Rolling horizon McCormick inequalities

a b s t r a c t Federal, state and local government-funded energy conservation and renewable energy projects are implemented by requesting large capital budgets at the beginning of a program. The technical and financial performance of these projects are uncertain given anticipated energy savings, varying energy costs, and sub- and super-additivity of energy projects costs and energy savings. The level of uncertainty is directly proportional to the length of the model’s planning horizon and are further exacerbated when these savings are used for investment in future projects. A rolling-horizon model that updates certain exogenous factors as well as optimal decision variable values for past times is presented. This model is run using illustrative cases showing its vast improvement in computational speed to solve, total stages required and total cost to implement all projects over a fixed-horizon, multistage model. Lastly, both suband superadditivity of the annual energy savings of the projects is considered making the problem more challenging but realistic. © 2016 Published by Elsevier B.V.

1. Introduction The federal government buildings are one of the largest energy consumers in the world. In 2014, 39% of all federal energy was consumed by federal facilities. Energy consumed in federal government facilities has generally been declining over the past four decades. However; the reduction stems from both the total square footage occupied by the federal government, which continues to fall from its peak in 1987, and from the energy consumed per square foot inside federal buildings, which has been declining since 1975 [23]. While significant reductions in building energy intensity have been made, many more are required, while tougher challenges exist in funding energy conservation and renewable projects. Facility energy intensity fell short of the 27% goals of Executive Order 13423 and Energy Independence and Security Act to reduce energy intensity (Btu/GSF) with only a 21% reduction [29]. The remaining conservation opportunities will require ingenuity to both fund and implement the projects. However, funding energy conservation continues to follow a multiple-year and risk-averse process.

∗ Corresponding author. E-mail address: [email protected] (B.R. Champion). http://dx.doi.org/10.1016/j.enbuild.2016.11.058 0378-7788/© 2016 Published by Elsevier B.V.

There are many approaches to the implementation of an energy or renewable project but most comprehensive energy programs begin with an assessment of current consumption and energy conservation opportunities at the individual building level. The initial assessment is the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) Level 2 Energy Audit. The audit is an onsite assessment and comprehensive energy analysis of the building’s energy-using components resulting in a list of proposed energy conservation measures (ECMs) which include the following attributes: • the proposed system or component description • an estimate of the investment required to implement the measure • an estimate of annual savings • an estimate of the annual cost savings in dollars • a performance measure such as simple payback ratio or savings to investment ratio A typical set of these measures are shown below in Table 1. The energy auditors determine the appropriate regulatory requirements as part of their scope of work in the contract with the Agency. The energy auditors then conduct audits to recommend the projects necessary to save the required energy. Projects that do not meet specific savings-to-investment ratios are not considered.

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Table 1 Typical energy conservation projects attributes. Project Description

Investment Cost ($)

Annual Energy Savings (KBTU)

Energy Rate ($/KBTU)

Annual Savings ($)

Estimated Useful Life (years)

Payback Ratio (years)

Heating project Lighting project

250,000 50,000

2,375,000 625,000

0.011 0.015

26,125.00 9375.00

30 15

9.57 5.33

All reported projects must be completed. The Agency’s approach to implementing these projects is ultimately risk-averse. The Agency requests a conservative budget from direct appropriated funding2 in the first stage and seeks to fund the required energy conservation projects. A stage is a one-year time period in the current research. Agencies would greatly benefit from innovation and novel approaches to assist in project implementation, funding and timing. The technical and financial performance of these projects are uncertain and often managed by a, “wait-and-see” approach. Here we present original approaches that request reasonable budgets and allow for recourse actions. The savings from implemented projects are used for investment in future projects. However, anticipated energy savings, varying energy costs, and interaction between energy projects affect the ability of these models to predict future savings. A rolling-horizon model that updates the optimization model’s inputs and optimal decision variable values for past stages is presented. This model is run using illustrative cases showing its vast improvement over a fixed-horizon, multistage model. These improvements are: • a reduction in the total number of stages required to implement all projects • the total cost to implement all projects • the computational speed to solve a model with many decision and auxiliary variables The value of the current work is the novel application and combination of several concepts such as multistage stochastic programming and subadditivity and superadditivity of energy savings from energy conservation projects using McCormick Inequalities [22] at several stages to improve on the current industry practice as well as adopting endogenous uncertainty in order for the agency to minimize the total cost of implementing all the energy conservation projects that it is considering. The remainder of this paper is presented as follows. Section 2, discusses the current landscape project selection, stochastic optimization and rolling horizon methodologies, as well as provides context and highlights novelty of the current research. Section 3 presents the model formulation and Section 4 applies the model to experimental yet practical examples. Sections 5 and 6 continue with discussion of the results and conclusions, respectively. 2. Literature and context 2.1. Connection to project Selection/Knapsack problem and stochastic programming A novel way to meet U. S. reduction and renewable goals is by using existing savings to fund future projects while accounting for uncertainty in implementation yields and energy prices. This requires selecting energy projects in a method that allow agencies to account for and reduce uncertainty associated with long plan-

2 Financing energy projects through appropriations allows federal agencies to own their projects and immediately benefit from the cost savings. This type of financing should be an agency’s first consideration in pursuit of its renewable energy goals given the hierarchy of action items in Executive Order 13693 [15].

ning horizons. As such, the current approach has a resemblance to a multi-stage, stochastic knapsack or project selection problem. The current methodology leverages annual savings in later stages from projects previously implemented which is analogous to securities in Markowitz’s portfolio selection work [21]. The current problem also incorporates constraints on the cost of selecting projects, whereas the cost of the securities were not specifically limited in that earlier work. In Asadia et al. [3] the authors present a multi-objective optimization model to assist stakeholders in the definition of measures aimed at minimizing the energy use in the building in a cost effective manner while satisfying the occupants’ needs and requirements. However, the model described incorporates many subjective attributes, which make the quantification of value difficult. A multi-criteria knapsack model was proposed to help designers to select the most feasible renovation actions in the conceptual phase of a renovation project [1]. The additive knapsack model presented in that study was based on linear programming. The current research and the problem is much more complex as the benefits of each selection vary with time. Specifically, the period in which a project is selected, has a large impact on the benefit (annual savings) and the certainty of the benefit. Gustafsson used a mixed-integer, linear programming (MILP) model to minimize the life-cycle cost of retrofits subject to minimum space heating requirements [19]. The author showed that a building’s heating system could be described mathematically in the form of a MILP. The primary objective of the research here is energy savings with cost being a secondary consideration as well as a two-level optimization approach to model the ECM decision process more accurately. A two-level optimization approach is modeled in Champion and Gabriel [10]. However, in the current research, the budgets are funded by direct appropriation, which is best modeled by a single objective function. Cano et al. presented a fixed-horizon deterministic energy technology selection model with an objective function that minimizes the total cost of energy subsystems throughout a 16-year horizon [7]. The Cano et al. work considers the performance of the installed technology through the operational variables in that model. The deterministic model is extended to a stochastic approach [9]. The current research models retrofits for building system and components and does not include the purchase price for energy. The current research extends the Cano et al. model by leveraging short-term performance for funding of future projects and optimizing over shorter, rolling horizons. The above are just a small sample of some project selection papers that have relevance to the current work. For further details, see Models and Method for Project Management [17]. The current approach is based on a multistage stochastic program for project selection. This area has been well-studied with early developments in Dantzig [12], Beale [5], and Charnes and Cooper [11]. In terms of energy systems, in Cano et al. [8], a decision supports system to manage energy sub-systems in a more robust energy-efficient and cost-effective manner was presented. In this paper, a two-stage stochastic model is proposed, where some first-stage decisions regarding investments in new energy technologies have to be taken before uncertainties are resolved [9]. Later recourse (second-stage decisions) on how to use the installed technologies are taken once values for uncertain parameters become

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known, thereby providing a trade-off between long- and shortterm decisions. The current work extends [8] to multiple stages and considers risk management constraints beyond the Conditional Value at Risk discussed in Cano et al. [8] The current work also discusses the time inconsistent policies not present in the two-stage stochastic models. Another extension the current research is that it allows for subadditivity or superadditivity of energy savings, which is also a novel feature of the current work. In project selection, the question of subadditivity or superadditivity of the objective function (costs, profits, etc.) is relevant and makes the problem under review more challenging to solve. The next section describes these concepts and the use of the McCormick inequalities [22] to model them. 2.2. Subadditivity and superaddivity of the energy savings Subadditivity and superadditivity for energy conservation projects can be explained by the interactive effects of these projects in terms of energy savings or costs.3 For example, an energy conservation project that retrofits lighting may decrease the electricity consumption but also reduce the heat gain to the building. This project, in return, makes the building’s boiler work harder to provide the additional heat load to the building. The resulting savings of the projects together will be lower than if only one project was implemented. Using the typical set of these measures from Table 41, consider for example that the agency will implement projects expecting to spend the total cost of $300,000 and get repeating annual savings of $35,500. However, in specific cases, the results of Table 2 overestimate the total annual savings. The agency may only realize annual repeating savings of $26,625 as shown in Table 3, below. Similarly, superadditive effects of energy conservation projects are also possible. An example of this can be observed with the selection of an energy management system or controls projects in combination with higher-efficiency heat or cooling generation equipment. For example, a controls project has a long payback and, generally, is proposed for the existing generation equipment. A second project’s scope could replace the existing generation equipment with newer, higher-efficiency equipment. The higherefficiency heat or cooling generation equipment will use less energy when operating while the controls will minimize operating heating and cooling times based on demand for the load. The combined energy savings of these two projects will be greater than their individual savings. Table 4, provides an example of superadditivity in energy conservation project selection. In the current research, all projects are evaluated for subadditivity and superadditivity by comparing all combinations of projects selected in each stage. For example, Table 5, below gives attributes for four illustrative projects that are available for selection. Table 6 below gives attributes and totals with no interactive effects for the three projects that are selected in this stage (indicated with Xproject = 1). Note that the total savings are simply the sum of savings for each project. Next, consider possible subadditivity and super additivity for energy conservation annual savings. The comparison of each project for additivity is achieved by creating a matrix that can serve as a coefficient for pt (ω), the yield of the annual savings at each stage as realized through each project’s annual savings. This discount and premium Kp,p matrix, below, determines the subadditive or superadditive effect among all projects compared two projects at a time. Subadditivity or superadditivity is possible when both

3

It should be noted that other factors such usage patterns and changes in rates based on time of day impact additivity. These are not modeled as this paper as this is a strategic decision making model.

projects are selected or when xp = xp = 1 for this stage.4 The product of xp · xp and the Kp,p matrix provides a matrix of yields that address subadditive or superadditive effects on all projects’ energy savings. The sum of all “Per Project Additivity” is the total subadditivity or superadditivity for the projects selected in that stage. The total annual savings in this stage are $43,306.25 ($42,000 from Table 5 + $1306.25 from Table 6) which is greater than the sum of the individual annual savings. The key benefits of this approach are • all projects are compared for the potential of subadditivity or superadditivity, • subadditivity and superadditivity are addressed at the individual project level allow interactions to be additive with other projects, • and this approach allows the comparison to be made and the discount and or premium to be calculated in each stage. Annual savings will become subadditive or superadditive as they are selected in later stages. However, adding the interactive functionality to the model for the sake of realism, results in additional complexity. The product of variables makes the problem nonlinear and potentially harder to solve without some exact linearization, which is described next via the McCormick inequalities [22]. This approach of allowing for subadditivity or superadditivity of energy savings is also a novel feature of the current work. Subadditivity of stochastic processes as discussed above are the key organizing principle driving problems of nearly intractable difficulty [28]. Further solving models as proposed in the current research may not provide global optima with nonlinear models. McCormick developed a method for convex/concave relaxations of factorable functions that allow for vast improvements in goal finding and CPU speed to solve [22]. In some cases, like the one below, the linearization resulting from these inequalities is exact. The current research leverages McCormick’s auxiliary variable method (AVM) which employs auxiliary variables for each factor involved. More precisely, instead of relaxing functions, the nonconvex optimization problem is linearized. The nonconvex problem is reformulated introducing auxiliary variables in such a way that the intrinsic functions are decoupled and can be exactly linearized one by one [30]. The current research leverages this AVM which addresses the problem of characterizing the convex envelope (the smallest convex set that covers a set of points) of the bilinear function as in Sherali and Alameddine [27].5 Specifically, xp · xp ≡ Vp,p which collapses the decision variables into a single variable allows multiplication by data and parameters. Additional constraints on the auxiliary variable, Vp,p , establishing upper and lower bounds are discussed in Section 3, Model. 2.3. Rolling horizon perspective Budgeting for the entire planning horizon with perfect foresight can be overly optimistic. Perfect foresight assumes that the yields and budgets from savings are known for all stages, perfectly, when making the first-stage decisions. Models that assume perfect foresight of the time horizon have perfect information for the entire time horizon. In deterministic, perfect foresight models, parameters are assumed to be known with 100% certainty. Perfect foresight

4 Note, just two projects at a time are considered but one could imagine three or more relative to subadditivity or superadditivity. 5 A bilinear function is a function of two variables that are linear with respect to each other, for example f (x, y) = xy.

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Table 2 Additivity: no interactive effects on energy conservation projects attributes. Project Description

Investment Cost ($)

Annual Energy Savings (KBTU)

Energy Rate ($/KBTU)

Annual Savings ($)

Estimated Useful Life (years)

Payback Ratio (years)

Heating project Lighting project TOTAL Expected Additive Savings

250,000 50,000 300,000

2,375,000 625,000 3,000,000

0.011 0.015 0.012

26,125 9375 35,500

30 15

9.57 5.33 8.45

Table 3 Subadditivity: interactive effects on energy conservation projects attributes. Project Description

Investment Cost ($)

Annual Energy Savings (KBTU)

Energy Rate ($/KBTU)

Annual Savings ($)

Estimated Useful Life (years)

Payback Ratio (years)

Heating project Lighting project TOTAL Subadditive Savings

250,000 50,000 300,000

2,375,000 625,000 2,250,000

0.011 0.015 0.012

26,125 9375 26,625

30 15

9.57 5.33 11.27

Table 4 Superadditivity: interactive effects on energy conservation projects attributes. Project Description

Investment Cost ($)

Annual Energy Savings (KBTU)

Energy Rate ($/KBTU)

Annual Savings ($)

Estimated Useful Life (years)

Payback Ratio (years)

Heating project Controls project TOTAL Superadditive Savings

250,000 100,000 300,000

2,375,000 50,000 2,700,000

0.011 0.015 0.012

26,125 750 32,400

30 15

9.57 133 9.26

Table 5 Additivity: conservation projects attributes for simplifying example calculation. Project Description

Investment Cost ($)

Annual Energy Savings (KBTU)

Energy Rate ($/KBTU)

Annual Savings ($)

Estimated Useful Life (years)

Payback Ratio (years)

Heating project Lighting project Insulation Project Controls project

250,000 50,000 150,000 100,000

2,375,000 625,000 125,000 50,000

0.011 0.015 0.09 0.13

26,125 9375 11,250 6500

30 15 50 20

9.57 5.33 13.33 15.38

Table 6 Additivity of projects selected. Project Description

Investment Cost ($)

Annual Savings ($)

Xproject (=1 if selected)

Savings Generated ($) (Xproject * annual savings)

Heating project Lighting project Insulation Project Controls project Additive Total

250,000 50,000 150,000 100,000

26,125 9375 11,250 6500

1 1 0 1

26,125 9375 0 6500 42,000

models while useful as base cases are less realistic than ones that allow for stochastic elements and/or some rolling-horizon foresight [13]. In reality, energy project selection is often made under uncertainty with hedging of worst-case scenarios. Scenario-based models include non-anticipativity constraints to ensure that the worst yields and the budgets scenarios are included and observed to be the same at all successor nodes [6]. In the rolling-horizon approach which is more realistic than perfect foresight, decisions are taken one stage at a time, realizing and possibly updating the parameters between these stages. In a multistage problem, decisions made in the current stage influence the recourse decisions made in later stages. Rollinghorizon approaches solve multistage problems with a planning ¯ that horizon of nT = |T|, by looking at smaller rolling horizons, H models subsets of the full problem. Rolling horizon models have

been considered since at least [4]. That experimental study was designed to investigate the efficiency of decisions obtained from optimizing a finite, multistage model and implementing those decisions on a rolling basis. The results of the study suggest that rolling schedules are quite efficient. The typical practice with a rolling-horizon policy calls for establishing the “master schedule” for a certain number of future stages, known as the planning horizon, based on the currently available relevant information e.g. demand forecasts, available capacity, inventory and backlog records, etc. [2,26]. This terminology is used in the current research. In the current research, the “demand” is established in the first stage by the auditor after a review of the applicable regulations. This is a one-time activity for the program and as such is not continually forecasted. Further, in this research, the model is updated as more data (project performance) and variables (budgets) become available.

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,

0 .8 ⎡ .8 0 ⎢ = ⎢ 1 .7 ⎢−0 2 1 1 4

1 .7 0 0 1

−0.2 . 4 1 1⎤ ⎥ −0.1 1 ⎥ 0 1⎥ 1 0

Fig. 1. Sample K matrix of subadditive and superadditive multipliers.

A rolling-horizon approach as presented below only considers a smaller future set of stages and allows for learning in between each “roll” of the horizon. As such, the approach can be computationally quicker as well as more realistic. Additionally, such a rolling horizon approach also allows for learning (in between rolls) for the decision-maker and thus can be used to model “endogenous probabilities”. Endogenous uncertainty problems are described as discrete event dynamic systems where the underlying stochastic process depends on the optimization decisions [24]. Thus, for example, a scenario tree with probability p for one of the nodes really depends on the values of the optimal decision variables. As an example, consider [14], that describes project selection with endogenous variables for exploration of new oil fields. The possibility of investment for these projects may be initiated in each stage. The probability distributions of the uncertain characteristics of projects are discrete, within each scenario, but the capacity and delivery are realized only after the optimal decisions are made. In the current research, projects are also selected in each stage but the endogenous variables affect the returns determines future project selection similar to Dupacova [14]. Stochastic programming models can be classified into two broad categories [20]: exogenous uncertainty where stochastic processes are independent of decisions that are taken (e.g. demands, prices), and endogenous uncertainty where stochastic processes are affected by these decisions. Decisions can affect the stochastic processes by altering the probability distributions (type 1) or determining the timing when uncertainties in the parameters are resolved (type 2) [16]. A number of planning problems that involve very large investments at an early stage have endogenous (technical) uncertainty (type 2) that dominates the exogenous uncertainty [18]. In the current research, the endogenous uncertainty is modeled because decisions regarding timing of projects selected severely impact the overall objective through the realization of the yields. As such, consideration of endogenous uncertainty is another distinguishing feature of the current work. Devine et al. [13] present improvements that come with rolling horizons for mixed complementarity problems (MCP) in the context of natural gas market equilibria [13]. For example, one advantage is that each roll is a separate solving of an MCP, which allows the opportunity to adjust inputs in between these rolls. For example, a new scenario tree for the next roll can be endogenously changed, by one or more players, based on a solution from the previous roll so that the model has endogenous probabilities. That novel approach is adopted in this research but is applied to multistage stochastic programming rather than MCPs (Fig. 1). In the following figure, p is set of ECM projects, t is set of stages T and ω is the set of scenarios with given probability, ␲t (␻). The decision variables are xp , a first-stage binary variable representing selection of the project and y tp (ω) is a t-stage a binary recourse variable representing selection of the projects for scenario. The parameter pt (ω) is the yield of the annual savings at each stage as realized through each project’s annual savings. Fig. 2, above illustrates the comparison between the models presented in the current research. In Fig. 2a, the multistage stochastic model shows the probability of each scenario(ω), t (ω) , and yield, pt (ω). The yield is a low, medium or high return coefficient

related to the estimated annual savings. In the multistage stochastic model, all decisions are made with information known in this first stage. At every stage beyond the first, the agency has a set of fixed recourse decisions, with fixed probability and yields. This results in three possible nodes at t = 2 and nine nodes at t = 3. The model also contains non-anticipativity, which does not allow the agency to anticipate what node they will arrive at before recourse actions are taken. The improvement in Fig. 2b, comes from realizing the node from which to start the next roll and update with endogenous learning. In Fig. 2b, the agency makes a decision for the planning horizon, here four years, however, after the first year, then realizes the actual node, t (ω) , and yield, pt (ω). However, the model assesses its position in the tree, here, y tp (med) taken as a representative example because it represents the expected value. This results in three possible nodes at t = 2 and only three nodes at t = 3 and so forth. At each stage, the agency reruns the model assuming a horizon of H. The resulting treatment of the t (ω) for each model are illustrated in Table 7, below. Note that the endogenous learning in model presented in Fig. 2b, also updates the yields, t p (ω) in each stage per Table 7, below. 3. Model The following is the notation, variables, and parameters used in the general statement of the stochastic multistage energy conservation model (SM-ECM). Sets p

Set of ECM projects with P = {1, 2,. . .np } where np = |P|, P’ P A set of stages t (typically years) = {2,. . .nT } where nT = |T|, T’ t Set of scenarios with given probability, ␲t (␻), ␻ = {1, 2,. . .n␻ } where n␻ = ||

t ω

Main Primal Decision Variables xp y tp (ω)

A first-stage binary variable representing selection of the project p; variable = 1, if selected by the agency to be implemented at t = 1, = 0 otherwise A t-stage (T = {2,. . .nT }) binary recourse variable representing selection of the projects for scenario (␻); variable = 1, if selected by the agency to be implemented in stage T = {2,. . .nT }, = 0 otherwise

Intermediate Variables Bt (ω) V p,p W tp (ω)

The budget in dollars for implementing the agency’s projects at stage t > 0 McCormick envelope auxiliary variable for initial stage variables xp and xp V p,p = xp xp McCormick envelope auxiliary variable for first- and recourse-stage variables (xp + yp t (␻)) and ( xp + yp t (␻))

Parameters C

Ot p

A scalar representing the capital budget requested through direct appropriation by the agency for implementing the agency’s projects in the first stage, in dollars ($) The operating budget prescribed for the agency at stage t, in dollars ($) The estimated annual savings in dollars achieved by implementing project p, the energy savings, in dollars ($)

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343

Fig. 2. Comparison on multistage stochastic program and of rolling horizon approach.

Table 7 Probability and yields by model type. T=2 Fig. 2a, Multistage Stochastic Model

Fig. 2b, Rolling Horizon model

p t p

(ω)

t (ω) Kp,p

 (1) = 0.33 2 (2) = 0.33 2 (3) = 0.33 2 (4) = 0.33 2 (5) = 0.33 2 (6) = 0.33 2 (7) = 0.33 2 (8) = 0.33 2 (9) = 0.33 2 (low) = 0.33 2 (med) = 0.33 2 (high) = 0.33 2

T=3 (1) = 0.65 (2) = 0.90 (3) = 1.05  (4) = 0.33 2 (5) = 0.33 2 (6) = 0.33 2 (7) = 0.33 2 (8) = 0.33 2 (9) = 0.33 2 p (low) = 0.65 2 p (med) = 0.90 2 p (high) = 1.05 2 p 2 p 2 p 2

T=4

 (1) = 0.33 3 (2) = 0.33 3 (3) = 0.33 2 (4) = 0.33 2 (5) = 0.33 2 (6) = 0.33 3 (7) = 0.33 3 (8) = 0.33 3 (9) = 0.33 3 (low) = 0.33 3 (med) = 0.33 3 (high) = 0.33 3

which is equal to the product of annual energy savings (KBTU) and energy rate ($/KBTU) The estimated investment needed to implement project p in dollars ($) The yield of the annual savings at each stage as realized through each project’s annual savings The probability of the discrete energy price at each stage t An np × np matrix for pairwise comparison and multipliers of yield

3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p

(1) = 0.65 (2) = 0.65 (3) = 0.65 (4) = 0.90 (5) = 0.90 (6) = 0.90 (7) = 1.05 (8) = 1.05 (9) = 1.05 (low) = 0.70 (med) = 0.90 (high) = 1.00

Not Shown in Figure

4 (low) = 0.33 4 (med) = 0.33 4 (high) = 0.33

4 p 4 p 4 p

(low) = 0.75 (med) = 0.90 (high) = 0.95

3.1. General formulation The objective function minimizes the total cost to complete all energy conservation measures. The objection function is composed of the following terms:

C : The capital budget requested through direct appropriation by the agency for implementing the agency’s projects for the entire planning horizon, in dollars($)

(3a)

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nP 

The following constraints state that the investment in any stage must be within the budget. The sum of all projects starting at t = 2 must not exceed the budget.

p xp : The estimated first-stage investment cost to

np 

p=1

implement the agency − selected projects



 nP

t (ω)

ω∈˝

(3b)

p y tp (ω) : The estimated second- and

ypt (ω) = ypt (ω )∀ω, ω ∈ {1. . .˝}

p=1 t=2

Bt (ω) = Bt (ω )∀ω, ω ∈ {1. . .˝}

agency − selected projects

(3c)

The agency, by choosing the timing of when it undertakes each energy conservation project, p, is trying to minimize the sum of these three terms. The terms are costs in the problem solved when the objective function is maximized and, as such, are negated when presented in the minimization form of the problem as shown below. nP 

p xp −



t (ω)

ω∈

p=1

nP nT  

p y tp (ω)

(3d)

xp +

Bt (ω) = Bt (ω )∀ω, ω forwhich t (ω) =  t (ω ), t = 2. . .T

where t (ω)isthenodeatstage, tandscenario, ω For Eq. (3j), the values of the decision variable ypt (ω) and variable (ω) chosen at stage t, depend on the data  t available up to time t, but not future observations.6 This is the basic requirement of non-anticipativity [31]. The budget must remain nonnegative (no loans). Bt

Bt (ω) ≥ 0∀t ∈ T, ∀ω ∈ 

B1 (ω) = C

(3n)

The decision variables are binary. y tp

(ω) = 1∀p ∈ P, ∀ω ∈ 

(3e)

nP t−1  

xp , y tp (ω) binary

(3o)

3.2. Subadditivity and superadditivity

The second constraint states that the nonnegative available budget at the time, t, for scenario, ω, Bt (ω) is the sum of: t p







(ω) p xp + y tp (ω) : The estimated savings for

p=1 t  =2

energy conservation measures from all prior stages in dollars($)

(3f)

The model above is modified to include the possibility of subadditivity or superadditivity. It begins by noting that the key impact is to the annual savings as in the example of Table 2. In the model, the annual savings are repeating every year after project implementation (selections by xp ory tp (ω) = 1) and formulated in the budget equation as term (3j). The term in (3j), above determines the budget by multiplying the yield pt (ω) by the estimated annual saving p if the project is



chosen Ot : The operating budget in dollars($)

(3g)

Bt−1 (ω) : The budget from the previous stage in dollars($) (3h) nP 

(3m)

The budget in stage is the capital requested in dollars.

t=2

−

(3l)

ypt (ω) = ypt (ω )∀ω, ω forwhich t (ω) =  t (ω ), t = 2. . .T

p=1 t=2

The first constraint faced by the agency is that all projects must be selected. nT 

(3k)

p=1

The following constraints enforce non-anticipativity of the model [25].

nT

later − stage investmentcost to implement the

−C −

p y tp (ω) ≤ Bt (ω) ∀t ∈ T, ∀ω ∈ 

p y t−1 (ω) ∀t ∈ T, ∀ω ∈ : the cost of p

p=1

projects implemented in the prior stage in dollars($) (3i)

n

np p t−1   

−

Kp,p ·

t p

(ω) p





xp + y tp (ω)







xp + y pt  (ω)

: The

p=1 p =1 t  =2

j



xp + y p (ω)

= 1. However, for subadditivity and super-

additivity, we compare two projects p and p’ at a time. Letting Kp,p be a np bynp matrix of pairwise multipliers for yields, pt (ω) equal to the energy annuals saving interactive effect. The addition of subadditivity (as opposed to super-additivity) best models the practical approach to energy conservation measures. However, this change results in a mixed integer non-linear program (MINLP) which is computationally much more challenging and thus less likely that agencies will use it to find a global optimal solution. By contrast, the (exact) linearization via the McCormick inequalities shown below is a computational tool that makes solving such problems easier. In order to transform the nonlinearities introduced by the subadditivity of the model, we apply the  auxiliary  variable model  [22]. This is achieved by letting a = xp + y tp (ω) and letting b =







xp + y tp (ω) as discussed in Section 2 where p and p’ are indices

for two distinct projects. The linearization of the product of the terms a and b is given as follows. First, let W = ab and note that W = 1 if and only if a = 1 and b = 1.

subadditivity or superadditivity of the product of energy savings and rates of all projects chosen in dollars($) (3j)

6

For example,  2 (1, 2, 3) corresponds to the node labeled, “2” in Fig. 2a.

B.R. Champion, S.A. Gabriel / Energy and Buildings 135 (2017) 338–349

Consider the following McCormick inequality constraints: LetW = abwhere

t p

W ≥0 W ≥b+a−1

0 p (high)

W ≤a Note that in (3p) if a = 1 and b = 1 then the second inequality forces W to be greater than or equal to 1. The last two inequalities provide 1 as an upper bound for W, so taken together imply that W = 1. Conversely, if W = 0 then the second inequality shows at most one of a or b can be equal to 1 (other an infeasibility). The last two inequalities are still valid in this case. Thus, the nonlinear equation of W = ab has been exactly linearized in (3p). The budget equation below in (3q) replaces terms (3f)–(3j) using McCormick inequalities to linearize the decision variables. nP t−1  



t p

j

(ω) p xp + y p (ω)



p=1 t  =2

t

+O + B

t−1

(ω) −

nP 

n

p y t−1 p

(ω) −

p=1 t−1 

Kp,p ·

t p

np p   p=1 p =1

(ω) p · Wpt (ω) ∀t ∈ T, ∀ω ∈ 

(3q)

t  =2

The following constraints are added. Wpt (ω)

≥0

Wpt (ω)

≥ (xp + ypt (ω)) + (xp + ypt  (ω)) − 1

(3r)



Wpt  (ω) ≤ (xp + ypt  (ω)) 

Wpt (ω) ≤ (xp + ypt (ω)) The terms and equations above complete the model in its entirety. 3.3. Rolling horizon with endogenous learning The rolling horizon method involves making first-stage decisions, based on a stochastic forecast/estimation. At the beginning of the second stage, the first-stage decisions are apparent. In order to make these decisions, forecasts for additional stages into the future are required. In addition, existing forecasts can be revised or updated. This procedure repeats for every stage justifying the term rolling horizon decision making for the practice. Here, the term “horizon” refers to the number of stages in the future for which the forecast is made. It is this horizon that is “rolled over” each stage [26]. In this model, the endogenous learning is applied as such. The effect of (3s–3v) is percentage improvement in the yield based on experience which is proportional to the number of projects implemented in the prior stage.

t p

t p

(low) =

⎧ ⎪ ⎨ ⎪ ⎩

(med) =

⎧ ⎪ ⎨ ⎪ ⎩

⎪ ⎩

t−1 p

(high) ∗ 0.98, 0 ≤ ypt−1 (high) < 1

t−1 p

(high) ∗ 1.00, 1 ≤ ypt−1 (high) < 3

(3p)

W ≤b

Bt (ω) =

(high) =

⎧ ⎪ ⎨

t−1 p

(low) ∗ 1.00, 0 ≤ ypt−1 (low) < 1

t−1 p

(low) ∗ 1.04, 1 ≤ ypt−1 (low) < 3

t−1 p t−1 p t−1 p

(3s)

(low) ∗ 1.06, 3 ≤ ypt−1 (low)

(med) ∗ 1.00, 0 ≤

ypt−1

(med) < 1

(med) ∗ 1.02, 1 ≤

ypt−1

(med) < 3

t−1 p

(med) ∗ 1.04, 3 ≤ ypt−1 (med)

1 p (low)

345

t−1 p

(high) ∗ 1.01, 3 ≤

ypt−1

(3u)

(high)

= 0.00

= 0.65

1 p (med)

= 0.90

1 p (high)

= 1.05

(3v)

The energy conservation model described above is run as follows: ¯ for a subset of the total time periods is 1. The rolling horizon H specified. ¯ year model is run 2. The H− 3. At the end of year 1, first-stage decisions become input parameters for stage 2.7 a The budget is reduced by the cost of projects implemented in the previous phase. b The budget is increased based on annual savings realized in year 1. c The endogenous learning adjustment is applied to yields, 2 p (ω) per the learning above. d The probability of the yields, 2 (ω) are not adjusted. In these cases, the discrete uniform distribution will remain (there is an equal likelihood of all discrete yields). ¯ year 2, the model is rerun 4. The H− a the budget is reduced by the of cost of project in the previous stage b budget is increased based on annual savings realized in stage 1 and stage 2 c The endogenous learning adjustment to yields, p3 (ω) d The probability of the yields, 3 (ω) are not adjusted. In these cases, the discrete uniform distribution will remain (there is an equal likelihood of all discrete yields). ¯ where the final H ¯ – year 5. Repeat until the end of year nT − H model is run or all projects are complete. A flow chart for a 4-year rolling horizon is shown below in Fig. 3.

4. Illustrative example The model described above seeks the objective of selecting the lowest-cost energy program (all projects must be completed). The lowest-cost program will make the most efficient use of the annual savings realized by implementing projects in prior stages. A practical application of the model is demonstrated using data from an agency’s campus of buildings in the southeastern United States [10]. In the numerical example, there are 48 ECMs with varying characteristics and project attributes as shown in Table 8. The model presented earlier is applied to these data as follows. np p pt

= |48|, the total number of ECMs Energy rate in $/KBTU as shown in the third column of Table 8 Annual cost of energy saved in dollars ($) as shown in the fourth column of Table 8

(3t) 7

Recall that a stage is a one-year time period in the current research.

346

B.R. Champion, S.A. Gabriel / Energy and Buildings 135 (2017) 338–349

Fig. 3. Rolling horizon approach with update rules.

t p

(ω)

the static annual savings fluctuation at each stage is realized through each project’s annual savings as shown in Table 9, below.

Endogenous learning (updates) were modeled using three possibilities for the distribution of yields. The probability t (ω)of each discrete value of the yield, pt (ω) in the distribution was kept constant in all cases. Six cases were run to evaluate the efficiency of the model. Cases 1, 3, and 5 are models of 2-year, 4-years and 6-year rolling horizons, respectively, without any endogenous learning. Cases 2, 4, and 6 are models of 2-year, 4-years and 6-year rolling horizons, respectively, with the endogenous learning of Eqs. (3s)–(3v) applied.

all sources except the annual budget is $9,912,042. The capital requested in the first year is $6,731,889, which funds 32 projects, leaving the balance of 16 projects to be funded through annual savings. 4.2. Rolling horizon results The summarized results of the illustrative model using several horizons are shown below in Table 10. Table 11 illustrates the comparison between the multistage stochastic model and the rolling horizon model. The tradeoffs involving the shorter length of the horizon for the costs of the overall program are apparent. The endogenous learning was not impactful in the shortest horizons.

4.1. Multistage results The benchmark Multistage Stochastic 7-year MIP model was run for a 48-project model in GAMS Rev 23.6.5 with the XPRESS solver on an x86 64-bit Microsoft Windows machine. This model includes the subadditivity of the energy conservation savings in Eq. (3j), and the McCormick envelopes for variables xp and y tp (ω) in (3r). The reported model statistics are: The resulting solver status was 1, “Normal Completion” with a model status of 8, “Integer Solution.” The Resource Usage was 540.559 and the Iteration Count was 326,664. It should be noted that models with interactive effects affecting just 5 of the 48 projects took over 72 h to solve and thus represents a large-scale instance of the problem described above given 48 total projects and 7 years considered. The optimal objective function values for the multistage problem is $9,912,042, which satisfies the relative optimality tolerance of 0.0. This means that the total cost to complete all projects from

8 The Rate/Yield Factor (x Annual Savings), pt (ω)are estimated based on observations of energy conservation projects completed between 2010 through 2015.

5. Discussion The experimental examples were run for both two, four and six-year rolling horizons. There were three major finding from the results of the practical application and the several cases observed. These are: 1. Early selection of projects by the rolling horizon approach limited the ability to spread projects throughout the planning horizon 2. The benefit of the year-over-year savings are lost in shorter horizons 3. The impact of the rolling horizon length is greater than that of the endogenous learning 4. The computational gain over the fixed-horizon models are substantial specifically when subadditivity and superadditivity impact more than 5 projects 5. The rolling horizon model only outperforms (requires a lower total cost than) the multistage models with longer horizons regardless of learning

B.R. Champion, S.A. Gabriel / Energy and Buildings 135 (2017) 338–349

347

Table 8 ECM data in practical application. Investment Cost ($)

Annual Energy Savings (KBTU)

Energy Rate ($/KBTU)

Annual Cost of Energy Saved ($)

Estimated Useful Life (Years)

P project1 project2 project3 project4 project5 project6 project7 project8

␥p $ 710,354 $ 637,975 $ 468,071 $ 40,368 $ 8557 $ 15,328 $ 55,207 $ 59,355

␣p 5,334,857 1,849,047 1,768,079 445,600 213,025 124,584 287,971 416,045

␨p $ 0.015 $ 0.033 $ 0.023 $ 0.010 $ 0.012 $ 0.023 $ 0.027 $ 0.022

␪p $ 80,023 $ 61,019 $ 40,666 $ 4456 $ 2556 $ 2865 $ 7775 $ 9153

N 30 23 30 30 15 9 15 15

8.88 10.46 11.51 9.06 3.35 5.35 7.10 6.48

project9 project10 project11 project12 project13 project14 project15 project16

$ 84,738 $ 188,994 $ 142,377 $ 186,520 $ 165,932 $ 169,521 $ 95,238 $ 220,871

559,247 801,565 660,074 440,470 2,243,077 650,787 554,558 1,366,652

$ 0.015 $ 0.033 $ 0.023 $ 0.033 $ 0.012 $ 0.023 $ 0.027 $ 0.019

$ 8389 $ 26,452 $ 15,182 $ 14,536 $ 26,917 $ 14,968 $ 14,973 $ 25,966

30 40 30 30 15 20 15 15

10.10 7.14 9.38 12.83 6.16 11.33 6.36 8.51

project17 project18 project19 project20 project21 project22 project23 project24

$ 201,577 $ 119,351 $ 152,286 $ 95,631 $ 53,495 $ 276,920 $ 94,078 $ 228,071

793,782 724,725 488,525 632,278 518,592 1,551,851 1,135,237 784,038

$ 0.030 $ 0.033 $ 0.023 $ 0.010 $ 0.012 $ 0.023 $ 0.027 $ 0.026

$ 23,813 $ 23,916 $ 11,236 $ 6323 $ 6223 $ 35,693 $ 30,651 $ 20,385

30 23 30 30 15 20 20 15

8.46 4.99 13.55 15.12 8.60 7.76 3.07 11.19

project25 project26 project27 project28 project29 project30 project31 project32

$ 236,862 $ 438,530 $ 558,439 $ 84,237 $ 18,149 $ 64,378 $ 387,393 $ 266,812

2,103,902 1,678,580 3,212,065 2,054,672 138,751 420,774 2,743,397 937,263

$ 0.014 $ 0.023 $ 0.029 $ 0.020 $ 0.013 $ 0.017 $ 0.026 $ 0.030

$ 29,455 $ 38,607 $ 93,150 $ 41,093 $ 1804 $ 7153 $ 71,328 $ 28,118

10 23 12 10 26 20 15 25

8.04 11.36 6.00 2.05 10.06 9.00 5.43 9.49

project33 project34 project35 project36 project37 project38 project39 project40

$ 185,099 $ 205,145 $ 195,433 $ 184,600 $ 110,377 $ 252,736 $ 157,354 $ 247,218

2,236,000 1,664,432 3,599,559 750,238 1,045,732 1,533,356 2,043,132 1,573,358

$ 0.011 $ 0.017 $ 0.014 $ 0.019 $ 0.012 $ 0.021 $ 0.020 $ 0.028

$ 24,596 $ 28,295 $ 50,394 $ 14,255 $ 12,549 $ 32,200 $ 40,863 $ 44,054

20 10 23 28 23 37 18 20

7.53 7.25 3.88 12.95 8.80 7.85 3.85 5.61

project41 project42 project43 project44 project45 project46 project47 project48

$ 256,421 $ 152,886 $ 455,000 $ 473,225 $ 127,011 $ 492,782 $ 266,790 $ 115,006

1,806,445 2,399,913 2,448,183 3,500,838 883,506 1,802,085 1,010,352 741,117

$ 0.024 $ 0.012 $ 0.022 $ 0.017 $ 0.017 $ 0.016 $ 0.031 $ 0.025

$ 43,355 $ 28,799 $ 53,860 $ 59,514 $ 15,020 $ 28,833 $ 31,321 $ 18,528

25 28 36 33 14 32 10 20

5.91 5.31 8.45 7.95 8.46 17.09 8.52 6.21

Totals

$ 10,402,698

66,372,316

5.1. Early selection of projects limited the ability to spread projects throughout the planning horizon This is best observed in cases 1 and 2, where the first rolling horizons compressed the selection of all projects into two years. As a result, most projects were implemented in the first stage. The model could not anticipate additional rolls. All six cases were heavily influenced by the model’s early selection of projects. This result is apparent in Table 11 where more projects were selected in the first stage than any other stage, in all cases. This result can be explained by observing the projects returns. Greater than 60% of the projects listed in Table 8 have a simple payback greater than the planning horizon (7 years). This means that most projects could

Payback (Years)

$ 1,351,279

not fund themselves within the planning horizon, let alone a shorter one. The model selects projects early, as shorter horizons will not generate enough savings to fund many additional projects. If the projects could generate significant savings within the rolling horizon, their selections would be delayed and therefore available for later stages. 5.2. The computational gain over the fixed-horizon models are substantial specifically when subadditivity and superadditivity involves 12 projects interacting Cases 1 and 2 best show that benefit of the rolling-horizon approach concerning computational time. In longer horizons, the fixed model could not be solved in 128 h when many projects (>12

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B.R. Champion, S.A. Gabriel / Energy and Buildings 135 (2017) 338–349

Table 9 Probability of annual saving fluctuation based energy price and savings yield at t = 1.

Scenario 1 (␻1 ) Scenario 2 (␻2 ) Scenario 3 (␻3 )

Probability, t (ω)

Rate/Yield Factor,

0.33 0.33 0.33

0.65 0.90 1.05

t p

(ω)8

Table 10 Multistage stochastic energy program results at the planning horizon. Blocks of Equations Blocks of Variables Non Zero Elements

6618 36 9,994,223

Single Equations Single Variables Discrete Variables

2,895,041 2,155,824 210,000

projects) are interacting in later stages. Recall that all projects are compared for interaction. When twelve projects have nonzero coefficients, the model does not find an optimal solution in the fixed model. In all 2-year cases, the sequential rolling horizon models could be solved to optimality. 5.3. The benefit of the year-over-year savings are lost in shorter horizons

to the 6-year model’s ability to leverage five years of the repeated annual savings and the impact of endogenous learning. The longer length of the rolling horizon allows the model to realize more annual savings, thus funding for more projects efficiently. The endogenous learning is most valuable after the first stage where 34 projects are completed. The higher yields provide higher budgets in later years however; the mix of projects remains relatively unchanged. The result can be seen by comparing the number of projects implemented per year in Cases 5 and 6 in Table 11. An alternative application of the rolling horizon was modeled in Case 7. The last roll should have been in year 2, however; allowing additional rolls have only improved the model. If allowed to extend beyond the planning horizon, the model will attempt to delay the final project’s completion in order to take full advantage of year over year savings over implemented projects. This result violates the constraints of the model. If that project were forced to complete in the final year of the planning horizon (i.e. force project 5 to be completed in stage 7) then this model would be superior to cases 3 and 4. This result, though infeasible, show the driver to delay projects indefinitely to benefit from the annual savings. The rolling horizon model is superior to the multistage model in specific cases. The savings can be greater than 20% in these cases. 6. Conclusion

This key result is one of the disadvantages of the rolling horizon approach. In many cases, the multistage model provides a better model as it spans the length of the planning horizon. The annual savings are cumulative over time. Likewise, compressed horizons do not allow for most of the learning to make a significant impact. Most projects are completed in the earliest stages prior to the endogenous learning taking effect, reducing the ability to provide a significant impact. A relatively smaller subset of projects benefit for the learning in the prior stages. Cases 3 and 4 present the most profound and meaningful results. The rolling-horizon model implements all projects in four years (the years shorter than that of the planning horizon). In the fourth year, case 3 (no learning) completes one project (project 20). In the same year, Case 4 (endogenous learning) completes one project as well (project 24). 5.4. The impact of the rolling horizon length is greater than that of the endogenous learning and the rolling-horizon model only outperforms (requires a lower total cost than) the multistage models with longer horizons regardless of learning Cases 5 and 6 provide the best results of all cases as these cases are only the only ones that outperform the multistage model. The cost of the overall program is $7,929,082 (the sum of the budgets requested) and saves the agency over 20%. This result is attributable

In this research, we introduced the concept of rolling horizons with fixed improvement and with additivity. We incorporated McCormick’s auxiliary variable model to make this problem solvable. The subadditivity and superadditivity provided challenges with regard to the size of the problem. This model was compared to several experimental cases to a multistage stochastic program for energy project selection. While there are improvements to the results of the model from improved yield in each stage, the larger impacts to the objective were made by selecting the appropriate length of the horizon. The rolling horizon selected should start with the length of the planning horizon and reduced until the objective exceeds that of a comparable multistage model. Shorter horizons will allow for more endogenous learning but in these applications of this research, cumulative savings outweigh the ability to learn and better estimate yields. This model provides great improvement over the comparable stochastic model in the longer rolling horizons. The three main benefits are improved objective functions (greater than 20% lower cost to implement all projects), applicability, which allow agencies to model actual field conditions and speed allowing subadditivity and superadditivity to be solved in less than a few hours. Federal, state and local agencies will greatly benefits from this model in their strategic decision making and energy project selection.

Table 11 Resulting projects at “Med” yield and remaining budget.

Multistage Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Capital Outlay Projects Completed Capital Outlay Projects Completed Capital Outlay Projects Completed Capital Outlay Projects Completed Capital Outlay Projects Completed Capital Outlay Projects Completed Capital Outlay Projects Completed

Stage 1

Stage 2

Stage 3

Stage 4

Stage 5

Stage 6

Stage 7

$9,912,042 32 Projects $10,290,220 43 Projects $10,290,220 43 Projects $10,099,872 39 Projects $10,099,872 39 Projects $9,940,430 34 Projects $9,940,430 34 Projects

NA 0 Projects $1,170,225 5 Projects $1,170,225 5 Projects $2,087,085 3 Projects $2,122,803 5 Projects $958,627 4 Projects $958,627 4 Projects

NA 0 Projects All Projects Completed

NA 0 Projects

NA 0 Projects

NA 0 Projects

NA 16 Projects

$1,218,283 5 Projects $1,338,623 3 Projects NA 0 Projects NA 0 Projects

$1,252,253 1 Project $1,383,758 1 Project NA 0 Projects NA 0 Projects

All Projects Completed

NA 0 projects NA 0 projects

NA 10 Projects NA 10 Projects

NA 0 Projects NA 0 Projects

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