A two-level approach to large mixed-integer programs with application to cogeneration in energy-efficient buildings

Abstract

We study a two-stage mixed-integer linear program (MILP) with more than 1 million binary variables in the second stage. We develop a two-level approach by constructing a semi-coarse model that coarsens with respect to variables and a coarse model that coarsens with respect to both variables and constraints. We coarsen binary variables by selecting a small number of prespecified on/off profiles. We aggregate constraints by partitioning them into groups and taking convex combination over each group. With an appropriate choice of coarsened profiles, the semi-coarse model is guaranteed to find a feasible solution of the original problem and hence provides an upper bound on the optimal solution. We show that solving a sequence of coarse models converges to the same upper bound with proven finite steps. This is achieved by adding violated constraints to coarse models until all constraints in the semi-coarse model are satisfied. We demonstrate the effectiveness of our approach in cogeneration for buildings. The coarsened models allow us to obtain good approximate solutions at a fraction of the time required by solving the original problem. Extensive numerical experiments show that the two-level approach scales to large problems that are beyond the capacity of state-of-the-art commercial MILP solvers.

Appendices

Appendix 1: Periodic solutions

In this appendix, we show that the semi-coarse model is equivalent to the fine model under the idealistic assumption that the problem data is periodic and there is no coupling between time periods. Moreover, we show that the coarse model is equivalent to the semi-coarse model if the non-negative weights in the convex combination of constraints form a full rank matrix. This equivalence motivates the use of our algorithm on problems that have “nearly” periodic data.

Let us consider periodic coefficients in the objective function and constraints; that is, we can divide the second-stage data into $n=N/\delta$ identical parts,

$$\begin{aligned} b = \left( \begin{array}{c}\mathbf{b}\\ \vdots \\ \mathbf{b}\end{array}\right) , \quad c = \left( \begin{array}{c}\mathbf{c}\\ \vdots \\ \mathbf{c}\end{array}\right) , \quad d = \left( \begin{array}{c}\mathbf{d}\\ \vdots \\ \mathbf{d}\end{array}\right) , \quad f = \left( \begin{array}{c}\mathbf {f} \\ \vdots \\ \mathbf {f} \end{array}\right) , \quad A = \left( \begin{array}{c}\mathbf{A}\\ \vdots \\ \mathbf{A}\end{array}\right) , \end{aligned}$$

(6.1)

where $\mathbf{b}, \mathbf{c}, \mathbf{d}\in \mathbb {R}^\delta$, $\mathbf{f} \in \mathbb {R}^\gamma$, and $\mathbf{A}\in \mathbb {R}^{\gamma \times m}$. Similarly, we assume (without loss of generality) that the first-stage decisions impact all n time periods identically. We also assume that the second-stage matrices B, C, and D are block diagonal:

$$\begin{aligned} B = \left[ \begin{array}{ccc} \mathbf{B}&{}&{} \\ &{} \ddots &{} \\ &{}&{} \mathbf{B}\\ \end{array} \right] , \quad C = \left[ \begin{array}{ccc} \mathbf{C}&{}&{} \\ &{} \ddots &{} \\ &{}&{} \mathbf{C}\\ \end{array} \right] , \quad D = \left[ \begin{array}{ccc} \mathbf{D}&{}&{} \\ &{} \ddots &{} \\ &{}&{} \mathbf{D}\\ \end{array} \right] , \end{aligned}$$

(6.2)

where $\mathbf{B}, \mathbf{C}, \mathbf{D}\in \mathbb {R}^{\gamma \times \delta }$ with $N = n \delta$ and $M = n \gamma$.

Lemma 2

Under the periodic data assumption (6.1)–(6.2), there exists an optimal solution to (1.1) that satisfies

$$\begin{aligned} \mathbf{x}^\star = \mathbf{x}_i^\star , \quad \mathbf{v}^\star = \mathbf{v}_i^\star , \quad \mathbf{w}^\star = \mathbf{w}_i^\star , \quad \hbox { for } \; i = 1,\ldots ,n. \end{aligned}$$

(6.3)

Proof

Suppose that $(x^\star ,v^\star ,w^\star )$ is an optimal solution that does not satisfy (6.3). That is, there exist $(\mathbf{x}_q^\star , \mathbf{v}_q^\star , \mathbf{w}_q^\star ) \ne (\mathbf{x}_p^\star , \mathbf{v}_p^\star , \mathbf{w}_p^\star )$ such that $\mathbf{b}^T \mathbf{x}_q^\star + \mathbf{c}^T \mathbf{v}_q^\star + \mathbf{d}^T \mathbf{w}_q^\star \, \le \, \mathbf{b}^T \mathbf{x}_p^\star + \mathbf{c}^T \mathbf{v}_p^\star + \mathbf{d}^T \mathbf{w}_p^\star$ and $\mathbf{A}y + \mathbf{B}\mathbf{x}_i + \mathbf{C}\mathbf{v}_i + \mathbf{D}\mathbf{w}_i \le \mathbf{f}$ for $i=p$, q. Then one can construct another solution by replacing the set of variables $(\mathbf{x}_p^\star , \mathbf{v}_p^\star , \mathbf{w}_p^\star )$ with $(\mathbf{x}_q^\star , \mathbf{v}_q^\star , \mathbf{w}_q^\star )$, and repeat this process until the periodic solution (6.3) is obtained. $\square$

From Lemma 2, it follows that the periodic version of (1.1) can be equivalently written as

$$\begin{aligned} \begin{array}{ll} \displaystyle \mathop {\hbox {minimize}}_{\mathbf{v},\mathbf{w},\mathbf{x},y} &{} \displaystyle a^T y + n ( \mathbf{b}^T \mathbf{x}+ \mathbf{c}^T \mathbf{v}+ \mathbf{d}^T \mathbf{w}) \\ \hbox {subject to }&{} \mathbf{A}y + \mathbf{B}\mathbf{x}+ \mathbf{C}\mathbf{v}+ \mathbf{D}\mathbf{w}\le \mathbf {f} \\ &{} y \in \mathbb {Z}_+^m, \; \mathbf{x}\in \{ 0,1\}^\delta \\ &{} \mathbf{v}\in \mathbb {R}^\delta , \; L \mathbf{x}\le \mathbf{v}\le U \mathbf{x}\\ &{} \mathbf{w}\in \mathbb {R}^\delta , \; \mathbf{w}\ge 0 . \end{array} \end{aligned}$$

(6.4)

In other words, it suffices to solve the periodic version of (1.1) with the single period problem (6.4) under condition (6.1)–(6.2), that is, if the data is periodic and if there is no coupling between time periods.

In our algorithm, we obtain the $\delta$-profiles in (2.9) for $\bar{X},\bar{V},\bar{W}$ by solving the following short-horizon problem:

$$\begin{aligned} \begin{array}{ll} \displaystyle \mathop {\hbox {minimize}}_{\mathbf{v}_i,\mathbf{w}_i,\mathbf{x}_i,y} &{} \displaystyle a^T y + \frac{n}{h} \left( \,\, \sum _{i \in \mathcal {D}} (\mathbf{b}^T \mathbf{x}_i + \mathbf{c}^T \mathbf{v}_i + \mathbf{d}^T \mathbf{w}_i) \right) \\ \hbox {subject to }&{} \mathbf{A}y + \mathbf{B}\mathbf{x}_i + \mathbf{C}\mathbf{v}_i + \mathbf{D}\mathbf{w}_i \le \mathbf{f} , \quad i \in \mathcal {D} \\ &{} y \in \mathbb {Z}_+^m, \; \mathbf{x}_i \in \{ 0,1\}^\delta , \quad i \in \mathcal {D} \\ &{} \mathbf{v}_i \in \mathbb {R}^\delta , \; L \mathbf{x}_i \le \mathbf{v}_i \le U \mathbf{x}_i, \quad i \in \mathcal {D} \\ &{} \mathbf{w}_i \in \mathbb {R}^\delta , \; \mathbf{w}_i \ge 0 , \quad i \in \mathcal {D} , \end{array} \end{aligned}$$

(6.5)

where $\mathcal {D}$ is a small set of periods and h is the number of periods in $\mathcal {D}$. (A more detailed description of the moving-horizon approach is provided in Sect. 3.4.) Similar to Lemma 2, we can show that the solution of (6.5) is equivalent to (6.4), independent of the choice of $\mathcal {D}$.

We do not assume uniqueness of the periodic solution. However, since (6.4) and (6.5) are equivalent, we can assume that any solver returns the same solution no matter how we choose $\mathcal {D}$. Hence, we will assume without loss of generality that a single periodic profile, $({\bar{X}}, {\bar{V}}, {\bar{W}})$, is used for the semi-coarse model. By construction, we have

$$\begin{aligned} {\bar{X}}\in \{0,1\}^\delta , \quad L {\bar{X}}\le {\bar{V}}\le U {\bar{X}}, \quad {\bar{W}}\ge 0, \quad \hbox {and} \quad \mathbf{A}y + \mathbf{B}{\bar{X}}+ \mathbf{C}{\bar{V}}+ \mathbf{D}{\bar{W}}\le \mathbf{f}. \end{aligned}$$

(6.6)

Substituting $\mathbf{x}_i = x_{i} {\bar{X}}$, $\mathbf{v}_i = v_{i} {\bar{V}}$, and $\mathbf{w}_i = w_{i} {\bar{W}}$ into the periodic version of (1.1) yields the semi-coarse model

$$\begin{aligned} \begin{array}{ll} \displaystyle \mathop {\hbox {minimize}}_{v,w,x,y} &{} \displaystyle a^T y + \sum _{i=1}^{n} \left( x_{i} \mathbf{b}^T {\bar{X}}+ v_{i} \mathbf{c}^T {\bar{V}}+ w_{i} \mathbf{d}^T {\bar{W}}\right) \\ \hbox {subject to }&{} \displaystyle \mathbf{A}y + \left( x_{i} \mathbf{B}{\bar{X}}+ v_{i} \mathbf{C}{\bar{V}}+ w_{i} \mathbf{D}{\bar{W}}\right) \le \mathbf{f}, \quad i=1,\ldots ,n \\ &{} y \in \mathbb {Z}_+^m, \quad x_{i} \in \{ 0,1\}, \quad v_{i} = x_{i}, \quad w_{i} \in [0,1], \quad i=1,\ldots ,n. \\ \end{array} \end{aligned}$$

(6.7)

Proposition 4

The solution of the semi-coarse model (6.7) is the optimal solution of (6.4) under the periodic data assumption (6.1)–(6.2).

Proof

Since $({\bar{X}},{\bar{V}},{\bar{W}})$ is an optimal profile, that is, the periodic solution of the short-horizon problem (6.5), then setting $x_{i}=v_{i}=w_{i}=1$ in (6.7) in conjunction with (6.6) reduces the semi-coarse model to the short-horizon problem (6.5) with the optimal profile $({\bar{X}},{\bar{V}},{\bar{W}})$, which is equivalent to (6.4).

We next consider a convex combination of (row) constraints

$$\begin{aligned} \mathbf{A}y + x_{i} \mathbf{B}{\bar{X}}+ v_{i} \mathbf{C}{\bar{V}}+ w_{i} \mathbf{D}{\bar{W}}\le \mathbf{f}, \quad i = 1,\ldots ,n \end{aligned}$$

(6.8)

to define the coarse model constraints

$$\begin{aligned} \lambda _i^T \mathbf{A}y + x_{i} \lambda _i^T \mathbf{B}{\bar{X}}+ v_{i} \lambda _i^T \mathbf{C}{\bar{V}}+ w_{i} \lambda _i^T \mathbf{D}{\bar{W}}\le \lambda _i^T \mathbf{f} , \quad i = 1,\ldots ,n \end{aligned}$$

(6.9)

where the non-negative weights $\lambda _i \in \mathbb {R}^\delta$ sum up to 1 for $i=1,\ldots ,n$. We can easily show that if $\lambda _i$ in (6.9) is such that $\varLambda = [\lambda _1 \cdots \lambda _n] \in \mathbb {R}_+^{\delta \times n}$ is a full row-rank matrix, then the set of aggregated constraints (6.9) is equivalent to (6.8). This observation leads to the following result.

Proposition 5

Let the non-negative weights $\lambda$’s be such that $[\lambda _1,\ldots ,\lambda _n]$ has full row rank. Then the solution of the coarse model is feasible (hence optimal) to the solution of the semi-coarse model (6.7):

$$\begin{aligned} \begin{array}{ll} \displaystyle \mathop {\hbox {minimize}}_{v,w,x,y} &{} \displaystyle a^T y + \sum _{i=1}^{n} \left( x_{i} \mathbf{b}^T {\bar{X}}+ v_{i} \mathbf{c}^T {\bar{V}}+ w_{i} \mathbf{d}^T {\bar{W}}\right) \\ \hbox {subject to }&{} \displaystyle \mathbf{A}y + x_{i} \lambda ^T_i \mathbf{B}{\bar{X}}+ v_{i} \lambda ^T_i \mathbf{C}{\bar{V}}+ w_{i} \lambda ^T_i \mathbf{D}{\bar{W}}\le \lambda ^T_i \mathbf{f}, \quad i=1,\ldots ,n \\ &{} y \in \mathbb {Z}_+^m, \quad x_{i} \in \{ 0,1\}, \quad v_{i} = x_{i}, \quad w_{i} \in [0,1], \quad i=1,\ldots ,n. \\ \end{array} \end{aligned}$$

(6.10)

Thus, we have shown that if the data is periodic as in (6.1)–(6.2), then the full, semi-coarse, and coarse models are equivalent. Hence, our approach converges in one iteration. Our argument has assumed that there is no coupling between successive periods, resulting in a block-diagonal structure of B, C, and D. In the application of the cogeneration problem in Sect. 3, there exists coupling in time, for example, in the form of storage in the water tanks. In practice, the commercial buildings we consider indeed have periodic diurnal patterns, and hence coupling between successive days is relatively weak. We believe that our approach can be useful in cases where interdaily coupling is small.

Appendix 2: Full model for the cogeneration problem

For the sake of completeness, we describe here our cogeneration model.

Index sets. We use calligraphic upper-case letters for all sets. The set of technologies: batteries, boilers, CHP-SOFC, power SOFC, and water tank storage is denoted by $\mathcal {J}$. We use batt, boil, pow, chp, and stor for shortcuts. The subset $\mathcal {J}_g = \{ \hbox {power SOFC, CHP-SOFC} \}$ denotes SOFC generation technologies that require on/off operations at the second stage. Each technology $j \in \mathcal {J}_g$ has a number of identical units, and the index set is denoted by $\mathcal {U}_j$. $\mathcal {T}$ is the index set of time, $\mathcal {M}$ is the index set of months, and $\mathcal {T}_m \subset \mathcal {T}$ is the index set of time for each month $m \in \mathcal {M}$.

Variables. We use lower-case letters to denote variables. The first-stage variables in our model are the number of units of technology $j \in \mathcal {J}$, which we denote by $y_j \in \mathbb {Z}_+$. All other variables model operation of the installed units, and they are our second-stage variables. We use the subscript t to indicate the time period: $b_t$ and $b_t^\mathrm{IO}$ are the power storage and input/output for batteries, respectively; $s_t$, and $s_t^\mathrm{out}$ are the heat storage and output for water tanks, respectively; $p_{jt}$ and $q_{jt} \ge 0$ are the power and heat output of technology j, respectively; $u_t \ge 0$ is the power purchased from the utility and $u_m^\mathrm{max} \ge 0$ is the maximum power purchased in month m; $x_{ijt} \in \{0,1\}$ indicates whether unit i of technology j operates in time period t; and $w_{ijt} \in [0,1]$ is an auxiliary variable to indicate whether unit i of technology j switches from t to $t+1$. Note that we do not impose a binary constraint on $w_{ijt}$ because it takes a binary value at the solution due to the switching constraint (7.1g) and the binary constraint $x_{ijt} \in \{0,1\}$. We use the convention $t \in \mathcal {T}$, $j \in \mathcal {J}$, $i \in \mathcal {U}_j$, $m \in \mathcal {M}$ unless explicitly mentioned.

Parameters. We use upper-case letters to denote parameters and constants: $C_j$ is the capital and installation cost of technology j; H is the number of hours in the lifetime of technology (e.g., $H = 87600$ for a ten-year model); T is the number of hours in the problem horizon; Y is the hourly discount factor with $3\,\%$ annual interest rate (i.e., $Y = 1 - 0.03/8760$); I is the hourly increase factor with $10\,\%$ annual increase rate (i.e., $I^{8760} = 1.1$); $M_{j}$ is the operation and maintenance cost of technology j; $P_t$ and $G_t$ are the price for electricity and natural gas, respectively; $P_m^\mathrm{max}$ is the peak demand price for power from the utility in month m; $W_j$ is the cost of switching a unit of technology j on or off; $S^c_j$ is the thermal capacity per unit for technology j; $B^c$ is the power capacity per unit for batteries; $D_t^P$ and $D_t^Q$ are the power and heat demand in period t, respectively; $R^\mathrm{min}_j$ and $R^\mathrm{max}_j$ are the minimum and maximum power output when a unit of technology j is turned on, respectively; $E_j^P$ and $E_j^Q$ are the power and thermal efficiency of technology j, respectively; and $L^P$ and $L^Q$ are the average power loss in battery and heat loss in water tank storage, respectively.

Cost function. We now formulate the complete MILP model for the cogeneration problem in (7.1). The objective function consists of two parts: the capital and installation cost in the first stage and the operation cost in the second stage. The operation cost includes the peak power usage, operation and maintenance, switching units, gas, and purchased power costs. We discount the second-stage costs with an hourly discount factor based on a $3\,\%$ annual interest rate; hence $Y=1-0.03/8760$. Therefore, the cost incurred in hour t is multiplied by the discount factor $Y^t$.³ The peak demand charge at the end of month m is discounted with $Y^{t_m}$.We take into account a $10\,\%$ annually increase of the maintenance and switching costs over time. This is done by multiplying $I^t$ to cost terms in (7.1a) where $I = 1 + 1.088 \times 10^{-5}$. To compare cost over different time horizons T, we compute the average hourly cost and multiply the result with the lifetime of technologies, H, resulting in the factor H / T in the second-stage cost in (7.1).

Constraints. The constraints in the MILP model (7.1) can be divided into three groups. The first group of constraints couples first- and second-stage variables; in particular, the number of on-units for fuel cells and the capacity of battery, storage, and boiler are bounded by the number of units purchased in the first stage, as described in (7.1e), (7.1k), (7.1o), and (7.1p), respectively. The second group of constraints couples variables across technologies; in particular, technologies are coordinated to meet the power demand (7.1c) and the heat demand (7.1l). We note that dualizing these constraints results in decoupled variables for each technology; see, for example,  [3, 14]. However, MILP (7.1) does not lend itself to this Lagrangian relaxation approach because of coupling constraints in the third group. This group of constraints couples variables over time periods in the second stage. In particular, the battery and storage constraints (7.1i) and (7.1m) link variables over the entire horizon. The heat storage constraint (7.1m) is a combination of two constraints, namely, the storage equation $s_{t+1} = (1 - L^Q) s_{t} - s^\mathrm{out}_{t} + E_\mathrm{chp}^Q s_t^\mathrm{in}$ and the upper bound on the heat generated by CHP-SOFC, $s_t^\mathrm{in} \le p_{\mathrm{chp},t}/E_\mathrm{chp}^P$. Eliminating the heat input $s_t^\mathrm{in}$ yields (7.1m). In addition, constraint (7.1h), which models the maximum power purchased from the utility in each months, links variables over time instances in each month. Moreover, constraint (7.1g), which models on/off switches of the generation units, couples binary variables $x_{ijt}$ over two immediate time instances. Boundary conditions (7.1j) and (7.1n) for battery and storage couple variables at both ends of the entire horizon.

We conclude this subsection by explaining the rest of the constraints in (7.1). The number of purchased units is upper bounded in (7.1b). Since all units of the same technology are identical, we introduce a symmetry breaking constraint (7.1f) that states that if unit $i+1$ is on, then unit i must be on as well. When unit i of technology j is turned on, the power generated $p_{ijt}$ is bounded by the minimum and maximum capacity in (7.1d). The non-negativity constraint for variables is given by (7.1q).

$$\begin{aligned} \mathop {\hbox {minimize}}\quad&{} \displaystyle \sum _{j \in \mathcal {J}} C_j y_j + \frac{H}{T} \left\{ \sum _{m \in \mathcal {M}} Y^{t_m} P_m^{\mathrm{max}} u_m^{\mathrm{max}}\Bigg /\right. \nonumber \\&\left. + \displaystyle \sum _{t \in \mathcal {T}} Y^t \left[ \sum _{j \in \mathcal {J}_g} ({I^t \cdot M_{j} p_{jt} + \displaystyle \sum _{i \in \mathcal {U}_j} I^ t \cdot W_j w_{ijt}}) + G_t q_{t} + P_t u_t \right] \right\} \end{aligned}$$

(7.1a)

$$\begin{aligned} \hbox {subject to }\quad&y_j \in \mathbb {Z}, ~ 0 \le y_j \le |\mathcal {U}_j| \end{aligned}$$

(7.1b)

$$\begin{aligned}&\displaystyle \sum _{j \in \mathcal {J}_g} p_{jt} + u_t - b_t^\mathrm{IO} \ge D_t^P \end{aligned}$$

(7.1c)

$$\begin{aligned}&\displaystyle R_j^\mathrm{min} x_{ijt} \le p_{ijt} \le R_j^\mathrm{max} x_{ijt}, \quad p_{jt} = \displaystyle \sum _{i \in \mathcal {U}_j} p_{ijt} \qquad j \in \mathcal {J}_g \end{aligned}$$

(7.1d)

$$\begin{aligned}&x_{ijt} \in \{0,1\}, \quad \displaystyle \sum _{i \in \mathcal {U}_j} x_{ijt} \le y_j \qquad j \in \mathcal {J}_g \end{aligned}$$

(7.1e)

$$\begin{aligned}&x_{(i+1)jt} \le x_{ijt} \qquad j \in \mathcal {J}_g, i < |\mathcal {U}_j| \end{aligned}$$

(7.1f)

$$\begin{aligned}&x_{ij(t+1)} - x_{ijt} \le w_{ijt}, \, x_{ijt} - x_{ij(t+1)} \le w_{ijt}, \, 0 \le w_{ijt} \le 1, \; \; \qquad \nonumber \\&\quad j \in \mathcal {J}_g, t < |\mathcal {T}| \end{aligned}$$

(7.1g)

$$\begin{aligned}&0 \le u_t \le u_m^\mathrm{max}\qquad t \in \mathcal {T}_m \end{aligned}$$

(7.1h)

$$\begin{aligned}&b_{t+1} = (1 - L^P) b_{t} + b_t^\mathrm{IO} \qquad t < |\mathcal {T}| \end{aligned}$$

(7.1i)

$$\begin{aligned}&b_1 \,=\, b_{|\mathcal {T}|} \end{aligned}$$

(7.1j)

$$\begin{aligned}&0 \le b_t \le B^c y_\mathrm{batt} \end{aligned}$$

(7.1k)

$$\begin{aligned}&s^\mathrm{out}_{t} + q_{t} \ge D_t^Q \end{aligned}$$

(7.1l)

$$\begin{aligned}&s_{t+1} - (1 - L^Q) s_{t} + s^\mathrm{out}_{t} \le (E_\mathrm{chp}^Q /E_\mathrm{chp}^P) p_{\mathrm{chp},t} \qquad t < |\mathcal {T}| \end{aligned}$$

(7.1m)

$$\begin{aligned}&s_{1} = s_{|\mathcal {T}|} \end{aligned}$$

(7.1n)

$$\begin{aligned}&s^\mathrm{out}_{t} \le s_{t} \le S^c_\mathrm{stor} y_\mathrm{stor} \end{aligned}$$

(7.1o)

$$\begin{aligned}&q_{t} \le S^c_\mathrm{boil} y_\mathrm{boil} \end{aligned}$$

(7.1p)

$$\begin{aligned}&b_t, p_{jt}, q_{t}, s_t, s_t^\mathrm{out}, u_t \ge 0 \end{aligned}$$

(7.1q)

Appendix 3: Tables of numerical results

Table 7 Problem size and AMPL/CPLEX solution information for the cogeneration problem (7.1) for five buildings

Table 8 Problem size and AMPL/CPLEX solution information for the semi-coarse model (9.1) for five buildings

Table 9 Problem size and AMPL/CPLEX solution information for the first iteration of the coarse model (10.1) for five buildings

Table 10 Comparison between the full model (7.1) and the semi-coarse model (9.1) for five buildings

Table 11 Comparison between the full model (7.1) with and without fixing the first-stage solutions from the semi-coarse model (9.1) for five buildings

Appendix 4: Semi-coarse model for the cogeneration problem

$$\begin{aligned} \mathop {\hbox {minimize}}\quad&\displaystyle \sum _{j \in \mathcal {J}} C_j y_j + \frac{H}{T} \left\{ \sum _{m \in \mathcal {M}} Y^{t_m} P_m^\mathrm{max} u_m^\mathrm{max} + \displaystyle \sum _{d} \bar{Y}_d^T \left[ \sum _{j,i,k,l} \bar{p}_{ijdkl} M_{j} \bar{P}_{jkl} \right. \right. \nonumber \\&\quad \left. \left. + \displaystyle \sum _{d,j,i,k} \bar{x}_{ijdk} W_j \bar{W}_{jk} + \sum _{d, k \in \mathcal {P}_q} \bar{q}_{dk} \bar{Q}_{k} + \sum _{d,k \in \mathcal {P}_u} \bar{u}_{dk} \bar{U}_{k} \right] \right\} \end{aligned}$$

(9.1a)

$$\begin{aligned} \hbox {subject to }\quad&y_j \in \mathbb {Z}, ~ 0 \le y_j \le |\mathcal {U}_j| \qquad j \in \mathcal {J}\end{aligned}$$

(9.1b)

$$\begin{aligned}&\displaystyle \sum _{j,i,k,l} \bar{p}_{ijdkl} \bar{P}_{jkl} + \sum _{k \in \mathcal {P}_u} \bar{u}_{dk} \bar{U}_{k} - \sum _{k \in \mathcal {P}_b} \bar{b}_{dk} \bar{B}_{k}^\mathrm{IO} \ge \bar{D}_d^P \end{aligned}$$

(9.1c)

$$\begin{aligned}&\bar{x}_{ijdk} \in \{0,1\}, ~ \sum _{k \in \mathcal {P}_o} \bar{x}_{ijdk} \le 1 \end{aligned}$$

(9.1d)

$$\begin{aligned}&\displaystyle \sum _{i,k} \bar{x}_{ijdk} \bar{X}_{jk} \le y_j \mathbf{1} \end{aligned}$$

(9.1e)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_o} \bar{x}_{(i+1)jdk} \bar{X}_{jk} \le \sum _{k \in \mathcal {P}_o} \bar{x}_{ijdk} \bar{X}_{jk}\qquad i < |\mathcal {U}_j| \end{aligned}$$

(9.1f)

$$\begin{aligned}&\bar{p}_{ijdk} \in [0,1], ~ \sum _{l \in \mathcal {P}_p} \bar{p}_{ijdkl} = \bar{x}_{ijdk}, ~ \sum _{k \in \mathcal {P}_o} \sum _{l \in \mathcal {P}_p} \bar{p}_{ijdkl} \le 1 \end{aligned}$$

(9.1g)

$$\begin{aligned}&0 \le \sum _{k \in \mathcal {P}_u} \bar{u}_{dk} \bar{U}_{k} \le u_m^\mathrm{max} \mathbf{1}\qquad d \in \mathcal {D}_m \end{aligned}$$

(9.1h)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_b} \bar{b}_{(d+1)k} \bar{B}_k(1) = \sum _{k \in \mathcal {P}_b} \bar{b}_{dk} \big [ (1-L^P) \bar{B}_k(\delta ) + \bar{B}_k^\mathrm{IO}(\delta ) \big ]\qquad d < |\mathcal {D}| \end{aligned}$$

(9.1i)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_b} \bar{b}_{1k} \bar{B}_{k}(1) \,=\, \sum _{k \in \mathcal {P}_b} \bar{b}_{|\mathcal {D}|k} \bar{B}_{k}(\delta ) \end{aligned}$$

(9.1j)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_b} \bar{b}_{dk} \bar{B}_k \le B^c y_\mathrm{batt} \mathbf{1} \end{aligned}$$

(9.1k)

$$\begin{aligned}&\displaystyle \sum _{k \in \mathcal {P}_s} \bar{s}_{dk} \bar{S}^\mathrm{out}_{k} + \sum _{k \in \mathcal {P}_q} \bar{q}_{dk} \bar{Q}_{k} \ge \bar{D}_d^Q \end{aligned}$$

(9.1l)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_s} \bar{s}_{dk} \big [ S_\delta - (1-L^Q)I_\delta \big ] \bar{S}_{k} + \sum _{k \in \mathcal {P}_s} \bar{s}_{(d+1)k} E_\delta \bar{S}_{k}\qquad d < |\mathcal {D}| \nonumber \\&\quad + \sum _{k \in \mathcal {P}_s} \bar{s}_{dk} \bar{S}^\mathrm{out}_{k} \le (E_{j}^Q /E_{j}^P) \sum _{i,k,l} \bar{p}_{ijdkl} \bar{P}_{jkl} \qquad j = \mathrm{chp} \end{aligned}$$

(9.1m)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_s} \bar{s}_{1k} \bar{S}_{k}(1) \,=\, \sum _{k \in \mathcal {P}_s} \bar{s}_{|\mathcal {D}|k} \bar{S}_{k}(\delta ) \end{aligned}$$

(9.1n)

$$\begin{aligned}&\displaystyle \sum _{k \in \mathcal {P}_s} \bar{s}_{dk} \bar{S}^\mathrm{out}_{k} \le S^c_\mathrm{stor} y_\mathrm{stor} \mathbf{1} \end{aligned}$$

(9.1o)

$$\begin{aligned}&\displaystyle \sum _{k \in \mathcal {P}_q} \bar{q}_{dk} \bar{Q}_{k} \le S^c_\mathrm{boil} y_\mathrm{boil} \mathbf{1} \end{aligned}$$

(9.1p)

$$\begin{aligned}&\bar{u}_{dk}, ~\bar{b}_{dk}, ~\bar{s}_{dk}, ~\bar{q}_{dk} \in [0,1] \end{aligned}$$

(9.1q)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_u} \bar{u}_{dk} \le 1, \sum _{k \in \mathcal {P}_b} \bar{b}_{dk} \le 1, \sum _{k \in \mathcal {P}_s} \bar{s}_{dk} \le 1, \sum _{k \in \mathcal {P}_q} \bar{q}_{dk} \le 1. \end{aligned}$$

(9.1r)

Appendix 5: Coarse model for the cogeneration problem

$$\begin{aligned} \mathop {\hbox {minimize}}\quad&\displaystyle \sum _{j \in \mathcal {J}} C_j y_j + \frac{H}{T} \left\{ \sum _{m \in \mathcal {M}} Y^{t_m} P_m^\mathrm{max} u_m^\mathrm{max} + \displaystyle \sum _{d} \bar{Y}_d^T \left[ \sum _{j,i,k,l} \bar{p}_{ijdkl} M_{j} \bar{P}_{jkl}\right. \right. \nonumber \\&\quad \left. \left. + \displaystyle \sum _{d,j,i,k} \bar{x}_{ijdk} W_j \bar{W}_{jk} + \sum _{d,k \in \mathcal {P}_u} \bar{u}_{dk} \bar{U}_{k} + \sum _{d, k \in \mathcal {P}_q} \bar{q}_{dk} \bar{Q}_{k} \right] \right\}&\end{aligned}$$

(10.1a)

$$\begin{aligned} \hbox {subject to }\quad&y_j \in \mathbb {Z}, ~ 0 \le y_j \le |\mathcal {U}_j| \qquad j \in \mathcal {J}\end{aligned}$$

(10.1b)

$$\begin{aligned}&\displaystyle \sum _{j,i,k,l} \bar{p}_{ijdkl} \hat{\bar{P}}_{jkl} + \sum _{k \in \mathcal {P}_u} \bar{u}_{dk} \hat{\bar{U}}_{k} - \sum _{k \in \mathcal {P}_b} \bar{b}_{dk} \hat{\bar{B}}_{k}^\mathrm{IO} \ge \hat{\bar{D}}_d^{P} \end{aligned}$$

(10.1c)

$$\begin{aligned}&\bar{x}_{ijdk} \in \{0,1\}, ~ \sum _{k \in \mathcal {P}_o} \bar{x}_{ijdk} \le 1 \end{aligned}$$

(10.1d)

$$\begin{aligned}&\displaystyle \sum _{i,k} \bar{x}_{ijdk} \hat{\bar{X}}_{jk} \le \delta \cdot y_j \end{aligned}$$

(10.1e)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_o} \bar{x}_{ijdk} \hat{\bar{X}}_{jk} \ge \sum _{k \in \mathcal {P}_o} \bar{x}_{(i+1)jdk} \hat{\bar{X}}_{jk}\qquad i< |\mathcal {U}_j| \end{aligned}$$

(10.1f)

$$\begin{aligned}&\bar{p}_{ijdk} \in [0,1], ~ \sum _{l \in \mathcal {P}_p} \bar{p}_{ijdkl} = \bar{x}_{ijdk}, ~ \sum _{k \in \mathcal {P}_o} \sum _{l \in \mathcal {P}_p} \bar{p}_{ijdkl} \le 1 \end{aligned}$$

(10.1g)

$$\begin{aligned}&0 \le \sum _{k \in \mathcal {P}_u} \bar{u}_{dk} \hat{\bar{U}}_{k} \le \delta \cdot u_m^\mathrm{max} \qquad d\in \mathcal {D}_m \end{aligned}$$

(10.1h)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_b} \bar{b}_{(d+1)k} \bar{B}_k(1) = \sum _{k \in \mathcal {P}_b} \bar{b}_{dk} \left( (1-L^P) \bar{B}_k(\delta ) + \bar{B}_k^\mathrm{IO}(\delta )\right) \quad d < |\mathcal {D}| \end{aligned}$$

(10.1i)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_b} \bar{b}_{1k} \bar{B}_{k}(1) \,=\, \sum _{k \in \mathcal {P}_b} \bar{b}_{|\mathcal {D}|k} \bar{B}_{k} (\delta ) \end{aligned}$$

(10.1j)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_b} \bar{b}_{dk} \hat{\bar{B}}_k \le \delta \cdot B^c y_\mathrm{batt} \end{aligned}$$

(10.1k)

$$\begin{aligned}&\displaystyle \sum _{k \in \mathcal {P}_s} \bar{s}_{dk} \hat{\bar{S}}^\mathrm{out}_{k} + \sum _{k \in \mathcal {P}_q} \bar{q}_{dk} \hat{\bar{Q}}_{k} \ge \hat{\bar{D}}_d^Q \end{aligned}$$

(10.1l)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_s} \bar{s}_{dk} (L^Q\hat{\bar{S}}_{k}-\bar{S}_k(1)) + \sum _{k \in \mathcal {P}_s} \bar{s}_{(d+1)k} \bar{S}_{k}(1) \qquad d < |\mathcal {D}| \nonumber \\&+ \sum _{k \in \mathcal {P}_s} \bar{s}_{dk} \hat{\bar{S}}^\mathrm{out}_{k} \le (E_{j}^Q /E_{j}^P) \sum _{i,k,l} \bar{p}_{ijdkl} \hat{\bar{P}}_{jkl} \qquad j = \mathrm{chp}\end{aligned}$$

(10.1m)

$$\begin{aligned}&\sum _{k \in \mathcal {P}_s} \bar{s}_{1k} \bar{S}_{k}(1) \,=\, \sum _{k \in \mathcal {P}_s} \bar{s}_{|\mathcal {D}|k} \bar{S}_{k} (\delta ) \end{aligned}$$

(10.1n)

$$\begin{aligned}&\displaystyle \sum _{k \in \mathcal {P}_s} \bar{s}_{dk} \hat{\bar{S}}^\mathrm{out}_{k} \le \delta \cdot S^c_\mathrm{stor} y_\mathrm{stor} \end{aligned}$$

(10.1o)

$$\begin{aligned}&\displaystyle \sum _{k \in \mathcal {P}_q} \bar{q}_{dk} \hat{\bar{Q}}_{k} \le \delta \cdot S^c_\mathrm{boil} y_\mathrm{boil} \end{aligned}$$

(10.1p)

$$\begin{aligned}&\bar{u}_{dk}, ~\bar{b}_{dk}, ~\bar{s}_{dk}, ~\bar{q}_{dk} \in [0,1]\nonumber \\&\sum _{k \in \mathcal {P}_u} \bar{u}_{dk} \le 1, \sum _{k \in \mathcal {P}_b} \bar{b}_{dk} \le 1, \sum _{k \in \mathcal {P}_s} \bar{s}_{dk} \le 1, \sum _{k \in \mathcal {P}_q} \bar{q}_{dk} \le 1. \end{aligned}$$

(10.1q)