1. Introduction
Heating, ventilation and airconditioning (HVAC) systems typically utilize a large percentage of the total building energy consumption, amounting to approximately 25–30% for dwellings [
1] and up to more than 50% in industries relying on cleanrooms [
2]. Between 40% and 60% of this requirement is due to chillers, which are therefore responsible for a significant fraction of total energy use. Such numbers indicate that the efficiency of HVAC systems is closely dependent on the efficiency of the chiller unit. Since a multiple chiller system typically employs machines, there are usually several combinations of chillers’ part loads that are able to satisfy the load demand. The problem of determining the load fraction that each chiller has to deliver in order to minimize the system power consumption is known as optimal chiller loading (OCL) problem. In the last decade, several methods have been proposed to solve the OCL problem [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19]. Chang et al. [
3] assume that the chillers’ efficiency is a quadratic function of the partialization and maximize the sum of chillers’ coefficients of performance (COP) using the Lagrangian method. Of course, maximizing the sum of the efficiencies is not equivalent to the “canonical” OCL problem, where system power consumption is instead minimized. In a second paper, Chang [
4] has addressed the canonical OCL problem again by the Langrangian multipliers methods, assuming that all the chiller power consumption curves are convex and cubic in the part load ratio. The poor convergence properties of the lambda iteration at low cooling load demand were obviated by a suitable gradient method.
Nevertheless, it is wellknown that the chillers’ power consumption could be a concave function, which makes OCL an NPhard mixedinteger problem and poses a further challenge to the search of the optimal solution as global convergence of iterative methods cannot be guaranteed. This has motivated the development of several heuristic methods which do not guarantee optimality, but are effective to obtain a fair solution in a viable execution time.
Assuming quadratic models of power consumption, Geem [
5] resorts to the generalized reduced gradient (GRG) method. Salari et al. [
6] show that the mixed integer problem can be solved using the general algebraic modeling system (GAMS). In parallel, many natureinspired heuristic algorithms have been proposed, namely, genetic algorithm (GA) [
7,
8], simulated annealing (SA) [
9,
10], particle swarm optimization (PSO) [
11,
12], evolution strategy (ES) [
13], differential evolution (DE) [
14], cuckoo search algorithm using differential operator (DCSA) [
15], differential search (DS) [
16], improved firefly algorithm (IFA) [
17], teachinglearningbased optimization (TLBO) [
18], improved invasive weed optimization (EIWO) [
19], distributed chaotic estimation of distribution algorithm (DCEDA) [
20], imperialist competitive algorithm (ICA) [
21] and wild goat’s algorithm (WGA) [
22].
Many of these algorithms have been tested on the Hsinchu benchmark, a widely used case study consisting of a sixchiller system installed in a semiconductor factory located in the Hsinchu Scientic Garden (Taiwan). Hence, it is therefore natural to use the performances achieved on this benchmark to compare and rank alternative solution methods. A selection consisting of the six best algorithms is reported in
Table 1. In order of publication, they are IFA (2013), DCSA (2014), GAMS (2015), TLBO (2017), EIWO (2018), DCEDA (2020). Two recent algorithms, ICA (2020) and WGA (2020), claim to have improved over the existing literature. However, an inspection of the chillers’ cooling capacities used in these two papers shows that they are different from those of the standard benchmark so that the claim of improvement is void. It is worth noting that, since an exact solver of the Hsinchu benchmark is not available, the final word on the optimality of solutions found by heuristic algorithms still needs to be had.
In the practical management of a real chiller system, solving the OCL is not sufficient, because there exist further dynamic constraints, namely minimum uptime and downtime requirements on chillers’ operation. When these constraints are accounted for, the power consumption minimization problem goes under the name of optimal chiller sequencing (OCS). Again, this is a problem that is hardly tractable without resorting to some heuristics. In particular, the knowledge of all future cooling loads is required, which raises the problem of forecasting it with reasonable accuracy, a task that can be successfully addressed only on a finite prediction horizon. In the literature, OCS solvers resulting from the combination of heuristic OCL algorithms with dynamic programming schemes have been proposed [
8,
23,
24,
25].
The present paper addresses both the OCL and OCS problems. Concerning the former one, two main issues are investigated. First of all, we derive an exact algorithm when the chillers’ power consumption is a quadratic function of the partial load ratio (PLR), as it happens for the Hsinchu benchmark. Our XOCL algorithm allows one to have the final word on the existing heuristic methods, highlighting also some erroneous results reported in the literature. The second issue has to do with the practical applicability of XOCL to realworld plants. In particular, the execution time is compared with a stateofart mixed integer solver and the adequacy of the quadratic power consumption model is discussed.
Concerning the OCS problem, we exploit the XOCL algorithm to derive a lower bound on the minimum power consumption achievable by any OCS solver. Second, a greedy OCS algorithm leveraging on XOCL is proposed and compared with [
23]. Finally, the lower bound is used to quantitatively assess the degree of suboptimality ensuing from the lack of preview implicit in the greedy approach.
Literature benchmarks, i.e., the Hsinchu one and two OCS benchmarks are used for test and comparison. Moreover, an extensive dataset collected during two years in a semiconductor fab is used to build and demonstrate a comprehensive solution of both OCL and OCS, including the databased estimation of the chillers’ power consumption models. In particular, the potential energy saving with respect with the current HVAC energy management is assessed.
2. The Optimal Chiller Loading Problem
Assuming
n chillers operated in parallel, let
${Q}_{i}$$,\phantom{\rule{0.222222em}{0ex}}i=1,\dots ,n$, denote the cooling power delivered by the
ith chiller and
${P}_{i}$$={P}_{i}\left({Q}_{i}\right),\phantom{\rule{0.222222em}{0ex}}i=1,\dots ,n$ the associated power consumption. For a prescribed overall cooling load demand
${Q}_{load}$, the goal of the OCL problem is finding the cooling powers
${Q}_{i},\phantom{\rule{0.222222em}{0ex}}i=1,\dots ,n$ that the
n chillers have to deliver in order to minimize the system total energy consumption
${P}_{tot}$:
For each chiller, PLR (part load ratio) denotes the cooling load fraction, given by
where
${Q}_{100\%,i}$ is the maximum power supplied under full capacity operation. The vector of all PLRs is denoted by
When the
ith chiller is turned on, it should not operate under a minimum
$PLR$, denoted by
$PL{R}_{min,i},1$. For the subsequent derivations, it is convenient to introduce a binary variable
${\delta}_{i}$ that indicates the status of the
ith chiller and a realvalued variable
${x}_{i}$, such that:
The power consumption ${P}_{i}$ of the ith chiller is assumed to depend mainly on $PL{R}_{i}$ and the condenser inlet water temperature ${T}_{i}$, that is ${P}_{i}={P}_{i}(PL{R}_{i},{T}_{i})$.
The consumption surface ${P}_{i}(PL{R}_{i},{T}_{i})$ is either obtained from laboratory experiments or field data collected during operation.
Note that the problem of finding the optimal part load ratios
${\mathbf{PLR}}^{*}$ can be formulated as a mixedinteger nonlinear program (MINLP):
In the above problem, two types of constraints are present: the cooling demand constraint, Equation (1b) and a set of operational constraints Equation (1c–e) regarding the admissible operating regions of the chillers.
In the following, it is implicitly assumed that the cooling demand constraint is such that the admissible solutions set is nonempty. Then, given that the constraints define a closed set of admissible solution, the cost function admits a minimum.
2.1. Quadratic Power Consumption Model
With the exception of particular structures, mixedinteger programming problems are classified as NPhard, which means that in the worst case, the solution time grows at least exponentially with the problem size. Although its combinatorial nature might suggest the use of heuristics, we will show that a significant subclass of industrial OCL problems, characterized by a quadratic power consumption model quadratic in
$PL{R}_{i}$, may still be successfully attacked by a carefully designed exact method. It is worth asking whether the quadratic assumption of the power consumption is more or less realistic. We will discuss this point later. Even if the consumption is not quadratic, a good quadratic approximation is likely computable. Moreover, in the case in which it is not easy to workout a good quadratic approximation, it can still be addressed by means piecewise quadratic model, as suggested in
Section 6.
Assumption 1. The power consumption ${P}_{i}$ of the ith chiller obeys the following model:where ${\beta}_{p,i}$ are the model parameters and $f(\xb7)$ is a suitable function. Moreover, it is assumed that ${\beta}_{2,i}\ne 0$. When the OCL problem is solved in a given time slot, the condenser inlet water temperature
${T}_{i}$ can be assumed to be known. Then, for a given
${T}_{i}$, in the interval
$[PL{R}_{min},1]$, the consumption surface is a quadratic function of
$PL{R}_{i}$ alone:
with
${a}_{i}={\beta}_{0,i}+f\left({T}_{i}\right)$,
${c}_{i}={\beta}_{1,i}$, and
${q}_{i}={\beta}_{2,i}$.
In view of the quadratic nature of the cost function, the system total energy consumption can be expressed in matrix form as follows:
where
Moreover, the equality constraint (1b) can be rewritten as
where
Equation (4a,b) represent the standard OCL problem with quadratic power consumption, that so far has been attacked by either mixed integer programming solvers or a variety of heuristic algorithms. Differently from these approaches, below we derive an algorithm that computes the exact solution.
2.2. Partition of the Solution Space
In view of (4), the admissible set
$\mathcal{F}$ for
$PL{R}_{i}$ is
where each subset
${\mathcal{F}}_{\sigma}$ is associated with one of the following operating conditions: switched off (
$\sigma =0$), minimum part load (
$\sigma =1$), maximum part load (
$\sigma =2$), intermediate part load (
$\sigma =3$). In the sequel,
${\sigma}_{i}\in \{0,1,2,3\}$ will denote the state of the
$i\text{}$th chiller.
In order to satisfy the operational constraints of a multiple chiller system, the solution of the OCL problem must be searched within the cartesian product of the chillers’ admissible sets, i.e.,
Since the admissible set
$\mathcal{F}$ of a single chiller can be partitioned in four subsets, the overall admissible set
$\mathcal{S}$ can be partitioned in
${4}^{n}$ subsets
${\mathcal{S}}_{j},\phantom{\rule{0.222222em}{0ex}}j=1,\dots ,{4}^{n}$, each of which is in a onetoone correspondence with the
$n$digit multichiller code
formed by the state codes
${\sigma}_{i},\phantom{\rule{0.277778em}{0ex}}i=1,\dots ,n$, of the
n chillers.
To make an example, consider the case of
$n=3$ chillers. Then, the possible
${4}^{n}=64$ subsets
${\mathcal{S}}_{j}$ are associated to the multichiller codes as follows:
The
n elements of the set
${\mathcal{S}}_{j}$ are in a onetoone correspondence with the chiller number
$i,1\le i\le n$. Given the chiller number
i, the notation
${\mathcal{S}}_{j}\left[i\right]$ will denote the operating condition, either a point or a range, of the
ith chiller. For instance, within
${\mathcal{S}}_{8}$, whose multichiller code is
${\mathbf{s}}_{8}=[0,1,3]$, the three chillers
$i=1,2,3$ will operate at the following conditions:
2.3. Divide and Conquer Strategy
For $j=1,\dots ,{4}^{n}$, let $\mathbf{QP}\left(j\right)$ indicate the OCL problem (4) restricted to the subset ${\mathcal{S}}_{j}$.
Problem 1. $\mathbf{QP}\left(j\right)$ The ${4}^{n}$ problems $\mathbf{QP}\left(j\right)$ can be partitioned in two subsets, $\mathcal{C}$ and $\overline{\mathcal{C}}$:
 
$\mathcal{C}=\{j\mid {s}_{ji}\ne 3,\forall i\}$: all the partial load ratios are fixed so that ${\mathcal{S}}_{j}$ has cardinality one;
 
$\overline{\mathcal{C}}$, when there is at least one chiller operating at intermediate part load ($\exists i:{s}_{ji}=3$).
Consider, for example,
${\mathcal{S}}_{7}$, whose multichiller code is
${\mathbf{s}}_{\mathbf{7}}=[0,1,2]$: chiller #1 is switched off, chiller #2 is operating at minimum part load, and chiller #3 is operating at maximum part load. Within this subset, no optimization is actually needed because all the chillers’
$PLR$s are fixed and
has cardinality one. Therefore, only a feasibility check is required: if constraint (8b) is satisfied, then
${\mathcal{S}}_{7}$ is the optimal solution of
$\mathbf{QP}\left(7\right)$. Otherwise,
$\mathbf{QP}\left(7\right)$ does not admit a solution. It is easy to see that the number of elements of the subset
$\mathcal{C}$ is
${3}^{n}$.
As a second example, consider
${\mathcal{S}}_{8}$, whose multichiller code is
${\mathbf{s}}_{\mathbf{8}}=[0,1,3]$: here, the chiller #3 is operating at intermediate part load, i.e.
with
${x}_{3}\in {\mathcal{F}}_{3}$. In this case,
$\mathbf{QP}\left(8\right)$ is a constrained quadratic programming problem in the unknown
${x}_{3}$.
As will be shown later, for $j\in \overline{\mathcal{C}}$, the $\mathbf{QP}\left(j\right)$ problems enjoy a remarkable property: their optimal solution, if it exists, is a critical point and no more than one critical point exists. In the following, ${\mathbf{PLR}}^{*}\left(j\right)$ will denote:
the optimal solution of $\mathbf{QP}\left(j\right)$, if $j\in \mathcal{C}$;
the unique feasible critical point of $\mathbf{QP}\left(j\right)$, if $j\in \overline{\mathcal{C}}$.
Let
$\mathcal{A}$ denote the set of integers
j s.t.
${\mathbf{PLR}}^{*}\left(j\right)$ exists. The associated value of the cost function will be denoted by
Then, the key idea is to exploit the partition of the solution space by a twostep procedure:
Solve $\mathbf{QP}\left(j\right)$, obtaining ${\mathbf{PLR}}^{*}\left(j\right)$ and ${P}_{tot}^{*}\left(j\right)$ for $j=1,\dots ,{4}^{n}$;
Letting
where obtain the globally optimal part load vector as
${\mathbf{PLR}}^{*}={\mathbf{PLR}}^{*}\left({j}^{*}\right)$.
In the next subsection, it is shown how to reduce the inequalityconstrained problems associated with $j\in \overline{\mathcal{C}}$ to equalityconstrained quadratic problems (EQP), for which a closed form solution is available.
2.4. Reduction to EqualityConstrained Problems
For a given
$j\in \overline{\mathcal{C}}$, we denote by
${\mathcal{V}}_{j}=\{i\in \{1,2,\dots ,n\}\mid {s}_{ji}=3\}$ the set of chillers operating at intermediate part loads, i.e., between
$PL{R}_{min}$ and 1. We also let
$\kappa \left(j\right)$ denote the cardinality of
${\mathcal{V}}_{j}$. It is then possible to rewrite
$\mathbf{QP}\left(j\right)$ as a reducedorder quadratic problem in the
$\kappa \left(j\right)\le n$ unknowns
$PL{R}_{i},\phantom{\rule{0.222222em}{0ex}}i\in {\mathcal{V}}_{j}$.
above,
${\tilde{Q}}_{Load}$ is
is the cooling load that must be supplied by the chillers operating at intermediate part loads.
Now, we associate to each $\mathbf{QP}\left(j\right)$ the corresponding equalityconstrained quadratic problem $\mathbf{EQP}\left(j\right)$, that is obtained by removing the inequality constraints (10c).
Problem 2. $\mathbf{EQP}\left(j\right)$ It is convenient to rewrite $\mathbf{EQP}\left(j\right)$ in matrix form. For this purpose, we introduce a selection matrix $\mathbf{M}\left({\mathcal{V}}_{j}\right)\in {\mathbb{R}}^{\kappa \left(j\right)\times n}$ that selects $\kappa \left(j\right)$ elements out of n.
Note that, being integers, the elements of
${\mathcal{V}}_{j}$ admit an obvious ordering. Then,
In short,
$\mathbf{M}\left({\mathcal{V}}_{j}\right)$ will be denoted by
${\mathbf{M}}_{j}$. The reducedorder matrices are thus given by:
Then,
$\mathbf{EQP}\left(j\right)$ can be restated as:
By applying the firstorder Karush–Kuhn–Tucker (KKT) necessary condition to the
EQP$\left(j\right)$ problem, the following linear system is obtained:
where
${\tilde{\mathbf{x}}}^{*}$ identifies a critical point, either maximum, minimum or saddle, and
${\lambda}^{*}\in \mathbb{R}$ is the associated Lagrange multiplier. In order to guarantee the existence of the solution, a technical assumption is introduced.
Assumption 2. $\tilde{\mathbf{E}}{\tilde{\mathbf{Q}}}^{1}{\tilde{\mathbf{E}}}^{T}\ne 0$.
Note that Assumption 2 is immediately satisfied if
${q}_{i}>0,\forall i$, that is when the quadratic power consumption curves (
3) are all convex, although this is not necessary.
Theorem 1. Under Assumption 1, $\mathbf{EQP}\left(j\right)$ admits a unique critical point ${\mathbf{x}}^{*}$, given by Proof of Theorem 1. In view of Assumption 1, $det\left(\mathbf{Q}\right)\ne 0$, because ${q}_{i}\ne 0,\forall i$. Moreover, Assumption 2 guarantees that $\tilde{\mathbf{E}}{\tilde{\mathbf{Q}}}^{1}{\tilde{\mathbf{E}}}^{T}\ne 0$. Then, it is immediate to see that the KKT condition (15) admits (16)–(17) as a unique solution. □
Given the critical point
${\tilde{\mathbf{x}}}^{*}$, it is easy to obtain a critical point for
$\mathbf{QP}\left(j\right)$, as well. For this purpose, it is convenient to introduce an auxiliary vector
$\overline{\mathbf{x}}\in {\mathbb{R}}^{n\times 1}$, such that:
The candidate critical point for
$\mathbf{QP}\left(j\right)$ is thus given by:
The keystone of the solution procedure is the connection between the critical points of $\mathbf{QP}\left(j\right)$ and $\mathbf{EQP}\left(j\right)$, as stated in the following theorem.
Theorem 2. The critical point ${\mathbf{PLR}}^{*}\left(j\right)$ exists if ${\widehat{\mathbf{PLR}}}^{*}\left(j\right)\in {\mathcal{S}}_{j}$. In such a case, ${\mathbf{PLR}}^{*}\left(j\right)={\widehat{\mathbf{PLR}}}^{*}\left(j\right)$.
Proof of Theorem 2. Sufficiency. Assume that the critical point ${\widehat{\mathbf{PLR}}}^{*}$ for EQP(j) belongs to ${\mathcal{S}}_{j}$. Observe that EQP(j) has less contraints than QP(j). Therefore, if a critical point for EQP(j) is feasible for QP(j), it is ipso facto a critical point for QP(j), as well.
Necessity. Assume that QP(j) admits a critical point, say ${\mathbf{PLR}}^{*}\left(j\right)$. Given that EQP(j) and QP(j) differ only for strict inequality constraints, any critical point for QP(j) is critical also for EQP(j). Since EQP(j) admits at most one critical point, necessity is proven. □
2.5. Summary of the XOCL Algorithm
We are now in a position to summarize the steps of the proposed algorithm, hereafter named XOCL, for the exact solution of the OCL problem.
The partition of the solution set in the ${4}^{n}$ subsets ${\mathcal{S}}_{j},\phantom{\rule{0.222222em}{0ex}}j=1,\dots {4}^{n}$ allows one to divide the MINLP problem into ${4}^{n}$ subproblems $\mathbf{QP}\left(j\right)$ (8). For each subset ${\mathcal{S}}_{j}$, two situations can occur:
In the former case, only a feasibility check is required to decide whether the PLR configuration is to be kept as a candidate solution. In the latter case, the solution of the $\mathbf{QP}\left(j\right)$ problem is reduced to the solution of an $\mathbf{EQP}\left(j\right)$ problem that admits a unique critical point, easily computable in closed form. As stated in Theorem 2, if the critical point of $\mathbf{EQP}\left(j\right)$ belongs to ${\mathcal{S}}_{j}$, it coincides with the critical point of $\mathbf{QP}\left(j\right)$. Otherwise, $\mathbf{QP}\left(j\right)$ does not admit a critical point.
Once all the
$\mathbf{QP}\left(j\right)$ have been processed, the corresponding set of critical points
$\{{\mathbf{PLR}}^{*}\left(j\right),\phantom{\rule{0.222222em}{0ex}}j=1,\dots ,{4}^{n}\}$ include the optimal solution of the overall OCL problem, which can be found just by comparing the associated system power consumptions
${P}_{tot}^{*}\left(j\right)$. A pseudocode summary is reported in Algorithm 1.
Algorithm 1: XOCL. 
 1:
Input:$\mathit{a},\mathit{c},\mathit{q}\in {\mathbb{R}}^{n\times 1}$, $\mathit{s}\in {\mathbb{R}}^{{4}^{n}\times n}$, ${Q}_{load}\in \mathbb{R}$  2:
Output:${\mathit{PLR}}^{*},{P}_{tot}^{*}$  3:
 4:
for$j=1,\dots ,{4}^{n}$do  5:
${\mathcal{V}}_{j}=\{i\in \{1,2,\dots ,n\}:{s}_{ji}=3\}$, $\kappa =\phantom{\rule{4pt}{0ex}}{\mathcal{V}}_{j}$  6:
if${\mathcal{V}}_{j}=\varnothing $ then  7:
$\{\u25b9$ feasibility check}  8:
all the chillers part load ratios are fixed  9:
$\mathit{x}\leftarrow {\mathcal{S}}_{j}$  10:
if ${\sum}_{i=1}^{n}{x}_{i}\xb7{Q}_{100\%,i}={Q}_{load}$ then  11:
${\mathit{PLR}}^{*}\left(j\right)=\mathit{x}$  12:
${P}_{tot}^{*}\left(j\right)={\sum}_{i=1}^{n}{P}_{i}\left(PL{R}_{i}^{*}\left(j\right)\right)$  13:
else  14:
${\mathit{PLR}}^{*}\left(j\right)=\varnothing $  15:
${P}_{tot}^{*}\left(j\right)=\varnothing $  16:
end if  17:
else  18:
$\{\u25b9$ solve the EQP$\left(j\right)$ associated to QP$\left(j\right)$}  19:
${\mathcal{V}}_{j}\Rightarrow \tilde{\mathbf{Q}},\tilde{\mathbf{E}},\tilde{\mathbf{c}},\tilde{d}$  20:
${\lambda}^{*}={\displaystyle \frac{\tilde{d}+\tilde{\mathit{E}}{\tilde{\mathit{Q}}}^{1}\tilde{\mathit{c}}}{\tilde{\mathit{E}}{\tilde{\mathit{Q}}}^{1}{\tilde{\mathit{E}}}^{T}}},\phantom{\rule{0.277778em}{0ex}}{\lambda}^{*}\in \mathbb{R}$  21:
${\tilde{\mathit{x}}}^{*}={\tilde{\mathit{Q}}}^{1}(\tilde{\mathit{c}}+{\tilde{\mathit{E}}}^{T}{\lambda}^{*})$  22:
$\overline{\mathit{x}}=\left\{\begin{array}{cc}\hfill {\mathcal{S}}_{j}\left[i\right],\phantom{\rule{1.em}{0ex}}& \mathrm{if}i\in \overline{{\mathcal{V}}_{j}}\phantom{\rule{1.em}{0ex}}\hfill \\ \hfill 0,& \mathrm{otherwise}\hfill \end{array}\right.$  23:
${\widehat{\mathit{PLR}}}^{*}\left(j\right)=\overline{\mathit{x}}+{\mathit{M}}_{j}^{T}{\tilde{\mathit{x}}}^{*}$  24:
if ${\widehat{\mathit{PLR}}}^{*}\left(j\right)\in {\mathcal{S}}_{j}$ then  25:
${\mathit{PLR}}^{*}\left(j\right)=\widehat{\mathit{PLR}}\left(j\right)$  26:
${P}_{tot}^{*}\left(j\right)={\sum}_{i=1}^{n}{P}_{i}\left(PL{R}_{i}^{*}\left(j\right)\right)$,  27:
else  28:
${\mathit{PLR}}^{*}\left(j\right)=\varnothing $  29:
${P}_{tot}^{*}\left(j\right)=\varnothing $  30:
end if  31:
end if  32:
end for  33:
 34:
${j}^{*}={min}_{j}{P}_{tot}\left(j\right)$  35:
${\mathit{PLR}}^{*}={\mathit{PLR}}^{*}\left({j}^{*}\right)$  36:
${P}_{tot}^{*}={P}_{tot}\left({j}^{*}\right)$

Summarizing the difference between OCL and OCS is that the former one is a static problem, whereas the second one is intrinsically a dynamic problem. An OCS problem stripped of its dynamic constraints could be solved just as a sequence of OCL problems.
3. Test on Hsinchu Benchmark Model
3.1. Hsinchu Cooling Plant Model
The Hsinchu chiller system, originally described in [
7], has become a widely used benchmark for the testing and comparison of OCL algorithms [
6,
15,
17,
18,
19]. The case study involves six chillers installed in a semiconductor factory located in Hsinchu Scientific Garden (Taiwan) with a 7620 kW total cooling capacity. Quadratic models of the chillers’ energy consumption were obtained and validated from data collected every 5 min over a 5month period [
7]. The benchmark problem assumes that the condenser inlet water temperature is 24.5
${}^{\xb0}$C. The coefficients of the six chillers’ energy consumption models (
3) are reported in [
7] (Table 3) and the corresponding PPLR curves are displayed in
Figure 1. It is asked to solve the OCL problem for five different cooling loads, ranging from 70% to 90% of the system total cooling capacity. It is also required that the partial load ratio of each chiller never goes below 0.3. According to our notation, the following parameter settings are used:
 
${a}_{i},{c}_{i},$ and
${q}_{i}$ from [
7];
 
${Q}_{Load}=90\%,85\%,80\%,75\%$, and $70\%$ of the chillers’ maximum capacity (${\sum}_{i=1}^{n}{Q}_{nom}$);
 
$PL{R}_{min,i}=0.3,\phantom{\rule{1.em}{0ex}}\forall i$.
3.2. OCL Benchmark: Results
The comparison between the XOCL optimal solution and those obtained by the six literature methods in
Table 1 sheds light on some errors in the published results.
For the three highest cooling load demands (${Q}_{Load}=\{6858,6477,6096\}$ [kW]), the XOCL optimal solution coincides with the common solution provided by the six algorithms, thus confirming that they had reached the optimum. In the remaining two cases (${Q}_{Load}=5717$ and 5334 [kW]), the solution computed by XOCL coincides with those of GAMS, EIWO and DCEDA, which, however, are apparently outperformed by IFA and DCSA. The worst performance is that of TLBO.
However, concerning the solution provided by IFA and DCSA, four wrong values of power consumption (identified by asterisks in
Table 1) were published in [
15,
17]:
 
${P}_{tot,IFA}$ at load 75% and 70%;
 
${P}_{tot,DCSA}$ at load 75% and 70%.
In fact,
Table 1 reports
${P}_{tot,IFA}={P}_{tot,DCSA}\approx 3507.3$ [kW] at 70% cooling load, which is inconsistent with the
$PL{R}_{i}$ reported in the same table. Such inconsistency is easily verified by plugging the
$PL{R}_{i}$’s into the chillers’ power consumption Equation (
3) in order to obtain the individual chiller power consumptions of the
${P}_{i}$ [kW] column (the papers [
15,
17] describing algorithms IFA and DCSA do not report these individual consumptions). Apparently, the source of the error is the mechanical use of the chiller’s quadratic power consumption model out of its operational range, that is, in correspondence with the null partial load ratio. Obviously, the associated power consumption is null as well, but if the quadratic model has a negative constant term, as is the case for chillers 3–5, the formula will return a negative power consumption as if a turnedoff chiller could generate free power.
In
Table 2, the published chillers’ power consumption
${P}_{p,i}$ and the corresponding corrected values
${P}_{c,i}$ are reported for both IFA and DCSA. The values highlighted in red are the unfeasible chillers’ power consumption, of which the correction is reported in column
${P}_{c,i}$.
Once the errors have been corrected, IFA and DCSA are no more optimal at
$75\%$ and
$70\%$ loads. At load
$75\%$${P}_{tot,IFA}$ rises from
$3840.0690$ to
$3960.5709$ [kW] (
$+3.14\%$), while at
$70\%$, it rises from
$3507.2848$ to
$3627.7578$ [kW] (
$+3.43\%$). Analogously, for DCSA,
${P}_{tot,DCSA}$ goes from
$3840.0545$ to
$3960.5580$ [kW] (
$+3.14\%$) at
$75\%$ and from
$3507.2848$ to
$3627.7578$ [kW] (
$+3.43\%$) at
$70\%$. The corrected results for all the six literature methods are reported in
Table 3.
In conclusion, on the Hsinchu benchmark, GAMS, EIWO and DCEDA prove to be the best heuristic algorithms, as their solutions coincide with the optimal ones computed by XOCL, for all considered loads.
4. The Optimal Chiller Sequencing Problem
The cooling load demand of a building can be subject to significant variations during the day. Consequently, solving the OCL problem in each time step t (for example, 20 min), just ignoring minimum up/down time constraints on the chillers, could lead to frequent switchings (chiller startups and shutdowns). In order to preserve chillers from excessive mechanical stress and increase their operating life, each machine should not be switched off before a minimal uptime is reached. Analogously, it should not be switched on too quickly. To comply with these requirements, minimum up/downtime constraints must be enforced in the formulation of the socalled optimal chiller sequencing (OCS) problem. In its full formulation, OCS is a dynamical problem, because, in order to minimize the cumulative power consumption, the current decision should also take into account future constraints. As a consequence, the solution approaches proposed in the literature range from dynamic programming to heuristic methods were designed to alleviate the complexity of the problem.
4.1. A Lower Bound to the OCS Problem
Any solution to the OCS problem must face some level of approximation. Even when dynamic programming is used, there is the necessity of forecasting future loads, which introduces a suboptimality margin with respect to the ideal solution based on perfect knowledge of the future load profile. The availability of an easytocompute limit of performance against which the results of heuristic methods can be benchmarked is therefore of interest. In order to derive such a limit, one can consider the relaxed OCS problem (ROCS), that is, an OCS problem without up and downtime constraints. The relaxed OCS problem boils down to a sequence of independent OCL problems to be solved at each step in correspondence with the associated load. The availability of an exact OCL solver, such as XOCS, makes it possible to compute the exact solution of the relaxed OCS as well. Notably, in view of the independence of the OCL problems, what matters is not the load sequence but the load distribution, so that the ROCS bound could be easily derived based on statistical distributions reflecting different production and weather scenarios.
Such a bound can be used to quantitatively assess the existing margin of improvement for a given heuristic OCS solver. In fact, if the achieved power consumption is close enough to the bound, there is no scope for the search of further improvements. Along this direction, in the following section, the XOCL solver is used to derive a greedy OCS algorithm, whose performance is then assessed against the ROCS bound.
4.2. XOCS, a Greedy Approach to OCS
XOCS is a greedy algorithm that reduces OCS to a sequence of OCL problems, solvable through XOCL. The approach is greedy because at each time step, future constraints are ignored, and the optimal OCL solution, compatible with the current minimum up/down time constraint, is searched for. In the mathematical form, the greedy OCS problem can be written as a mixedinteger quadratic problem with linear constraints:
Problem 3. $\mathbf{greedy}\mathbf{OCS}$where
$MU{T}_{i}$ and
$MD{T}_{i}$ are the
ith chiller’s minimum uptime and minimum downtime limits, expressed in number of time steps. The time counters
${T}_{i}^{ON}\left(t\right)$ and
${T}_{i}^{OFF}\left(t\right)$ are expressed as:
It is easy to observe that the greedy OCS problem is an OCL problem with the two additional constraints (19e–f) which force some chillers to be online/offline depending on their previous states $\delta \left(\tau \right),\tau =t1,t2,\dots $.
At each time step t, two situations can occur:
 
all the chillers’ states ${\delta}_{i}\left(t\right)$ are free, i.e., all the chillers have been online/offline for more time steps than those prescribed by MUT/MDT;
 
there is at least one chiller, say the ith one, whose state ${\delta}_{i}\left(t\right)$ is constrained to be online or offline (${\delta}_{i}\left(t\right)=1$ or ${\delta}_{i}\left(t\right)=0$) by the MUT or MDT.
Concerning the first case, the minimum up/downtime constraints are not active, therefore the step of the greedy OCS boils down to an OCL problem and its optimal solution can be found by the XOCL algorithm. In the second case, instead, at each time
t, the optimal solution is found by applying the XOCL algorithm to a suitable subset of the OCL solution space
$\mathcal{S}$. At each time step
t, the solution space of the greedy OCS is therefore given by:
where
are the sets of chillers which must be on and off, respectively.
The idea is to exploit the partitions of the solution space ${\mathcal{P}}_{j}$ by the typical twostep procedure of XOCL:
Herein, $\mathcal{A}\left(t\right)$, which denotes that the set of integers j s.t. ${\mathbf{PLR}}^{*}\left(j\right)$ exists, is a function of t because the load changes with time.
4.3. OCS Simulated Example
In this section, the performance of the XOCS method is assessed on two case studies taken from the literature [
23]. Case study 1 involves a hotel in Taipei with two 450 refrigeration tons (RT) chillers and two 1000 RT chillers, while case study 2, again from the Hsinchu Science Industrial District, features nine 1250 RT chillers. In both cases, chillers are described by their COPPLR curves, expressed by a secondorder polynomial model:
where
${\alpha}_{i}$,
${\beta}_{i}$ and
${\gamma}_{i}$ are the chiller’s coefficients, reported in
Table A1.
The first step was the identification of quadratic power consumption models (
3) using data sampled from the COPPLR curves. Further details regarding the identification procedure are reported in
Appendix A. The coefficients of the quadratic PPLR curves in
Figure 2 are reported in
Table A2.
The aim of both benchmarks is to compute the sequence of chillers’ partializations over one day, assuming 20min stages, so as to minimize the cumulative power consumption, while satisfying the cooling demand constraint at each stage. The load demand profiles are displayed in
Figure 3.
The parameters’ settings were as follows:
 
$PL{R}_{min}=0.5$;
 
$MU{T}_{i}=3\phantom{\rule{1.em}{0ex}}\forall i$;
 
$MD{T}_{i}=1\phantom{\rule{1.em}{0ex}}\forall i$.
Recall that $MU{T}_{i}=3$ means that the ith chiller must be on at least 3 consecutive steps before being turned off. Likewise, $MD{T}_{i}=1$ indicates that the ith chiller, once turned off, must remain off at least 1 step.
The results obtained via XOCS were compared with those obtained by dynamic programming, as reported in [
23]. Where optimization was carried out under the ideal condition that all future loads are known in advance.
For sake of comparability with DP, the chillers power consumption associated with XOCS was evaluated by plugging the PLRs computed by XOCS into the original benchmark’s COP model [
23] and not via the approximate quadratic power consumption curves used by XOCS.
4.4. OCS Benchmark: Results
The results are shown in
Table 4 and
Table 5. In case study 1, the power consumption obtained by the XOCS method was, at each stage, lower than or equal to that obtained by the DP method. From stage 1 to stage 29, the DP and XOCS methods selected the same chillers. Since the MUT and MDT constraints were not active, the solutions coincided with the ROCS ones. At stage 30, the ROCS solution had chillers 4 and 3 switched on. However, the MUT constraint forced the DP and XOCS to leave chillers 1 and 2 switched on. The same goes for stage 31. At stage 32, 44 and 59 the XOCS method performed better than the DP one. Although the chillers were not constrained by minimum up/down time limits, apparently the DP method, as implemented in [
23], could not find the global minimum. Notably, the greedy XOCS method achieves a total electric power consumption (46,495.91 [kW]) which is less than 0.1 % greater than the minimum achievable bound ROCS (64,432.56 [kW]), meaning that there is no practical margin of improvement left).
In the second case study, the cooling load demand varied slowly over time, so that the MUT and MDT constraints were never active. Therefore, the XOCS, notwithstanding its greedy nature, attains the best achievable performance bound ROCS. For the majority of the cooling loads, DP and XOCS gave the same results, the only exceptions being at stages 22, 48–62, 68–72, where XOCS performed slightly better. For both the case studies, the cumulative daily power consumptions obtained by DP (marked by asterisks) had been reported incorrectly in [
23]:
 
Case study 1: ${P}_{day}=$ 645,220.08 [kW] instead of 64,883.84 [kW]
 
Case study 2: ${P}_{day}=$ 298,425.69 [kW] instead of 289,525.25 [kW]
As a matter of fact, the daily power consumption values reported in the paper did not match with the sums of the power consumptions at each step, which we used for the correction.
So far, the performances of alternative methods have been compared on OCL and OCS benchmark problems whose quadratic consumption models were taken from the literature. Moreover, a limited number of loads were considered.
In this section, the feasibility of HVAC efficient management based on the exact solution of the OCL problem is validated against a realworld scenario that includes the estimation of the chillers’ consumption models from the field data.
4.5. Field Data
The study was based on experimental data from a large HVAC system installed in a semiconductor fab located in Austria. The chiller unit is composed by five watercooled centrifugal chillers, see
Table 6, connected in parallel, in a constant primary flow chilled water system. Quarterhourly data were recorded at different working conditions over a period of almost two years, from February 2017 to January 2019. Collected data include: temperatures, PLR, power consumption.
The time series of the cooling load demand is shown in
Figure 4.
The chillers were subject to the following operating constraints:
 
$PL{R}_{min}=0.2$;
 
$MU{T}_{i}=4$ (one hour), $\phantom{\rule{1.em}{0ex}}\forall i$;
 
$MD{T}_{i}=2$ (half an hour), $\phantom{\rule{1.em}{0ex}}\forall i$.
4.6. Chiller Energy Consumption Models
The chillers’ power consumption models were estimated using the evaporator cooling capacity ${Q}_{evap}$ [kW] and the condenser inlet water temperature T [${}^{\xb0}$C], as covariates, and the compressor power consumption data P [kW], as target.
For each chiller, the dataset was randomly partitioned in two subsets: 70% for training and 30% for testing, respectively. The parameters
${\beta}_{p,i}$ of the model (
2) were estimated via least squares fitting of the training data, discarding data with
$PLR<PL{R}_{min}=0.2$. The values of the estimated parameters
${\beta}_{p,i}$ are reported in
Table 6, together with the percentage coefficient of variation, defined as
$\mathrm{CV}\%=100\times \mathrm{SE}\left({\beta}_{p,i}\right)/\left{\beta}_{p,i}\right$. The cooling capacity
${Q}_{nom}$ are the nominal values provided by the manufacturer.
In
Figure 5, the fitted surfaces
$P(PLR,T)$ are displayed against the validation data for each of the five chillers. It is seen that the the quadratic model, in spite of a few outliers that are not uncommon when data are collected in industrial frameworks, predicts the consumptions at different operating conditions well, as also confirmed by the goodnessoffit (GOF) plots in
Figure 6.
The 3D histogram of the covariates
$({Q}_{evap},T)$ sketched in
Figure 7 shows that
T, i.e., the condenser inlet water temperature depicted in
Figure 8, is mainly concentrated in a narrow range centered around the setpoint, namely 21.5
${}^{\xb0}$C. Following what is usually done in the literature benchmarks, one could neglect temperature variations around the set point and solve the OCL and OCS problems using the chillers consumption curves at 21.5
${}^{\xb0}$C, displayed in
Figure 9. However, the inspection of
Figure 5 shows that for chiller 4, the power consumption is significantly affected by the temperature, especially in summer. Therefore, differently from other literature studies, OCL and OCS solutions, we computed this based on the complete model
$P(PLR,T)$.
4.7. Real HVAC System: Assessment of Potential Savings
The field data were used to perform a retrospective analysis of the efficiency of the HVAC system management. More precisely, starting from the historical decisions and the associated cumulative power consumption, two comparisons were performed for the 2year OCS problem. First of all, the lower bound ROCS on the best achievable consumption was computed in order to quantitatively assess the potential improvement margin. Being a lower bound, ROCS may be overly optimistic, so that it is important to evaluate the performance that can be obtained in practice. This was done by running the XOCS solver, whose energy consumption could then be compared with the (ideal) ROCS bound and the historically recorded power consumption.
The cumulative energy consumption recorded during the 2year monitoring was $1.758\times {10}^{6}$ [kWh]. This figure can be compared with the ROCS lower bound, equal to $1.600\times {10}^{6}$ [kWh]. This means that the potential margin of improvement is not larger than $8.97\%$.
When the XOCS algorithm was applied, the cumulative energy power consumption was
$1.601\times {10}^{6}$ [kWh]. As a matter of fact, for this HVAC system, the loss of performance due to the suboptimality of the greedy algorithm is definitely negligible (it is less than
$0.1\%$). On average, the power saving is
$0.157\times {10}^{6}/(2\times 365\times 24)=8.96$ [kW]. In
Figure 10, the difference between the consumption achieved by XOCS and the lower bound is plotted on a weekly basis. The black and blue lines overlap almost perfectly, so that they cannot be distinguishable.
5. Execution Time
The XOCL and XOCS algorithms, coded in Matlab
^{®}, were executed on a standard laptop (Intel(R) i77500U dualcore with hyperthreading, RAM 16GB, 2.7 GHz). While no explicit parallelization of the algorithm was implemented, the solution of the
${4}^{n}$ QP problems was formulated as a unique algebraic computation using sparse matrices. This means that the algorithm may have benefited from some optimization automatically enforced by the Matlab
^{®} compiler. In order to compare the numerical efficiency, the Hsinchu benchmark was also solved using the CPLEX solver for costrained mixed integer problems as implemented in the GAMS environment, used by [
6]. The results concerning the computational times are summarized in
Table 7.
Concerning the Hsinchu OCL benchmark, the computational cost of XOCL was negligible ($0.11$ s) and, more importantly, definitely lower than the time ($1.991$ s) spent by GAMS.
Concerning the three OCS benchmarks, in all cases, the XOCS computation time was larger than that for ROCS, because the greedy algorithm performs the additional task of looking for solutions that satisfy the up/downtime constraints.
In the OCS examples, the larger number of loads makes it possible to assess the average computation time per load. For a small number of chillers, the average computation time per load of XOCS is remarkably small: $0.368/72=5\times {10}^{3}$ s for OCS benchmark 1 (4 chillers) and $573.96/\mathrm{68,110}=8.4\times {10}^{3}$ s for the OCS on field data (5 chillers). In the latter case, XOCS took less than 6 min to solve the OCS problem over 2year data with quarterhour sampling.
As expected, in view of the exponential growth of the number of QP problems, the maximal average computation time per load is found in correspondence with the OCS benchmark 2, where 9 chillers are present. Nevertheless, the average time per load amounts to $228.29/72=3.17$ s, which is totally affordable.
Finally, a simulated benchmark was set up in order to better assess the average computation time per load as a function of the number of chillers. In particular, nine OCL simulated experiments, characterized by a variable number of chillers, ranging from two to ten, were considered. The tenchiller system was composed by the six Hsinchu chillers, whose models are reported in [
7] (Table 3), with the first four replicated. For each experiment, the OCL problem was solved for five different cooling loads ranging from 70% to 90% of the chiller’s total cooling capacity. In
Figure 11, the average computation time per load is displayed as a function of the number of chillers. The plot highlights the exponential growth but, at the same time, it shows that, even for a large HVAC system made of 10 chillers, the computation time per load (10 s) is all but prohibitive.
6. Discussion
The main purpose of the paper was the derivation of an exact algorithm for the solution to the OCL problem. This goal was successfully completed by a decomposition approach that exploits a suitable partition of the solution space. The availability of an exact method has been immediately exploited along two directions. First, it has become possible to give a definitive assessment on the performances of different literature methods that had been applied to some consolidated benchmark problems. By the way, the comparison with the exact solution revealed that some unrealistic performances had been declared in the literature, due to erroneous extrapolations of the power consumption curve. The exact method has been exploited also in relation with the optimal chiller sequencing problem. For a given cooling demand profile, if the dynamic up/downtime constraints are neglected, a sequence of OCL problems can be exactly solved to yield a lower bound limit, called ROCS, to the performance of any feasible solution.
Concerning the practical applicability of the exact solution in place of heuristic approaches discuss in the literature, two main objections may be raised: (i) the explosion of the computational cost with the number of chillers, (ii) the need to make overly restrictive assumptions on the shape of the power consumption curves.
On the first side, it is indeed true that the OCL problem as formulated in (4) is NPhard, as confirmed by
Figure 11, where the exponential growth of the computations time is apparent. At the same time, the figure shows that, even for a medium/largesized chiller system, e.g., 6–10 chillers, the computation time per load is less than 2 s. This suggests that, even for chiller systems used in large semiconductor factories, the cost of the exact solution is all but prohibitive. The numerical efficiency is a direct consequence of the tiny number of computations required to solve the elementary EQP problems, see (12). Moreover, the partition strategy underlying XOCL implies that it is totally parallelizable into
${4}^{n}$ threads, a feature that has not been explicitly exploited and that could further speed up the solution.
The second objection to the general applicability of the exact solution has to do with the quadratic assumption made on the power consumption curve. Even if the majority of benchmarks satisfy this assumption, there is no stringent reason to rule out other functional descriptions. Nevertheless, when confronted with real data (see the field data benchmark), we found that a quadratic power consumption fitted well the recorded data, see
Figure 6. Moreover, even when a single quadratic function is not adequate, it is still possible to switch to a piecewise quadratic description. In that case, it would be immediate to generalize the exact algorithm by just increasing the number of partitions
${\mathcal{S}}_{j}$, in such a way that the problem still boils down to a set of easytosolve EQP problems.
In view of its numerical efficiency, the application of XOCS to the solution of the OCS problem also appears very promising. Indeed, in the OCS benchmarks taken from the literature and in the field data OCS benchmark, the performance of XOCS is very close to the lower bound, implying that there is no scope for the use of more sophisticated algorithms.
Author Contributions
Conceptualization, F.A., M.R., G.D.N.; methodology, F.A.; software, F.A.; validation, F.A., M.R., G.D.N.; formal analysis, F.A.; data curation, F.A.; writing—original draft preparation, F.A.; writing—review and editing, M.R., G.D.N.; project administration, G.D.N.; funding acquisition, G.D.N.; supervision, G.D.N. All authors have read and agreed to the published version of the manuscript.
Funding
This work has been partially supported by the Italian Ministry for Research in the framework of the 2017 Program for Research Projects of National Interest (PRIN), Grant no. 2017YKXYXJ. A part of the work has been performed in the project Power Semiconductor and Electronics Manufacturing 4.0 (SemI40), under grant agreement No 692466. The project is cofunded by grants from Austria, Germany, Italy, France, Portugal and Electronic Component Systems for European Leadership Joint Undertaking (ECSEL JU).
Conflicts of Interest
The authors declare no conflict of interest.
Acronyms
${P}_{i}$  Electric power consumption of the ith chiller [kW] 
${P}_{tot}$  Multichiller system total power consumption [kW] 
${Q}_{i}$  Cooling power delivered by the ith chiller [kW] 
${Q}_{100\%,i}$  Maximum cooling power supplied under full capacity by the ith chiller [kW] 
${Q}_{load}$  Cooling load demand [kW] 
$PL{R}_{i}$  Part load ratio of the ith chiller 
$PL{R}_{min,i}$  Minimum part load ratio of the ith chiller 
${T}_{i}$  Condenser inlet water temperature on the ith chiller [${}^{\xb0}$C] 
$OCL$  Optimal Chiller Loading problem 
$OCS$  Optimal Chiller Sequencing problem 
$X\text{}OCS$  Exact Optimal Chiller Sequencing algorithm 
$X\text{}OCL$  Exact Optimal Chiller Loading algorithm 
$R\text{}OCS$  Relaxed Optimal Chiller Sequencing problem 
$HVAC$  Heating, Ventilation and AirConditioning 
Appendix A. Derivation of Power Consumption Models from the COP Ones
Assuming that the available COPPLR curves represent the true model of chillers’ efficiency, quadratic approximated power consumption models can be easily derived as follows:
Sample N data points from the COPPLR curve of the ith chiller, uniformly in the range [$PL{R}_{min,i}$,1] obtaining the pairs $\{PL{R}_{i}\left(k\right),CO{P}_{i}\left(k\right)\},\phantom{\rule{0.277778em}{0ex}}k=1,\dots N$;
Compute the chiller’s power consumptions as
Using the training set made of inputoutput paired samples $\{PL{R}_{i}\left(k\right),{P}_{i}\left(k\right)\},\phantom{\rule{0.277778em}{0ex}}k=1,\dots N$ estimate the parameters of the quadratic model via ordinary least squares (OLS).
The above procedure was applied to the case studies presented in
Section 4.3. For both the case studies, 50 data points were randomly sampled (uniformly in [0.5,1]) from the COPPLR curves reported in
Table A1 to obtain the training datasets represented in the left panels of
Figure A1 and
Figure A2 as red dots. The estimated PkW curves are shown in the right panels of
Figure A1 and
Figure A2, and their parameters are reported in
Table A2.
Table A1.
Coefficients of chillers’ COPPLR curves.
Table A1.
Coefficients of chillers’ COPPLR curves.
Systems  Chiller  ${\mathit{\alpha}}_{\mathit{i}}$  ${\mathit{\beta}}_{\mathit{i}}$  ${\mathit{\gamma}}_{\mathit{i}}$  ${\mathit{Q}}_{\mathit{nom}}$ 

Case 1  1  0.1561  3.9023  −2.5909  450 
2  0.9000  1.8432  −1.4188  450 
3  0.2932  3.0419  −2.0054  1000 
4  0.1415  3.6376  −2.2469  1000 
Case 2  1  1.5652  1.8094  −0.9803  1250 
2  1.0519  4.1471  −2.4173  1250 
3  0.5703  3.1602  −2.0912  1250 
4  0.3257  2.3513  −1.4265  1250 
5  0.5438  1.8668  −1.2361  1250 
6  1.5271  1.0634  −0.7238  1250 
7  0.7865  1.8473  −1.1633  1250 
8  0.8499  3.7768  −2.2859  1250 
9  1.1191  1.0228  −0.7542  1250 
Table A2.
Estimated parameters of chillers’ PPLR curves.
Table A2.
Estimated parameters of chillers’ PPLR curves.
Systems  Chiller  ${\mathit{a}}_{\mathit{i}}$  ${\mathit{c}}_{\mathit{i}}$  ${\mathit{q}}_{\mathit{i}}$  ${\mathit{Q}}_{\mathit{nom}}$ 

Case 1  1  243.58  −398.01  504.00  450 
2  130.81  −103.53  309.65  450 
3  417.51  −444.57  771.99  1000 
4  383.79  −347.84  611.64  1000 
Case 2  1  95.54  321.92  103.60  1250 
2  170.68  41.43  235.46  1250 
3  371.09  −307.99  693.76  1250 
4  477.85  −217.09  733.35  1250 
5  433.17  −186–20  810.89  1250 
6  104.21  358.47  205.50  1250 
7  272.33  116.22  457.96  1250 
8  218.68  −20–94  333.72  1250 
9  191.69  276.56  429.51  1250 
Figure A1.
(Left): COPPLR curves of chillers in case study 1. Red dots are the 50 samples used to identify the quadratic approximate power consumption model. (Right): Identified quadratic PPLR curves of chillers in case study 1. Red dots are the energy consumption data used for the training.
Figure A1.
(Left): COPPLR curves of chillers in case study 1. Red dots are the 50 samples used to identify the quadratic approximate power consumption model. (Right): Identified quadratic PPLR curves of chillers in case study 1. Red dots are the energy consumption data used for the training.
Figure A2.
(Left): COPPLR curves of chillers in case study 2. Red dots are the efficiency sampled data. (Right): PPLR curves of chillers in case study 2. Red dots are the energy consumption data used for the training.
Figure A2.
(Left): COPPLR curves of chillers in case study 2. Red dots are the efficiency sampled data. (Right): PPLR curves of chillers in case study 2. Red dots are the energy consumption data used for the training.
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Figure 1.
Hsinchu benchmark: PPLR curves at $T=24.5{\phantom{\rule{3.33333pt}{0ex}}}^{\xb0}$C.
Figure 1.
Hsinchu benchmark: PPLR curves at $T=24.5{\phantom{\rule{3.33333pt}{0ex}}}^{\xb0}$C.
Figure 2.
Estimated chillers’ PPLR curves on case study 1 (left) and case study 2 (right).
Figure 2.
Estimated chillers’ PPLR curves on case study 1 (left) and case study 2 (right).
Figure 3.
Cooling load demand profile for case study 1 (left) and case study 2 (right).
Figure 3.
Cooling load demand profile for case study 1 (left) and case study 2 (right).
Figure 4.
Cooling load demand time series.
Figure 4.
Cooling load demand time series.
Figure 5.
Threedimensional representation of power consumption models. Red dots represent the experimental data.
Figure 5.
Threedimensional representation of power consumption models. Red dots represent the experimental data.
Figure 6.
GoodnessofFit (GOF) plots of the quadratic power consumption models. Blue dots are the validation data.
Figure 6.
GoodnessofFit (GOF) plots of the quadratic power consumption models. Blue dots are the validation data.
Figure 7.
3D histogram of the pairs $({Q}_{load},T)$ recorded from February 2017 to January 2019.
Figure 7.
3D histogram of the pairs $({Q}_{load},T)$ recorded from February 2017 to January 2019.
Figure 8.
Condenser inlet water temperature time series.
Figure 8.
Condenser inlet water temperature time series.
Figure 9.
PPLR curves of Field data benchmark at $T=$ 21.5 ${}^{\xb0}$C.
Figure 9.
PPLR curves of Field data benchmark at $T=$ 21.5 ${}^{\xb0}$C.
Figure 10.
Weekly power consumption time series: comparison bewteen actual recorded data (red), the lower bound ROCL (black) and the consumption associated with XOCS (blue).
Figure 10.
Weekly power consumption time series: comparison bewteen actual recorded data (red), the lower bound ROCL (black) and the consumption associated with XOCS (blue).
Figure 11.
Simulated XOCL experiment: average computation time per load.
Figure 11.
Simulated XOCL experiment: average computation time per load.
Table 1.
Comparison of the results of IFA, DCSA, GAMS, TLBO, EIWO and DCEDA on Hsinchu benchmark.
Table 1.
Comparison of the results of IFA, DCSA, GAMS, TLBO, EIWO and DCEDA on Hsinchu benchmark.
${\mathit{Q}}_{\mathit{load}}$ [kW]  i  IFA (2013)  DCSA (2014)  GAMS (2015)  TLBO (2017)  EIWO (2018)  DCEDA (2020) 

${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW] 

$\mathbf{6858}(\mathbf{90}\%)$  1  0.812774    0.812726    0.8127  808.9736  0.8186    0.8127    0.8126   
2  0.749527    0.749619    0.7496  740.7275  0.7523    0.7492    0.7489   
3  1.000000    1.000000    1.0000  903.3450  1.0000    1.0000    1.0000   
4  1.000000    1.000000    1.0000  781.4890  1.0000    1.0000    1.0000   
5  1.000000    1.000000    1.0000  755.2010  1.0000    1.0000    1.0000   
6  0.838603    0.838559    0.8386  748.8392  0.8297    0.8390    0.8395   
∑   $\mathbf{4738.576}$   $\mathbf{4738.575}$   $\mathbf{4738.5753}$   $\mathbf{4738.54}$   $\mathbf{4738.575}$   $\mathbf{4738.58}$ 
$\mathbf{6477}(\mathbf{85}\%)$  1  0.727803    0.727731    0.7277  718.5040  0.727731    0.7275    0.7280   
2  0.656174    0.656132    0.6561  641.1960  0.656132    0.6563    0.6564   
3  1.000000    1.000000    1.0000  903.3450  1.000000    1.0000    1.0000   
4  1.000000    1.000000    1.0000  781.4890  1.000000    1.0000    1.0000   
5  1.000000    1.000000    1.0000  755.2010  1.000000    1.0000    1.0000   
6  0.716408    0.716524    0.7165  621.9135  0.716524    0.7166    0.7160   
∑   $\mathbf{4421.649}$   $\mathbf{4421.649}$   $\mathbf{4421.6486}$   $\mathbf{4421.65}$   $\mathbf{4421.649}$   $\mathbf{4421.65}$ 
$\mathbf{6096}(\mathbf{80}\%)$  1  0.642725    0.642735    0.6427  639.1411  0.6431    0.6427    0.6431   
2  0.562642    0.562645    0.5626  553.8955  0.5621    0.5628    0.5622   
3  1.000000    1.000000    1.0000  903.3450  1.000000    1.0000    1.0000   
4  1.000000    1.000000    1.0000  781.4890  1.000000    1.0000    1.0000   
5  1.000000    1.000000    1.0000  755.2010  1.000000    1.0000    1.0000   
6  0.594504    0.594490    0.5945  510.6347  0.5946    0.5944    0.5946   
∑   $\mathbf{4143.706}$   $\mathbf{4143.706}$   $\mathbf{4143.7064}$   $\mathbf{4143.64}$   $\mathbf{4143.706}$   $\mathbf{4143.71}$ 
$\mathbf{5717}(\mathbf{75}\%)$  1  0.842218    0.843697    0.0000  0.0000  0.55765    0.0000    0.0000   
2  0.781365    0.783794    0.7150  702.4809  0.46918    0.7151    0.7144   
3  0.000002    0.000001    1.0000  903.3450  0.99995    1.0000    1.0000   
4  0.999995    1.000000    1.0000  781.4890  1.00000    1.0000    1.0000   
5  1.000000    1.000000    1.0000  755.2010  1.00000    1.0000    1.0000   
6  0.887053    0.883049    0.7934  700.0373  0.47250    0.7933    0.7941   
∑   ${\mathbf{3840.063}}^{*}$   ${\mathbf{3840.055}}^{*}$   $\mathbf{3842.5532}$   $\mathbf{3904.70}$   $\mathbf{3842.553}$   $\mathbf{3843.07}$ 
$\mathbf{5334}(\mathbf{70}\%)$  1  0.759350    0.749969    0.0000  0.0000  0.64179    0.0000    0.0000   
2  0.691121    0.682477    0.5835  572.3074  0.66219    0.5834    0.5831   
3  0.000021    0.000012    1.0000  903.3450  0.33009    1.0000    1.0000   
4  1.000000    1.000000    1.0000  781.4890  0.99059    1.0000    1.0000   
5  1.000000    1.000000    1.0000  755.2010  0.99900    1.0000    1.0000   
6  0.757897    0.776363    0.6217  534.0950  0.58047    0.6218    0.6221   
∑   ${\mathbf{3507.286}}^{*}$   ${\mathbf{3507.270}}^{*}$   $\mathbf{3546.4375}$   $\mathbf{3642.51}$   $\mathbf{3546.438}$   $\mathbf{3546.48}$ 
Table 2.
Published chillers’ power consumptions and the corresponding corrected values.
Table 2.
Published chillers’ power consumptions and the corresponding corrected values.
${\mathit{Q}}_{\mathit{load}}$ [kW]  i  IFA  DCSA 

${\mathit{PLR}}_{\mathit{i}}[]$  ${\mathit{P}}_{\mathit{p},\mathit{i}}$ [kW]  ${\mathit{P}}_{\mathit{c},\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}[]$  ${\mathit{P}}_{\mathit{p},\mathit{i}}$ [kW]  ${\mathit{P}}_{\mathit{c},\mathit{i}}$ [kW] 

$\mathbf{5717}(\mathbf{75}\%)$  1  0.842218  843.0026  843.0026  0.843697  844.7431  844.7431 
2  0.781365  777.3214  777.3214  0.783794  780.1789  780.1789 
3  0.000002  −120.5019  0.0000  0.000001  −120.5035  0.0000 
4  0.999995  781.4855  781.4855  1.000000  781.4890  781.4890 
5  1.000000  755.2010  755.2010  1.000000  755.2010  755.2010 
6  0.887053  803.5604  803.5604  0.883049  798.9460  798.9460 
∑   $3840.0690$  $3960.5709$   $3840.0545$  $3960.5580$ 
$\mathbf{5334}(\mathbf{70}\%)$  1  0.759350  750.8680  750.8680  0.749969  741.1047  741.1047 
2  0.691121  677.0164  677.0164  0.682477  668.0074  668.0074 
3  0.000021  −120.4730  0.0000  0.000012  −120.4867  0.0000 
4  1.000000  781.4890  781.4890  1.000000  781.4890  781.4890 
5  1.000000  755.2010  755.2010  1.000000  755.2010  755.2010 
6  0.757897  663.1834  663.1834  0.776363  682.1826  682.1826 
∑   $3507.2848$  $3627.7578$   $3507.4980$  $3627.9847$ 
Table 3.
Comparison of the results of IFA, DCSA, GAMS, TLBO, EIWO, DCEDA and XOCL on Hsinchu benchmark.
Table 3.
Comparison of the results of IFA, DCSA, GAMS, TLBO, EIWO, DCEDA and XOCL on Hsinchu benchmark.
${\mathit{Q}}_{\mathit{load}}$ [kW]  i  IFA (2013)  DCSA (2014)  GAMS (2015)  TLBO (2017)  EIWO (2018)  DCEDA (2020)  XOCL 

${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW]  ${\mathit{PLR}}_{\mathit{i}}$  ${\mathit{P}}_{\mathit{i}}$ [kW] 

$\mathbf{6858}(\mathbf{90}\%)$  1  0.812774  809.0541  0.812726  808.9999  0.8127  808.9736  0.8186  815.6654  0.8127  808.9705  0.8126  808.8595  0.8127  809.0002 
2  0.749527  740.6313  0.749619  740.7353  0.7496  740.7275  0.7523  743.7705  0.7492  740.2618  0.7489  739.9230  0.7496  740.7352 
3  1.000000  903.3450  1.000000  903.3450  1.0000  903.3450  1.0000  903.3450  1.0000  903.3450  1.0000  903.3450  1.0000  903.3450 
4  1.000000  781.4890  1.000000  781.4890  1.0000  781.4890  1.0000  781.4890  1.0000  781.4890  1.0000  781.4890  1.0000  781.4890 
5  1.000000  755.2010  1.000000  755.2010  1.0000  755.2010  1.0000  755.2010  1.0000  755.2010  1.0000  755.2010  1.0000  755.2010 
6  0.838603  748.8520  0.838559  748.8035  0.8386  748.8392  0.8297  739.0666  0.8390  749.2903  0.8395  749.8425  0.8386  748.8030 
∑   4738.576   4738.575   4738.5753   4738.54   4738.575   4738.66   4738.5733 
$\mathbf{6477}(\mathbf{85}\%)$  1  0.727803  718.5742  0.727731  718.5023  0.7277  718.5040  0.727731  718.5023  0.7275  718.2715  0.7280  718.7731  0.7277  718.5010 
2  0.656174  641.2379  0.656132  641.1959  0.6561  641.1960  0.656132  641.1959  0.6563  641.3638  0.6564  641.4637  0.6561  641.1960 
3  1.000000  903.3450  1.000000  903.3450  1.0000  903.3450  1.000000  903.3450  1.0000  903.3450  1.0000  903.3450  1.0000  903.3450 
4  1.000000  781.4890  1.000000  781.4890  1.0000  781.4890  1.000000  781.4890  1.0000  781.4890  1.0000  781.4890  1.0000  781.4890 
5  1.000000  755.2010  1.000000  755.2010  1.0000  755.2010  1.000000  755.2010  1.0000  755.2010  1.0000  755.2010  1.0000  755.2010 
6  0.716408  621.8000  0.716524  621.9132  0.7165  621.9135  0.716524  621.9132  0.7166  621.9874  0.7160  621.4020  0.7165  621.9136 
∑   4421.649   4421.649   4421.6486   4421.65   4421.649   4421.6739   4421.6466 
$\mathbf{6096}(\mathbf{80}\%)$  1  0.642725  639.1269  0.642735  639.1356  0.6427  639.1411  0.6431  639.4526  0.6427  639.1052  0.6431  639.4546  0.6427  639.1359 
2  0.562642  553.8980  0.562645  553.9006  0.5626  553.8955  0.5621  553.4276  0.5628  554.0352  0.5622  553.5144  0.5626  553.9010 
3  1.000000  903.3450  1.000000  903.3450  1.0000  903.3450  1.000000  903.3450  1.0000  903.3450  1.0000  903.3450  1.0000  903.3450 
4  1.000000  781.4890  1.000000  781.4890  1.0000  781.4890  1.000000  781.4890  1.0000  781.4890  1.0000  781.4890  1.0000  781.4890 
5  1.000000  755.2010  1.000000  755.2010  1.0000  755.2010  1.000000  755.2010  1.0000  755.2010  1.0000  755.2010  1.0000  755.2010 
6  0.594504  510.6442  0.594490  510.6324  0.5945  510.6347  0.5946  510.7257  0.5944  510.5561  0.5946  510.7257  0.5945  510.6325 
∑   $\mathbf{4143.706}$   $\mathbf{4143.706}$   $\mathbf{4143.7064}$   $\mathbf{4143.64}$   $\mathbf{4143.706}$   $\mathbf{4143.7297}$   $\mathbf{4143.3704}$ 
$\mathbf{5717}(\mathbf{75}\%)$  1  0.842218  843.0026  0.843697  844.7431  0.0000  0.0000  0.55765  570.8354  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000 
2  0.781365  777.3214  0.783794  780.1780  0.7150  702.4809  0.46918  478.8657  0.7151  702.5557  0.5831  701.7987  0.7150  702.4808 
3  0.000002  0.0000  0.000001  0.0000  1.0000  903.3450  0.99995  903.3189  1.0000  903.3450  1.0000  903.3450  1.0000  903.3450 
4  0.999995  781.4855  1.000000  781.4890  1.0000  781.4890  1.00000  781.4890  1.0000  781.4890  1.0000  781.4890  1.0000  781.4890 
5  1.000000  755.2010  1.000000  755.2010  1.0000  755.2010  1.00000  755.2010  1.0000  755.2010  1.0000  755.2010  1.0000  755.2010 
6  0.887053  803.5604  0.883049  798.9460  0.7934  700.0373  0.47250  414.9916  0.7933  699.9229  0.7941  700.7682  0.7934  700.0374 
∑   $\mathbf{3960.579}$   $\mathbf{3960.558}$   $\mathbf{3842.5532}$   $\mathbf{3904.70}$   $\mathbf{3842.553}$   $\mathbf{3842.602}$   $\mathbf{3842.5532}$ 
$\mathbf{5334}(\mathbf{70}\%)$  1  0.759350  750.8680  0.749969  741.1047  0.0000  0.0000  0.64179  638.3158  0.0000  0.0000  0.000  0.0000  0.0000  0.0000 
2  0.691121  677.0164  0.682477  668.0074  0.5835  572.3074  0.66219  647.2751  0.5834  572.2236  0.5831  571.9545  0.5835  572.3068 
3  0.000021  0.0000  0.000012  0.0000  1.0000  903.3450  0.33009  328.4962  1.0000  903.3450  1.0000  903.3450  1.0000  903.3450 
4  1.000000  781.4890  1.000000  781.4890  1.0000  781.4890  0.99059  774.8702  1.0000  781.4890  1.0000  781.4890  1.0000  781.4890 
5  1.000000  781.2010  1.000000  755.2010  1.0000  755.2010  0.99900  754.7026  1.0000  755.2010  1.0000  755.2010  1.0000  755.2010 
6  0.757897  663.1834  0.776363  682.1826  0.6217  534.0950  0.58047  489.8474  0.6218  534.1803  0.6221  534.4433  0.6217  534.0957 
∑   $\mathbf{3627.758}$   $\mathbf{3627.98}$   $\mathbf{3546.4375}$   $\mathbf{3642.51}$   $\mathbf{3546.438}$   $\mathbf{3546.4327}$   $\mathbf{3546.6437}$ 
Table 4.
Case study 1.
$\mathit{Stage}$  $\mathit{Load}$  ROCS  DP  XOCS 

$\mathit{kW}$  $\mathit{Chiller}$  $\mathit{kW}$  $\mathit{Chiller}$  $\mathit{kW}$  $\mathit{Chiller}$  $[{\mathit{PLR}}_{\mathbf{1}},{\mathit{PLR}}_{\mathbf{2}},{\mathit{PLR}}_{\mathbf{3}},{\mathit{PLR}}_{\mathbf{4}}]$ 

1  700  441.13  4  441.15  4  441.13  4  [0.00, 0.00, 0.00, 0.70] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
20  890  556.53  4  556.55  4  556.53  4  [0.00, 0.00, 0.00, 0.89] 
21  1100  695.32  4 2  695.56  4 2  695.32  4 2  [0.00, 0.63, 0.00, 0.82] 
22  1250  801.66  4 2  801.78  4 2  801.66  4 2  [0.00, 0.73, 0.00, 0.92] 
23  1100  695.32  4 2  695.56  4 2  695.32  4 2  [0.00, 0.63, 0,00, 0.82] 
24  1000  634.42  4 2  636.73  4 2  634.42  4 2  [0.00, 0.50, 0,00, 0.77] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
28  920  579.96  4  579.98  4  579.96  4  [0.00, 0.00, 0.00, 0.92] 
29  1610  1048.59  4 2 1  1048.43  4 2 1  1048.59  4 2 1  [0.75, 0.75, 0.00, 0.94] 
30  1650  1081.63  4 3  1082.07  4 2 1  1082.07  4 2 1  [0.76, 0.77, 0.00, 0.96] 
31  1670  1096.43  4 3  1099.69  4 2 1  1099.43  4 2 1  [0.77, 0.78, 0.00, 0.97] 
32  1700  1118.96  4 3  1122.12  4 2 3  1118.96  4 3  [0.00, 0.00, 0.78, 0.91] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
36  1900  1246.74  4 3 2  1247.61  4 2 3  1246.74  4 3 2  [0.00, 0.67, 0.74, 0.86] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
43  1750  1150.68  4 3 2  1151.60  4 2 3  1150.68  4 2 3  [0.00, 0.61, 0.69, 0.79] 
44  1575  1020.82  4 2 1  1030.56  4 3  1020.82  4 2 1  [0.74, 0.73, 0.00, 0.92] 
45  1900  1246.81  4 3 2  1250.32  4 3 1  1249.42  4 3 1  [0.70, 0.00, 0.73, 0.85] 
46  1790  1175.08  4 3 2  1180.75  4 3 1  1179.56  4 3 1  [0.67, 0.00, 0.69, 0.79] 
47  1725  1135.32  4 3 2  1142.76  4 3 1  1135.32  4 3 2  [0.00, 0.50, 0.70, 0.80] 
48  1775  1165.89  4 3 2  1166.78  4 3 2  1165.89  4 3 2  [0.00, 0.62, 0.69, 0.80] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
57  1900  1246.75  4 3 2  1247.61  4 3 2  1246.75  4 3 2  [0.00, 0.67, 0.74, 0.86] 
58  2100  1385.03  4 3 2 1  1386.08  4 3 2 1  1385.03  4 3 2 1  [0.68, 0.63, 0.70, 0.81] 
59  2000  1319.66  4 3 1  1326.20  4 3 2 1  1319.66  4 3 1  [0.73, 0.00, 0.77, 0.90] 
60  2100  1385.03  4 3 2 1  1386.08  4 3 2 1  1385.03  4 3 2 1  [0.68, 0.63, 0.70, 0.81] 
61  1900  1246.75  4 3 2  1247.61  4 3 2  1246.75  4 3 2  [0.00, 0.67, 0.74, 0.86] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
67  1750  1150.68  4 3 2  1151.60  4 3 2  1150.68  4 3 2  [0.00, 0.61, 0.69, 0.79] 
68  1475  948.04  4 2 1  948.08  4 2 1  948.04  4 2 1  [0.71, 0.67, 0.00, 0.85] 
69  1200  763.59  4 2  765.20  4 1  765.45  4 1  [0.72, 0.00, 0.00, 0.88] 
70  1050  664.47  4 2  671.50  4 1  670.46  4 1  [0.65, 0.00, 0.00, 0.76] 
71  1100  695.32  4 2  700.10  4 1  695.32  4 2  [0.00, 0.63, 0.00, 0.82] 
72  800  495.80  4  495.82  4  539.23  4 2  [0.00, 0.50, 0.00, 0.57] 
${P}_{day}$  64,432.56 kW  64,883.84 *
kW  64,495.91 kW 
Table 5.
Case study 3.
$\mathit{Stage}$  $\mathit{Load}$  XOCL  DP  XOCS 

$\mathit{kW}$  $\mathit{chiller}$  $\mathit{kW}$  $\mathit{Chiller}$  $\mathit{kW}$  $\mathit{Chiller}$  $[{\mathit{PLR}}_{\mathbf{1}},{\mathit{PLR}}_{\mathbf{2}},{\mathit{PLR}}_{\mathbf{3}},{\mathit{PLR}}_{\mathbf{4}},{\mathit{PLR}}_{\mathbf{5}},{\mathit{PLR}}_{\mathbf{6}},{\mathit{PLR}}_{\mathbf{7}},{\mathit{PLR}}_{\mathbf{8}},{\mathit{PLR}}_{\mathbf{9}}]$ 

1  6210  2899.68  8 6 3 2 1  2899.69  8 6 3 2 1  2899.68  8 6 3 2 1  [1.00 1.00 0.97 0.00 0.00 1.00 0.00 1.00 0.00] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
21  6150  2848.50  8 6 3 2 1  2852.34  8 6 3 2 1  2848.50  8 6 3 2 1  [1.00 1.00 0.92 0.00 0.00 1.00 0.00 1.00 0.00] 
22  6280  2974.84  8 6 3 2 1 9  3039.86  8 6 3 2 1 7  2974.84  8 6 3 2 1 9  [1.00 1.00 0.72 0.00 0.00 0.80 0.00 1.00 0.50] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
27  7355  3658.93  8 6 3 2 1 7  3672.69  8 6 3 2 1 7  3658.93  8 6 3 2 1 7  [1.00 1.00 0.93 0.00 0.00 1.00 0.95 1.00 0.00] 
28  7520  3768.28  8 6 3 2 1 7 9  3847.81  8 6 3 2 1 7 9  3768.28  8 6 3 2 1 7 9  [1.00 1.00 0.79 0.00 0.00 1.00 0.73 1.00 0.50] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
34  8720  4659.67  8 6 3 2 1 7 9  4659.65  8 6 3 2 1 7 9  4659.67  8 6 3 2 1 7 9  [1.00 1.00 0.98 0.00 0.00 1.00 1.00 1.00 1.00] 
35  8910  4791.00  8 6 3 2 1 7 9 4  4891.13  8 6 3 2 1 7 9 4  4791.00  8 6 3 2 1 7 9 4  [1.00 1.00 0.85 0.74 0.00 1.00 0.83 1.00 0.70] 
36  9090  4917.89  8 6 3 2 1 7 9 4  5015.07  8 6 3 2 1 7 9 4  4917.89  8 6 3 2 1 7 9 4  [1.00 1.00 0.88 0.77 0.00 1.00 0.87 1.00 0.74] 
37  9245  5032.60  8 6 3 2 1 7 9 4  5125.26  8 6 3 2 1 7 9 4  5032.60  8 6 3 2 1 7 9 4  [1.00 1.00 0.91 0.80 0.00 1.00 0.91 1.00 0.78] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
47  9820  5511.19  8 6 3 2 1 7 9 4  5552.25  8 6 3 2 1 7 9 4  5511.19  8 6 3 2 1 7 9 4  [1.00 1.00 1.00 0.90 0.00 1.00 1.00 1.00 0.96] 
48  9870  5552.25  8 6 3 2 1 7 9 4 5  5582.17  8 6 3 2 1 7 9 4  5552.25  8 6 3 2 1 7 9 4 5  [1.00 1.00 0.87 0.76 0.67 1.00 0.86 1.00 0.73] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
63  8780  4703.10  8 6 3 2 1 7 9 4  4804.11  8 6 3 2 1 7 9 4  4703.10  8 6 3 2 1 7 9 4  [1.00 1.00 0.83 0.73 0.00 1.00 0.80 1.00 0.66] 
64  8555  4511.80  8 6 3 2 1 7 9  4543.81  8 6 3 2 1 7 9  4511.80  8 6 3 2 1 7 9  [1.00 1.00 0.96 0.00 0.00 1.00 1.00 1.00 0.88] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
67  7740  3907.96  8 6 3 2 1 7 9  3985.58  8 6 3 2 1 7 9  3907.96  8 6 3 2 1 7 9  [1.00 1.00 0.81 0.00 0.00 1.00 0.76 1.00 0.62] 
68  7455  3728.52  8 6 3 2 1 7 9  3745.20  8 6 3 2 1 7  3728.52  8 6 3 2 1 7 9  [1.00 1.00 0.77 0.00 0.00 0.98 0.71 1.00 0.50] 
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 
72  6425  3055.91  8 6 3 2 1 9  3115.53  8 6 3 2 1 7  3055.91  8 6 3 2 1 9  [1.00 1.00 0.74 0.00 0.00 0.89 0.00 1.00 0.50] 
${P}_{day}$  296,440.08 kW  289,525.25 * kW  295,605.88 kW 
Table 6.
Estimated parameters of the quadratic power consumption model (
2) and corresponding
$CV\%$.
Table 6.
Estimated parameters of the quadratic power consumption model (
2) and corresponding
$CV\%$.
Chiller  ${\mathit{\beta}}_{0,\mathit{i}}$  ${\mathit{\beta}}_{1,\mathit{i}}$  ${\mathit{\beta}}_{2,\mathit{i}}$  ${\mathit{\beta}}_{3,\mathit{i}}$  ${\mathit{Q}}_{\mathit{nom}}$ 

($\mathit{CV}\%$)  ($\mathit{CV}\%$)  ($\mathit{CV}\%$)  ($\mathit{CV}\%$)  [kW] 

1  49.6367  124.7681  269.5769  3.1950  2700 
(5.24)  (2.50)  (0.82)  (3.21) 
2  56.2047  154.9900  278.0211  2.2997  2700 
(11.97)  (7.98)  (2.95)  (8.32) 
3  −10.9883  520.3125  −45.1983  −0.3309  2700 
(22.49)  (0.85)  (6.98)  (24.57) 
4  −159.0637  112.2988  15.7524  14.1461  2700 
(6.15)  (19.80)  (8.21)  (0.74) 
5  46.7748  461.7221  −8.3474  −1.1835  2700 
(27.74)  (1.84)  (79.87)  (47.23) 
Table 7.
Execution times for OCL and OSC benchmarks.
Table 7.
Execution times for OCL and OSC benchmarks.
 N${}^{\xb0}$ Chillers  N${}^{\xb0}$ Loads  XOCL [s]  ROCS [s]  XOCS [s]  GAMS [s] 

OCL Hsinchu benchmark  6  5  0.110      1.991 
OCS benchmark 1  4  72    0.302  0.368   
OCS benchmark 2  9  72    226.77  228.29   
OCS Field data  5  68,110    354.03  573.96   
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