Applied Energy 144 (2015) 51–63

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Applied Energy

journal homepage: www.elsevier .com/locate /apenergy

Optimization of a network of compressors in parallel: Real TimeOptimization (RTO) of compressors in chemical plants – An industrialcase study

http://dx.doi.org/10.1016/j.apenergy.2015.01.0100306-2619/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: d.xenos@imperial.ac.uk (D.P. Xenos).

Dionysios P. Xenos a,⇑, Matteo Cicciotti b,c, Georgios M. Kopanos a,d, Ala E.F. Bouaswaig c, Olaf Kahrs c,Ricardo Martinez-Botas b, Nina F. Thornhill a

a Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, UKb Department of Mechanical Engineering, Imperial College London, UKc Advanced Process Control, Automation Technology, BASF SE, Ludwigshafen, Germanyd Cranfield University, School of Energy, Environment & Agrifood, Bedfordshire, UK

h i g h l i g h t s

� Presentation of integrated optimization framework of a network of compressors.� Application to an industrial case study.� Presentation and demonstration of a real time application for optimizing compressors.� Inclusion of a comprehensive literature review of the optimization of networks of compressors.

a r t i c l e i n f o

Article history:Received 18 July 2014Received in revised form 2 January 2015Accepted 4 January 2015Available online 6 February 2015

Keywords:Real time optimizationIndustrial compressorsOptimal load sharingMathematical programmingRegression modelsEnergy savings

a b s t r a c t

The aim of this paper is to present a methodology for optimizing the operation of compressors in parallelin process industries. Compressors in parallel can be found in many applications for example in compres-sor stations conveying gas through long pipelines and in chemical plants in which compressors supplyraw or processed materials to downstream processes. The current work presents an optimization frame-work for compressor stations which describe integration of a short term and a long term optimizationapproach. The short-term part of the framework suggests the best distribution of the load of the compres-sors (where the time scale is minutes) and the long-term optimization provides the scheduling of thecompressors for large time periods (where the time scale is days). The paper focuses on the short-termoptimization and presents a Real Time Optimization (RTO) framework which exploits process data insteady-state operation to develop regression models of compressors. An optimization model employsthe updated steady-state models to estimate the best distribution of the load of the compressors toreduce power consumption and therefore operational costs. The paper demonstrates the application ofthe RTO to a network of parallel industrial multi-stage centrifugal compressors, part of a chemical processin BASF SE, Germany. The results from the RTO application showed a reduction in power consumptioncompared to operation with equal load split strategy.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction assuming different levels of operational control tasks (e.g. process

Compressors are machines which are used in industrial pro-cesses to provide air for combustion, to recirculate fluids througha process and to convey a gas through a pipe. Many researchershave studied the optimal operation of compressors (i) consideringdifferent applications (e.g. natural gas or supply gas systems), (ii)

control or planning), (iii) studying different scales of systems (e.g. alocal compressor station or large gas networks involving compres-sor stations in series) and (iv) examining of compressors using dif-ferent type of gas (e.g. air or natural gas).

The purpose of this paper is to present a general methodology tooptimize compressors in parallel. This methodology consists of theintegration of the scheduling of compressors and a real time optimi-zation approach. The paper introduces the overall methodology andfocuses on the demonstration of the Real Time Optimization (RTO)

Fig. 1. A compressor map with Inlet Guide Vanes (IGVs).

52 D.P. Xenos et al. / Applied Energy 144 (2015) 51–63

method to optimally share the load among parallel compressors.The methodology is applied to an industrial case study. The mainaim is to show that the suggested RTO framework can be appliedonline, and can reduce the power which is utilized from the motorsof the compressors compared to conventional industrial policies ofoperation. Moreover, the paper introduces the scheduling of thecompressors: optimal selection of the compressors taking intoaccount aspects of the operation, such as maintenance plans andminimum, and maximum running times.

The structure of the manuscript is as follows. Section 2 providesan introduction to the operation of compressors attached to adownstream system and the description of the management andoperational tasks taking place in a plant. The latter part will helpto identify the control actions which take place in a compressorstation. Section 3 provides a literature review to the optimizationof compressor stations or systems which involve compressors.The integrated framework and the methodology of the RTO forcompressors is described in Section 4. Section 5 provides thedescription of the case study. Section 6 presents the results anddiscussions. The paper ends with a summary and conclusions.

2. Introduction to the operation of compressors

2.1. Operation of compressors attached to a downstream system

It is generally accepted that compressors consume largeamounts of energy in various industrial sectors. Compressors dri-ven by gas turbines are reported to consume 5% of the transportedgas in pipeline networks of natural gas [1]. Moreover the utility forcompressed air is considered to be one of the most expensive inmany industries [2]. Compressors are assumed to be among themajor energy consumers in many intensive chemical processessuch as air separation.

The paper focuses on industrial multi-stage centrifugal com-pressors. Centrifugal compressors are used in applications whichrequest high mass flow rates and low pressure ratios [3]. Toachieve high ratios of compression, several single stages of centrif-ugal compressors are arranged in series. The single stages in seriesare attached to a rotating shaft. This structure is called multi-stagecentrifugal compressor and it is known as compressor train. Acompressor train is employed to increase the total discharge pres-sure compared to the pressure a single-stage compressor canachieve.

More than one single centrifugal stage or compressor train canbe connected in parallel to increase the total mass flow rate. Com-pressors operating in parallel are known as compressor stations.

A compressor is connected with a downstream load and, usu-ally, with an upstream process (in the air separation process, how-ever, the compressor is supplied with ambient air, therefore thereis no upstream process). A standard graphical representation usedto describe the operation of a compressor is a compressor map,which can be seen in Fig. 1.

A compressor map shows the characteristics, performance andoperational limits of a compressor, i.e. surge, choke, minimumand maximum speeds [4]. Surge restricts the operation when acompressor works at lower mass flow rates and higher pressures.When the compressors work at higher flows and lower pressureschoke restricts the operation. Moreover, the operation of a com-pressor is restricted due to minimum and maximum power pro-vided from its driver. These restrictions can be identified in acompressor map considering a minimum and a maximum speed.In the case of a constant speed compressor with variable InletGuide Vanes (IGVs) the operation is restricted between minimumand maximum opening of the IGVs. Fig. 1 illustrates the opera-tional region of a compressor between surge and choke for theminimum (10�) and maximum angle (100�) of the IGVs.

The operating point of a compressor is the intersection of itscharacteristic curve at a fixed rotational speed (or angle of IGV)and the characteristic curve of a downstream system. The charac-teristic of the downstream system is known as a load curve ordemand curve [5]. The reason for introducing these concepts isto explain that the downstream process influences the operationof the compressor, and therefore its performance and power con-sumption. Moreover, the data to be used for developing modelsof compressors correspond to various operating points.

Fig. 2a illustrates three load curves of different downstreamprocesses. It is assumed that the upstream process does not changethe inlet conditions of the compressor. Kurz et al. [5] describedthree main categories of downstream processes a compressor isusually connected with. Load curve A describes a pipeline systemin which the pressure becomes greater when the mass flowthrough the pipes increases. Load curve B is used to represent sys-tems which the pressure does not change significantly with thechange in mass flow. Refrigeration systems and process systemsin which gas is fed into at a specific discharge pressure are typicalexamples of this category [6]. Finally, load curve C describes gasstorage applications [7].

Fig. 2a also shows an Operating Point (OP) of the compressordescribed from this compressor map. The compressor is connectedwith a downstream system described by load curve A. The OP onthe compressor map gives information about the mass flowðmOPÞ, pressure ðpOPÞ of the gas supplied, opening of the Inlet GuideVanes (hOP) and isentropic efficiency ðgis;OPÞ.

There are several methods to control a compressor if an operat-ing point has to be modified. Lipták [8] and Kurz et al. [5] reportedfive main control methods: (1) suction throttling, (2) dischargethrottling, (3) flow recycling, (4) adjustment of the speed of themotor and (5) modification of the Inlet Guide Vanes. The paperexamines compressors with constant speed and adjustable IGVs.Fig. 2b shows an example of changing operating point from OP1to OP2. It shows that by increasing the opening of the IGVS by20� the OP changes. The compressor at OP2 provides higher massflow at a relatively higher pressure than in the case of OP1 andmoreover the efficiency of the compressor drops to 0.7. This infor-mation, pressure, mass flow and efficiency, can be used to estimatethe power consumed from the compressor (power in the shaft).Finally, by knowing the efficiency of the driver, the power con-sumed by the driver can be computed.

2.2. Management and control operational tasks

Fig. 3a shows a typical decision (or control tasks) pyramid ofplant-wide automation according to the ANSI/ISA-95 standard[9]. It involves the Process Control System (PCS) which includesreal time set point control and real time monitoring of the process.

(a) (b)

Fig. 2. Various types of load curves and Operating Point (OP) of a compressor connected with load curve A (a) and operating point change by increasing the position of theInlet Guide Vanes (b).

Fig. 3. Decision pyramids of a plant (a) and of a compressor station (b) according to the ANSI/ISA-95 [9].

D.P. Xenos et al. / Applied Energy 144 (2015) 51–63 53

The time scale of PCS is seconds or fractions of seconds. The Man-ufacturing Execution System (MES) deals with the manufacturingoperations and considers decisions for example the scheduling ofthe units of the operations for the next days or hours and the opti-mization of the process by improving its performance at real time.The optimization suggests the best set points of the control system.On the top of the pyramid is the Enterprise Resource Planning(ERP) considering business planning and logistics. The top levelof ERP deals with markets, production targets and sales.

Fig. 3b presents a decision pyramid of the operations of a com-pressor station. At the top of this pyramid, in Level 4, there are theproduction targets from the Central Dispatch Department [10].These targets are the input of Level 3. Level 3 includes the sched-uling and maintenance of the compressors. The same level alsodeals with the optimal distribution of the load among the compres-sors which provides set points, for example the set points of thespeeds of the motors of the compressors, to the controllers of thecontrol system. The role of the controllers is to operate the com-pressors at these given set points. The lowest levels, Level 0, 1, 2,involve the control of the individual units and the monitoring ofthe process. The RTO and the scheduling of compressors are con-sidered as control tasks which belong in Level 3 of the decision pyr-amids in Fig. 3a and b.

3. Literature review

3.1. Optimization of compressor stations

The optimization of compressor stations is a topic which manyresearchers have studied in the recent years. Compressors are usedin various applications where the nature of each application

influences the objectives and the constraints of the optimizationproblem. For instance, problems examining natural gas systemswhich transport gas through long pipes can consider the phenom-enon of the linepack [5]. The linepack is a time-dependent phe-nomenon which describes the storage of a gas inside a pipe. Thestored gas can be used for example when the operation cannot sat-isfy the demand due to an unexpected failure of a compressor. Onthe other hand, a supply compressor station of a chemical processprovides gas at an approximate constant pressure.

3.1.1. Optimization of compressors regarding the applicationFig. 4 presents the classification of the optimization of compres-

sor stations considering four main categories of applications: (i)natural gas networks [11–14], (ii) utilities [15–17], (iii) gas storageapplications [7] and (iv) other applications [18]. Furthermore, theexamination of the optimization problem can consider differenttime horizons, continuous operation or/and discrete events.

3.1.2. Optimization of compressors regarding the time horizonMany authors examined the optimization of different types of

gas compressors in different applications considering differenttime horizons. According to Fig. 4, there is the classification of sin-gle- and multi-period optimization. The single-period optimizationconsiders information of one time interval (steady-state) and themulti-period optimization employs information from the futurebased on forecasting methods. The forecast of the demand andparameters of the constraints are used in the optimization frame-work. The optimization framework suggests decisions for the cur-rent operation and these decisions are based on future information.An example of a forecasted parameter (i.e. future information) is a

Fig. 4. Classification of optimization of compressor stations. The current paper focuses on the optimal load sharing of utilities applications in steady-state conditions, i.e.single-period optimization.

54 D.P. Xenos et al. / Applied Energy 144 (2015) 51–63

scheduled maintenance of a compressor, which might influencethe decisions for the current operation.

The single-period optimization can be further classified assteady-state pipeline optimization [12,14,19,20], optimal loadsharing [10,15,21] and optimal selection of compressors[10,13,16,18,22]. The steady-state pipeline optimization examinesthe optimal operation of the fuel cost minimization problem [14]considering information of one period [20]. The optimal load shar-ing is the focus on this paper and it will be presented in Section 3.2.The third category involves discrete events in the formulation ofthe optimization problem, for example variables which representwhich compressor stations operate (on or off) [13] or binary vari-ables for deciding assignments of gas-lift compressors to end-users(wells) [18].

The multi-period optimal operation of compressors (consider-ing solution for more than one time period) includes (1) pipelineoptimization with a fixed number of operating compressors[20,23,24] and (2) optimal selection (or scheduling) of compressors[25–27].

3.2. Optimal load sharing

Kurz et al. [5] and Garcia-Hernandez and Brun [28] reportedthat the installation of spare stand-by compressors in a stationincreases its flexibility. Spare compressors are used when thecapacity of a station is not enough to satisfy the demand whichis requested due to changes in the demand side of the plant. Thesechanges are mainly caused because of changes in the markets(prices of products or electricity) and changes in the internal pro-duction strategies of the company (products specifications, amountof production).

Many authors and practitioners reported that it is difficult orimpossible for different compressors in a compressor station tohave identical characteristics and efficiencies [8,21,29,30]. More-over, these characteristics and efficiencies change over time dueto fouling and erosion [31], and nonuniform maintenance planswhich result in dissimilar compressor maps for the same compres-sor at different time periods [10,32,33]. The use of surrogate mod-els and process data can predict the performance andcharacteristics of the compressors [34].

Many strategies have been used to share the load of compres-sors. Kurz et al. [5] commented that if compressors have identicalcompressor maps, the load can be equally split or they can

operate at the same surge margin. Surge margin is the distancebetween operating point and surge. In the same work it wasreported that if two compressors have different sizes or differentefficiencies the more efficient should provide the base load andthe second compressor, the less efficient, has to deal with thefluctuations of the load. On the other hand Twohig [35] reportedthat parallel compressors in a pharmaceutical fermentation pro-cess were decided to operate so as the most aged (and less effi-cient) compressor worked at its maximum flow rate (where itsoptimal point was) and the more efficient compressors shouldvary their operation to cover the fluctuations of the demand.Moreover, Lipták [8] suggested a method in which the first stepis to estimate the efficiencies of the compressors in units of flowper units of power and the second is to load the units in order totheir efficiencies.

Another option to distribute optimally the load among the com-pressors is to formulate an optimization problem. A few research-ers [15,21,36] studied the optimal load sharing of compressorsoperating in parallel in order to minimize the fuel consumed bythe gas turbines drivers. However, these works did not presentan online application which considers the practical aspects of theimplementation of the actual optimization, for example update ofthe maps and steady-state detection. Paparella et al. [10] presentedan online optimization framework which updates the parametersof the models of the compressors online. The optimization frame-work applied to a gas boosting station. The authors showed thatsurrogate models can be used to predict the performance of thecompressors.

There is significant research contribution in the optimization ofutility systems for steam production [37,38]. The optimization inthese works involved the estimation of the distribution of load ofeach turbine to minimize operational costs. The methodology usedin these works is very similar to the optimization of the utilities forcompressed air. However, there is limited research regarding theoptimization of compressor stations for utilities.

3.3. Gap of knowledge and contribution of the paper

Study of the literature revealed a lack of a systematic way tooptimally share the load of compressors online considering varyingoperational conditions, such as atmospheric temperature and pres-sure, and demand requested from downstream processes, suchas air separation and compressed air for utilities. One of the

D.P. Xenos et al. / Applied Energy 144 (2015) 51–63 55

assumptions mainly used is that individual compressors have thesame characteristics and the same performance behaviour. Regard-ing this assumption, the conventional practice is to distribute theload evenly among the compressors or to apply other similar strat-egies described previously. In addition, a few works presentedoptimization strategies to share the load, however to the best ofthe authors’ knowledge there is no approach which considers acomprehensive online application with aspects such as update ofmaps and steady-state detection.

The current work suggests a comprehensive framework whichdeals with the optimization of compressors in parallel in two dif-ferent timescales, long- and short-term optimization. After the pre-sentation of the general framework, one of the aims of this paper isto present a methodology for generating models of constant speedmulti-stage centrifugal compressors with a water cooling systemand an Inlet Guide Vane (IGV) control system. Real industrial dataare used to generate these models.

In the literature, studies of compressor modelling focus on per-formance map-based or simulation-based cases [39]. The paperfocuses on a real application involving industrial compressors.The maps of the compressors are not available. Therefore, the effi-ciencies for different operational conditions are not given.

Moreover, the compressors are multi-stage involving intercool-ers between the stages. The lack of measurements, for example thenon availability of all the temperatures of the intercoolers, is themain reason the efficiency of the multi-stage compressors cannotbe explicitly estimated through thermodynamic calculations. Forthe requirements of the formulation of the optimization the useof the power of the motors of the compressors considers implicitlythe efficiency of the compressors as this is illustrated in Section 6.2.The paper, therefore, suggests a method to minimize the opera-tional costs of compressors in parallel when the efficiency of thecompressors cannot be explicitly estimated.

Another aim of the paper is to present the part of the frameworkwhich deals with the short-term optimization. The data-drivenmodels are implemented into an optimization model which com-putes the distribution of the load among the compressors in orderto achieve reduced operational costs. The proposed optimization isformulated in a general way and can be applied to a compressorstation with parallel nonidentical single-stage or multi-stage com-pressors in utilities or in process systems when operational dataare available.

Fig. 5. Integrated framework for op

4. Methodology

4.1. General framework for optimizing compressors in parallel

Fig. 5 presents the integrated framework for the optimization ofcompressors in parallel. This framework connects several decisiontasks from Level 0, 1, 2 and 3 of the automation pyramid in Fig. 3b:optimal scheduling and maintenance, optimal load sharing andupdate of the maps, and control and monitoring of the process.The optimal scheduling and maintenance focuses on decisions ofdiscrete events such as the switching on or off of compressors.The optimal load sharing considers the updates of compressormaps which corresponds to the asset monitoring in Level 3 inthe MES of pyramid in Fig. 3a. The process control applies the setpoints, given from the optimal load sharing to the process. Finally,the monitoring system collects data, analyses them, detects anddiagnoses faults.

A basic Real Time Optimization (RTO) scheme [40] can be tai-lored to the operation of parallel compressors. The sensors of themonitoring system collect process data of the operation such asmass flows, pressures and temperatures. A steady-state identifica-tion algorithm examines key process variables and identifies whenthe operation is in steady-state. If the system is in steady state thecollected data are validated [40,41]. The validated data are used toupdate the models of the compressors.

A NonLinear Programming (NLP) model employs these data-driven models and estimates the optimal load sharing, i.e. the setpoints of the controlled variables, i.e. mass flow rates of the com-pressors. The set points are given to the control system which roleis to apply and keep these points to the process until the next runof the RTO.

The scheduling problem is a Mixed Integer Linear Programming(MILP) problem which involves both continuous and binary vari-ables as degrees of freedom of the optimization. The schedulinggives high-level decisions which involve discrete events (for exam-ple switching on or off a compressor) to the RTO. This input is givenin a relatively large time interval compared to the timescale of theRTO.

When the actual demand given in the RTO is significantly differ-ent from this which is predicted in the scheduling problem, theNLP problem may result in an infeasible problem. This is becausethe online compressors may not be able to meet the requirements

timizing compressor stations.

Fig. 6. General methodology for the development of data-driven models.

Fig. 7. Moving time data set window.

56 D.P. Xenos et al. / Applied Energy 144 (2015) 51–63

of the demand side. In this case the scheduling problem updates itsmodels and a new schedule of the compressors is estimated. Thenew schedule gives the new configuration of the system whichcan satisfy the demand, for instance by bringing another compres-sor online. After these adjustments, the RTO loop is activated con-sidering the new configuration. Kopanos et al. [42] presents thestudy of the scheduling topic.

4.2. Modelling of compressors with data-driven models for RTOapplication

This section describes the methodology to develop models of ageneric centrifugal multi-stage compressor, with a cooling system,driven by an electrical motor with constant rotational speed. Themodels are developed from historical data from operation. Theuse of data-driven models allows the modelling of a multi-stagecompressor with available measurements only at the inlet of thefirst and at the exit of the last stage. A rigorous model requiresthe availability of measurements between the stages and analyticalmodels of the intercoolers. Moreover, the use of a data-drivenmodel in the optimization reduces the computational burden com-pared to the use of a rigorous model which considers aerodynamicsand thermodynamics of the fluid. The computational time of theoptimization plays an important role in an online application suchas the RTO.

4.2.1. Steady-state detectionThe steps of the development of data-driven models consist of

several procedures illustrated in Fig. 6. Xenos et al. [17] presentedthe description of each step of the methodology applied to anindustrial compressor. Data-driven models are black boxes whichhold a relationship between input and output variables. Theseinput and output variables should be close to steady state to holdthe validity of the mass and energy balances implied in the blackbox [43]. Therefore, the purpose of a steady-state detection algo-rithm is to identify the steady states of the operation to developreliable models.

In this work, a steady-state identification algorithm based on amoving window was developed. Fig. 7 shows the application of themoving window to a data-set of a single process variable. A movingdata-set window is defined from a fixed number of data points ofthe process variable, ns. The data included in the window areupdated at each step s, recent data are added and old data are dis-carded. The window moves every S number of data points.

A process variable j has value xði;j;tÞ, where i 2 I (I is the numberof compressors), j 2 J (J is the number of variables) and t 2 T (T isthe total number of data points which corresponds to the totalsample time). Variable s corresponds to a window with dataðt � ns þ 1; tÞ. The sample rate of the data set (discretization of T),the ns and the S are parameters which have to be tuned for adesired function of the steady-state identification algorithmregarding to the application.

At each step s the data-set window moves to t0 ¼ t þ S and thestandard deviation of the included data points in the window,rði;j;sÞ, of compressor i and variable j is calculated from:

rði;j;sÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ns

Xt02ðt�nsþ1;tÞ

ðxði;j;t0 Þ � lði;j;sÞÞ2

sð1Þ

with lði;j;sÞ is the mean of the data in the window:

lði;j;sÞ ¼1ns

Xt02ðt�nsþ1;tÞ

ðxi;t0 Þ ð2Þ

The steady-state algorithm detects a steady-state episode of aprocess variable j when a particular condition holds true, for exam-ple if the three times the standard deviation 3rði;j;sÞ is less than apredefined value, hði;jÞ. This bound is chosen by the users accordingto their engineering judgement as there are not standardized val-ues for this type of application.

The developed steady-state algorithm is multivariate andinvolves the examination of more than one variables to assess ifthe system is in steady-state. According to Mansour and Ellis[40] a system is in steady state when all the considered variablesare in steady state.

Hence, if a process variable j of compressor i is in steady state att then a binary variable Y ðss;i;j;tÞ is equal to 1, otherwise the variabletakes the value 0. Therefore, the steady state of the system of Icompressors considering j 2 J0 # J variables is estimated from thevalue of the variable Yss;system:

Yss;system ¼Yi2I

Yj2J0

Y ðss;i;j;tÞ ð3Þ

The output of the steady-state detection algorithm is a matrix ofdata with J0 rows (variables) and T� columns (number of finalsteady states of the system) where T�# T .

4.2.2. Development of modelsThe methods for preprocessing the data and generating the cal-

ibration and validation set of data are the same used in Xenos et al.[17]. For example data have to be normalized before the

D.P. Xenos et al. / Applied Energy 144 (2015) 51–63 57

development of the models. This is because different variableshave different units with different orders of magnitude. For exam-ple power is measured in kW and pressure in bar.

The top diagram in Fig. 8 illustrates a generic multi-stage com-pressor driven by an electrical motor. The development of a blackbox model of this compressor system includes: the multi-stagecompressor, the cooling system, the rotating shaft, gearbox andmotor. According to the step tests in an industrial centrifugal com-pressor presented in Xenos et al. [44], it was shown that the powerconsumption of the motor mainly depends on the mass flow enter-ing the compressor, ma, the ambient conditions, Tin; pin, and thepressure at the exit of the compressor (discharge pressure), pout .

The lower panel in Fig. 8 shows the procedure for the develop-ment of the model of the operation of a multi-stage compressor. Ablack box is used to predict the power (output of the model) con-sumed from the electrical driver, Pel, as a function of process vari-ables (input of the model) of the operation, mða;i;tÞ; Tin; pin; pout . Apolynomial regression model is used to develop the black box modelof each compressor i. By defining xði;k;tÞ ¼ ½mða;i;tÞ; T ðin;tÞ; pðin;tÞ; pðout;tÞ�with k ¼ 4, number of input variables, and yði;tÞ ¼ Pðel;i;tÞ, a blackbox model of compressor i is given by the following polynomial:

y�ði;tÞ ¼ bði;0Þ þ bði;1Þ � x�ði;1;tÞ þ bði;2Þ � x�ði;2;tÞ þ bði;3Þ � x�ði;3;tÞ þ bði;4Þ � x�ði;4;tÞþ bði;5Þ � x�2ði;1;tÞ þ bði;6Þ � x�2ði;2;tÞ þ bði;7Þ � x�2ði;3;tÞ þ bði;8Þ � x�2ði;4;tÞþ bði;9Þ � x�ði;1;tÞ � x�ði;2;tÞ þ bði;10Þ � x�ði;1;tÞ � x�ði;3;tÞ þ bði;11Þ � x�ði;1;tÞ � x�ði;4;tÞ

ð4Þ

where y�ði;tÞ ¼ yði;tÞ=ymaxi ; x�ði;j;tÞ ¼ xði;j;tÞ=xmax

ði;jÞ are the scaled variables ofthe regression models of compressors I. The xmax

ði;jÞ ; ymaxi are the max-

imum variables of their respective calibration and validation sets.The parameters of the models, bm;m ¼ 1; . . . ;12 are calculated withregression methods [45].

4.2.3. Assessment of the accuracy of the prediction of the modelsTo evaluate the accuracy of the prediction of the models the

Coefficient of Variation of the Root Mean Square Error, CV(RMSE)is used [46]:

CVðRMSEÞ ¼ RMSE

y� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPt2T�ðyðmeas;i;tÞ�yði;tÞÞ2

T�

qy�

and the coefficient of determination, known as R squared (RSQ)expresses how well the data fit the model [47]:

Sres ¼ 1�P

t2T� ðyðmeas;i;tÞ � yði;tÞÞ2

� �ðT� � 1Þ � vy

where �y is the mean and vy is the variance of the predicted values ofy and ymeas are the measured variables.

Fig. 8. Regression models of a multi-stage compressor driven by an electricalmotor.

4.3. Real Time Optimization (RTO) and optimization model

The offline steady-state identification algorithm, presented inSection 4.2 was modified for online applications. The onlinesteady-state identification examines if the Yss;system is 1 at the cur-rent moment, tr , considering J0 variables of each compressor i atthe window ðtr � ns � 1; trÞ.

According to Fig. 6, raw data are collected from the operationafter the steady-state identification and they are validatedthrough data reconciliation. The validated data are used to updatethe parameters of the models which are used from the optimiza-tion block and to provide the input parameters of the optimiza-tion model. The set points of the mass flows, which are theoutput of the optimization model, are the input of the control sys-tem. The controller can be a feedback controller. The controllerdeals with the application of the set points and adjusts the posi-tion of the actuators, Inlet Guide Vanes (IGVs), to achieve thedesired flows.

There is usually a mismatch between models and reality due tofitting errors and performance changes due to fouling and erosionas previously mentioned. These errors influence the shape of theobjective function of the optimization problem and consequentlythe estimation of the minimum of the total power consumed. Toreduce the influence of these errors, the mass flows of the com-pressors were chosen as degrees of freedom of the optimizationproblem. The position of the IGVs can be then adjusted from thecontrollers. An analysis on the developments of the models showedthat the accuracy in prediction of the models does not improve sig-nificantly when the position of the IGVs is considered as an inputvariable in the models.

Defining the input parameter vector of the optimization block(Fig. 5) z� ¼ ½x�2;tr

x�3;trx�4;tr� and the optimization variables (degrees

of freedom or manipulated variables of the optimization) massflows x�ði;tr Þ, then the optimal load distribution at a steady-stateepisode which starts at point t ¼ tr is computed by the followingoptimization formulation:

minx�Xi2I

Pðel;i;trÞ ð5Þ

subject to:

P�ðel;i;trÞ ¼ f iðz�;x�ði;trÞ;biÞ; i 2 I ð6ÞPðel;i;trÞ ¼ P�ðel;i;trÞ � y

maxi ; i 2 I ð7Þ

mða;i;trÞ ¼ x�ði;trÞ � xmaxði;1Þ; i 2 I ð8ÞX

i2I

mða;i;trÞ ¼ mðD;trÞ ð9Þ

mmini 6 mða;i;tr Þ; i 2 I ð10Þ

mða;i;trÞ 6 xmaxði;1Þ; i 2 I ð11Þ

Pminðel;iÞ 6 Pðel;i;trÞ; i 2 I ð12Þ

Pðel;i;trÞ 6 ymaxi ; i 2 I ð13Þ

Eq. (5) describes the objective function which is the minimization ofthe power consumption at steady state that starts at t ¼ tr and Eq.(6) describes the constraints which give the power of each compres-sor i as function of the z� and normalized mass flows x�ði;tr Þ. Eqs. (7)and (8) refer to the normalization of the powers and mass flows. Eq.(9) provides the mass balance between summation of the flows ofthe compressors and the demand, mðD;tr Þ, requested. Finally, Eqs.(10)–(13) define the regression domain.

The above optimization formulation was implemented inMatlab™ using the optimization function fmincon [48].

Fig. 9. The operation of three air multi-stage centrifugal compressors working inparallel to supply compressed air a downstream process (air separation column).The suggested control scheme can be seen here.

58 D.P. Xenos et al. / Applied Energy 144 (2015) 51–63

5. Description of the industrial case study

This section presents the application of the methodology to anair compressor station of multi-stage compressors in parallel, inBASF SE, Germany, which distribute compressed air to differentend-users. The end-users are air separation columns and plant-siteutilities for compressed air. The air compressor station consumesthe major part of the total energy in the air separation plant. More-over, the power rating of the plant is several tens of MWs.

The case study to be examined involves three air multi-stagecentrifugal compressors similar to the compressor depicted inFig. 8. The compressors operate in parallel to supply an air separa-tion column with compressed air. The air separation columnrequests compressed air of mass flow rate, mD, at a constant pres-sure, pop. The three compressors are assumed to have the samespecifications (power rate, minimum and maximum capacity andefficiencies) at the point of the commissioning of the plant. How-ever, the compressors are not in the same condition during thetime period of the study as will be shown in Section 6.

A description of the implemented control structure in theindustrial plant follows. The set point of the demand mD is givenas an input of the system to controller FC4. FT4 measures the totalmass flow of the compressed air provided by the compressors.Three controllers FC1, FC2 and FC3 give the same opening of theInlet Guide Vanes (IGVs) to each compressor in order to controlthe mass flows and meet the demand. By applying this controlstrategy the load should be shared equally among the compressors.The summation of the flows, total flow, measured from FT4 has tomatch with the mD. If there is a mismatch then the IGVs of all thecompressors are adjusted uniformly (open or close for all the com-pressors) to reach the desired mD. Referring to Fig. 9, this controlstructure does not include the RTO block and instead sends thesame point signal to each compressor.

The control strategy that the current paper suggests for optimalload sharing can be seen in Fig. 9. Three flow transmitters FT1, FT2and FT3 are used to transmit the individual flows at the exit of thecompressors. The RTO is placed between the FC4 and these flowtransmitters. In this case controllers FC1, FC2 and FC3 work inde-pendently and they receive the values of the flows of the compres-sors according to the RTO computations. The RTO receives thedemand from FC4 and estimates the best set points for FC1, FC2and FC3 based also on measurements of Tin; pin and pop. The indi-vidual feedback controllers FC1-FC3 have to independently adjustthe position of the IGVs of the compressors to reach the set pointsof the mass flows given from the RTO.

5.1. Practical challenges of the case study

The case study of the industrial air separation process hasrevealed several practical challenges which are summarized below.These challenges should be considered in order to achieve a realis-tic approach of the proposed methodology.

The operators did not operate the compressors over their fullrange in the collected data set. Therefore, only a partial compressormap is captured in the regression models. Hence, the feasible win-dow of operation of the model used for the optimization is notdefined from the actual physical limits of the compressor, i.e.surge, choke, and minimum and maximum Inlet Guide Vane open-ing. Instead, this feasible operational window of the compressor isdefined from the domain of the regression model [49]. The regres-sion domain is a part of the actual operational window.

Fig. 10a demonstrates that an operating point of the compres-sors comes from the intersection of the Compressor System (CS)curve and the load curve of the downstream process. The CS curveis defined as the merged individual compressor characteristics of

the parallel compressors assuming that the inlet conditionsare the same of all compressors. The set of all the operating pointsof the corresponding data set defines the regression domain of themodel of a compressor as can be seen in Fig. 10b. It is known fromthe plant that the operators operated well within the physical lim-its (surge and choke) of the compressors during the past operation.

A model is expected to be more accurate when the data set iscollected over a shorter time period, for example one week thana model derived from a data set of several months because forlonger period the compressor might have been in various statesduring this period. For example, a compressor is efficient immedi-ately after maintenance and less efficient after many hours of oper-ation. However, the range of the regression domain of the model issmaller in the case of a more accurate model.

In the case that the compressor was operated close to the surgeor choke line, the regression domain cannot be assumed rectangu-lar (see Fig. 10b) due to the physical restrictions. There are twooptions to deal with this issue: (a) use a convex hull for describingthe regression domain, or (b) add an extra constraint in the optimi-zation model described by Eqs. (6)–(13). The use of a convex hullcan be seen in Brooks et al. [49] and in Mitra et al. [50]. The convexhull ensures a more tight regression domain which increases thevalidity of the model in its boundaries [51]. The extra constraintcan be a regression black box model of the outlet pressure whichrelates the mass flow and other parameters such as ambient tem-perature and pressure, pi;out ¼ giðz;mða;iÞ; ciÞ; i 2 I where ci is thevector of the fitted parameters of the new regression model.

The data reconciliation step requires redundant measurements,for example extra measurements of the flow apart from these atthe exit of the compressors. Unfortunately the industrial case studydid not have these measurements, and hence the data reconcilia-tion step could not be implemented.

6. Results

6.1. Models of compressors

Data from fifteen days of operation were used to develop themodels of the compressors. The case study assumed that the per-formance of the models deteriorates gradually and that therewas no event which has caused a relatively high discontinuousdecrease in performance. Data of 129600 continuous operatingpoints with 10 s sample interval (0.1 Hz sample rate) were col-lected. The steady-state algorithm detected 7430 steady-state epi-sodes in the examined data set for the system of the compressors.

(a) (b)

Fig. 10. An operating point of the system defined by intersection between load curve and characteristic of the system (CS curve) (a) and the feasible window of operation of acompressor, i.e. regression domain (b).

D.P. Xenos et al. / Applied Energy 144 (2015) 51–63 59

Of these episodes, 80% were used for the fitting of the model andthe remaining 20% were used to validate the developed model.

Table 1 presents the statistics of the fitting, RSQ, and the valida-tion, RSQ and CV(RMSE), of the models of the three compressors.The standard function LinearModel.fit of Matlab™ [52] was usedto fit the data into the polynomial of Eq. (4). Fig. 11 shows the pre-diction of the normalized power consumption of each compressorversus the measurements of the power. The axes present normal-ized values of the power due to confidentiality restrictions. Thenormalization of the power consumption of each compressor iscalculated by dividing the power with the ymax

i .Compressor i1 shows the most accurate match between predic-

tion and actual measurements of the power. The fitting of themodel has also higher RSQ value than in the other cases. In the caseof compressor i2 the accuracy of the prediction is relatively lessthan the other two compressors. The reason for this is that themeasurements of the mass flow and power of compressor i2 arerelatively more noisy than in the case of the other two compres-sors. The CV(RMSE) of the models ranges between 0.54% and0.67%. By comparing the mean RMSE results with other resultsfrom similar case studies [15,17] these models can be consideredof high accuracy for predicting power consumption ofcompressors.

Table 2 presents the minimum and maximum bounds of themass flows and powers of the three compressors. The valuesshown in Table 2 represent the boundaries of mass flows and pow-ers where they are divided by two scaling factors to keep the con-fidentiality agreement with the provider of the case study. Thetable shows that compressors have different regression domainswith compressor i1 having a larger feasible operational windowthan the other two.

6.2. Illustrative example with industrial compressors

An illustrative example of the two compressors i1 and i2 explainsthe optimal load sharing using the optimization presented inSection 4.3. In this example, only two compressors are taken intoaccount into the optimization model. The rate of total mass flowwhich has to be delivered from the two compressors is m�D. The sum-mation of the mass flows of the two compressors mða;i1Þ and mða;i2Þhas to be equal to the mass flow rate of the demand.

Table 1Statistics of the fitting and validation of the regression models.

Compressors Fitting Validation ValidationRSQ RSQ CV(RMSE)

i1 0.992 0.992 0.54i2 0.968 0.967 0.67i3 0.987 0.988 0.57

Fig. 12 presents the normalized power consumed from the twocompressors individually and the combined normalized powerconsumption on vertical axis as a function of the normalized massflow of the first compressor, mða;i1Þ on horizontal axis. Therefore,given the mða;i1Þ, the mða;i2Þ equals to m�D �mða;i1Þ. By increasing themass flow of the first compressor, the compressor i1 consumesmore and on the other hand compressor i2 consumes less power,assuming that all the other parameters are kept fixed, e.g. inlettemperature and pressure of the downstream process. Moreover,the compressors are restricted to operate above a minimum massflow rate boundary, corresponding to compressor i1 equal tomaxfmmin

ða;i1Þ;m�D �mmax

ða;i2Þg ¼ 0:670 normalized units (n.u) and a

maximum equal to minfmmaxða;i1Þ;m

�D �mmin

ða;i2Þg ¼ 0:795 n.u. when thevalue of the mass flow of the demand is 1.471 n.u.

The actual operation (point described in Fig. 12) is defined as theoperation which took place in reality and the power consumed fromthe compressors is a result of the mass flows from the existing con-trol scheme. Compressors i1 and i2 operated at mða;i1Þ = 0.735 n.u.and mða;i2Þ = 0.736 n.u. which can be assumed that the load was splitevenly. As can be seen from Fig. 12, compressor i1 consumes morepower than i2 by 2.1% under these conditions. This means that thecompressors are in a different condition.

From the combined curve (Comp. i1þ i2), it can be observed thatin the actual compressor operation the mass flow of i1 ismða;i1Þ ¼ 0:735 n.u. In this case, the total consumption is higher thanoperating at the point which compressor i1 has mass flow ratemða;i1Þ ¼ 0:795 n.u. The reduction in power is 1.54% in this case. Inother words, this observation demonstrates that compressor i1 ismore efficient than compressor i2 at these conditions. The morethe compressor i1 operates against compressor i2 the higher is thetotal power reduction in the available search space. The upperboundary results due to the minimum mass flow boundary of com-pressor i1.

The above analysis and graphical representation is feasible fortwo compressors but if more than two compressors are involvedin a network then optimization (i.e. mathematical programming)deals with the estimation of the minimum value of the objectivefunction while ensuring that the constraints hold valid.

6.3. Demonstration of Real Time Optimization (RTO) application inparallel with real operation

Section 6.3 examines the application of the developed RTOmethodology on a simulation of real time operation. The historicaldata are simulated as if given in real time and the RTO runs in par-allel with the operation of the system in Fig. 9. The RTO estimatesthe optimal load sharing and the results coming from these com-putations are compared with the operation that took place in real-ity. The RTO methodology was applied to the compressor system ofthe three parallel compressors for more than 12 h.

Fig. 11. Prediction versus actual values of mass flow of compressors i1; i2 and i3 in the validation set.

Table 2Boundaries of mass flows and power consumptions. The units are dimensionless andscaled.

Compressors mminða;iÞ xmax

ði;1Þ Pminðel;iÞ

ymaxi

i1 0.670 0.924 0.304 0.406i2 0.676 0.836 0.295 0.365i3 0.624 0.842 0.286 0.374

60 D.P. Xenos et al. / Applied Energy 144 (2015) 51–63

The steady-state detection is configured to examine when thethree compressors are in steady state simultaneously for 40 s.The inputs of the optimization are collected during this period.After the RTO calculations, the system is examined if it is still insteady state and if this holds true the RTO results are given tothe proposed control system (see Fig. 9). The online steady-state

Fig. 12. Example of the optimization

identification algorithm detected 50 steady state episodes wherethe first 16 can be seen in Fig. 13.

It was assumed that the compressor conditions do not changesignificantly during the 15 days period of the collected data in Sec-tion 6.1 and the updated compressor maps from the 15 days timewindow were used in the optimization. Fig. 14 shows the normal-ized power consumption of the three compressors in three differ-ent cases for the examined 12 h: (a) actual operation, (b) equalsplit operation and (c) optimal operation. Table 3 describes theassumptions to be taken to estimate the power consumed in thesethree different cases.

The power consumption in Fig. 14 is a calculated quantitywhose errors depend on errors in the quantities used in the calcu-lation. The difference between the actual operation and equal split

of two compressors in parallel.

Fig. 13. The first sixteen steady-state episodes of the system of the compressors.

5 10 15 20 25 30 35 40 45 500.94

0.95

0.96

0.97

0.98

0.99

1

Number of steady state episodes

Nom

raliz

ed p

ower

con

sum

ptio

n (%

)

Actual operationEqual split operationOptimal operation

Fig. 14. Normalized total power consumption of the three compressors from actual,equal split and optimal operation.

Table 3Three different cases of operation.

Case Mass flows to estimate power using Eq. (4)

Actual operation The mass flows are given from the real dataEqual split operation The demand is split to equal mass flow ratesOptimal operation The mass flows are given from the optimization

5 10 15 20 25 30 35 40 45 500.65

0.7

0.75

0.8

0.85

0.9

Number of steady state episodes

Com

pres

sor i1

norm

aliz

ed m

ass

flow

(%)

Actual operationEqual split operation

Min/max boundariesOptimal operation

Fig. 15. Compressor i1 normalized mass flow rate from three different cases.

5 10 15 20 25 30 35 40 45 500.65

0.7

0.75

0.8

0.85

0.9

Number of steady state episodes

Com

pres

sor i2

norm

aliz

ed m

ass

flow

(%)

Actual operationEqual split operation

Min/max boundariesOptimal operation

Fig. 16. Compressor i2 normalized mass flow rate from three different cases.

5 10 15 20 25 30 35 40 45 500.65

0.7

0.75

0.8

0.85

0.9

Number of steady state episodes

Com

pres

sor i3

norm

aliz

ed m

ass

flow

(%)

Actual operationEqual split operation

Min/max boundariesOptimal operation

Fig. 17. Compressor i3 normalized mass flow rate from three different cases.

D.P. Xenos et al. / Applied Energy 144 (2015) 51–63 61

operation might be attributed to random variability in the data.However the power consumption for optimal operation is system-atically lower and the difference cannot be accounted for by ran-dom statistical variation.

The measurement error in the mass flow is less than ±0.005 onthe normalized mass flow scale. This applies to the results inFigs. 14–20.

Fig. 14 shows that the actual operation and the equal split strat-egy do not show an important difference in the total power con-sumed from the compressors. Moreover, Figs. 15–17 show thatthe compressors are not equally split in the actual operation. Espe-cially in the case of compressor i3 it can be seen that there is a dif-ference between the mass flow from equal split operation andactual operation. Although the values of the mass flows in the caseof the equal split are different from the mass flows from the actual

operation, there is not much difference in the total powerconsumption.

On the other hand, Fig. 14 shows that the optimization achievesreduction in the total power consumption in all 50 steady-state

Fig. 18. Power consumption of compressors i1 and i2 from optimization and actual operation (equal split).

Fig. 19. Compressor i1 normalized mass flow rate from actual and optimaloperation.

Fig. 20. Compressor i2 normalized mass flow rate from actual and optimaloperation.

62 D.P. Xenos et al. / Applied Energy 144 (2015) 51–63

periods. Fig. 15 shows that compressor i1 should work at highermass flows than in the case of the actual and equal split operationand Fig. 16 shows that compressor i2 should work at lower massflows. Fig. 17 demonstrates that the optimal operation suggeststhat compressor i3 has to work at mass flow rates very close tothe mass flows the actual operation which took place. In otherwords, the optimization estimated that compressor i3 was

operated well during the actual operation but compressor i1 asmore efficient should be loaded more than compressor i2 whichseems to be less efficient under the current operational conditions.

It was observed that in actual operation compressor i1 and i2were operated with equal load and compressor i3 dealt with theremaining load. The RTO method was applied exclusively to thetwo compressors i1 and i2 in the case of the 50 steady-state epi-sodes. This is because the previous results from the optimizationshowed that compressor i3 should not change operating pointfrom this of the actual operation, therefore the mass flow of com-pressor i3 does not have any degree of freedom in the optimization.

In this new case study, Fig. 18a shows the power consumptionof the two compressors from the actual and optimal operation. Itcan be seen that compressor i1 consumes more power comparedto compressor i2 in the actual operation in all 50 cases where theequal split strategy was applied. This shows again that compres-sors are in different performance conditions. Moreover, compres-sor i1 is less efficient than compressor i2 when the load isequally shared.

The results of the optimization give a lower total power con-sumption of the compressors compared to the power consumptioncoming from the actual operation. Compressor i1 is loaded moreand compressor i2 less. This strategy results in giving a lower totalpower consumption. This means that compressors i1 and i2 workmore efficiently when they operate at the mass flows that theRTO suggests. These mass flows can be seen in Figs. 19 and 20.

The suggested RTO framework can be applied to multiple com-pressors in parallel. The RTO can provide the automation of theoptimization of compressors in real-time. The online steady-statedetection algorithm detects when the operation is constant andthe RTO scheme computes the optimal distribution of the loadamong the online compressors considering their updated charac-teristics and performances. The duration of transient operationbetween two steady states is much smaller compared to the lengthof time the compressors stay at their new operating point, hencethe optimization of the transients of the system can be neglected.

The configuration of the online compressors is given from thesecond part of the framework in Fig. 5. Therefore, the schedulingof the compressors is also needed to provide the best selection ofcompressors. The study from Kopanos et al. [42] deals with theoptimal scheduling of the compressors.

7. Conclusions

The paper presented the state-of-the-art of the optimization ofcompressor stations involving multiple industrial compressors

D.P. Xenos et al. / Applied Energy 144 (2015) 51–63 63

operating in parallel. It also suggested an integrate framework tooptimize the operation of compressors for short and long timeperiods. The contribution of this paper is the presentation of anintegrate framework focusing on an online Real Time Optimization(RTO) method which collects raw data from the process. The dataare used to update the models of the compressors which are usedinto an optimization framework which reduces the power con-sumption in a steady-state period. The optimization deals with dif-ferent operational conditions such as inlet temperature andpressure.

A real industrial case study of an air compressors station, part ofan energy-intensive chemical process in BASF SE, Germany con-suming several tens of MW was optimized using the developedRTO methodology. The comparison between the RTO applicationand the actual operation taken place in reality showed that theRTO method has the potential to reduce the total power consump-tion of the compressors.

Acknowledgments

Financial support from the Marie Curie FP7-ITN project ‘‘Energysavings from smart operation of electrical, process and mechanicalequipment – ENERGY SMARTOPS’’, Contract No: PITN-GA-2010-264940 is gratefully acknowledged. The authors would like tothank BASF SE for providing a case study and technical support.

References

[1] DeMarco FCG, Elias, GP. Fuel consumption model on natural gas compressionstations driven by two-shaft gas turbine. In: PSIG 1107, PSIG annual meeting,Napa Valley, California, 24 May–27 May 2011; 2011.

[2] Saidur R, Rahim NA, Hasanuzzaman M. A review on compressed-air energy useand energy savings. Renew Sustain Energy Rev 2010;14:1135–53.

[3] Boyce MPPE. Centrifugal compressors: a basic guide. Oklahoma: Pen WellCorporation; 2003.

[4] Dixon SL, Hall CA. Fluid mechanics and thermodynamics of turbomachinery.6th ed. Oxford (UK): Elsevier; 2010.

[5] Kurz R, Lubomirsky M, Brun K. Gas compressor station economic optimization.Int J Rotating Mach 2012;12:1–9. http://dx.doi.org/10.1155/2012/715017.

[6] Bloch HP. A practical guide to compressor technology. 2nd ed. Hoboken(NJ): John Wiley & Sons, Inc.; 2006.

[7] Kurz R, Brun K. Assessment of compressors in gas storage applications. J EngGas Turbines Power 2010;132:062402–1:7. http://dx.doi.org/10.1115/1.4000147.

[8] Lipták BG. Process control and optimization, compressor control andoptimization. 2006 Instrument Engineer’s Handbook, vol. II, 4th ed., p. 1763–92 [chapter 8.15].

[9] Harjunkoski I, Nyström R, Horch A. Integration of scheduling and control –theory and practice. Comput Chem Eng 2009;33:1909–18.

[10] Paparella F, Domínguez L, Cortinovis A, Mercangöz M, Pareschi D, Bittanti S.Load sharing optimization of parallel compressors. In: European controlconference (ECC), July 17–19, 2013, Zürich, Switzerland; 2013.

[11] Shaw DC. Pipeline system optimization: a tutorial. Houston: ScientificSoftware-Intercomp; 1994.

[12] Wu S, Rios-Mercado RZ, Boydm EA, Scott LR. Model relaxations for the fuel costminimization of steady-state gas pipeline networks. Math Comput Model2000;31:197–220.

[13] Cobos-Zaleta D, Ríos-Mercado RZ. A MINLP model for minimizing fuelconsumption on natural gas pipeline networks. In: Memorias del XICongreso Latino Iberoamericano de Investigacion de Operaciones (CLAIO),27–31 October 2002, Chile; 2002.

[14] Borraz-Sánchez C. Optimisation methods for pipeline transportation of naturalgas. PhD dissertation. Norway: University of Bergen, Department ofInformatics; 2010.

[15] Han I-S, Han C, Chung C-B. Optimisation of the air- and gas-supply network of achemical plant. Trans IChemE, Part A, Chem Eng Res Des 2004;82(A10):1337–43.

[16] Widell KN, Eikevik T. Reducing power consumption in multi-compressorrefrigeration systems. Int J Refrig 2010;33:88–94.

[17] Xenos DP, Cicciotti M, Bouaswaig AEF, Thornhill NF, Martinez-Botas R.Modeling and optimization of industrial centrifugal compressor stationsemploying data-driven methods. In: Proceedings of ASME Turbo Expo 2014,GT2014-25089, June 16–20, 2014, Düsseldorf, Germany; 2014.

[18] Camponogara E, Nazari LF, Meneses CN. A revised model for compressordesign and scheduling in gas-lifted oil fields. IIE Trans 2012;44(5):342–51.

[19] Wong JP, Larson ER. Optimisation of tree-structured natural-gas transmissionnetworks. J Math Anal Appl 1968;24:613–26.

[20] Carter R, Reisner M, Sekirnjak E. Transient optimization – examples anddirections. In: PSIG 1011, PSIG annual meeting, Florida 11 May–14 May 2010;2010.

[21] Abbaspour M, Chapman KS, Krishnaswami P. Nonisothermal compressorstation optimisation. J Energy Resour Technol, Trans ASME 2005;127:131–41.

[22] Wright S, Somani M, Ditzel C. Compressor station optimization. Denver (CO):Pipeline Simulation Interest Group (PSIG); October 28–30, 1998.

[23] Marques D, Morari M. On-line optimisation of gas pipeline networks.Automatica 1988;24(4):455–69.

[24] Abbaspour M, Krishnaswami P, Chapman KS. Transient optimisation in naturalgas compressor stations for linepack operation. J Energy Resour Technol, TransASME 2007;129:314–24.

[25] van den Heever SA, Grossmann IE. A strategy for the integration of productionplanning and reactive scheduling in the optimisation of a hydrogen supplynetwork. Comput Chem Eng 2003;27:1813–39.

[26] Nguyen HH, Uraikul V, Chan CW, Tontiwachwuthikul P. A comparison ofautomation techniques for optimization of compressor scheduling. Adv EngSoftw 2008;39:178–88.

[27] Mahlke D, Martin A, Moritz S. A mixed integer approach for time-dependentgas network optimisation. Optim Methods Softw 2009;25(4):625–44.

[28] Garcia-Hernandez A, Brun K. Energy usage in natural gas pipeline applications.ASME J Eng Gas Turbines Power 2012;134:022402-1–2-9.

[29] Milum R. Multi-compressor capacity optimization. In: Petrotech, 7th pipelinetechnology conference 2012. A Petrotch, Inc. White paper; 2012.

[30] Rolls Royce. <www.rolls-royce.com/energy/energy_products/> [accessed06.02.14].

[31] Kurz R, Brun K. Fouling mechanisms in axial compressors. J Eng Gas TurbinesPower 2012;134:032401-1–1-9. http://dx.doi.org/10.1115/1.4004403.

[32] Forsthoffer WE. Forsthoffer’s best practice handbook for rotatingmachinery. Boston: Butterworth-Heinemann; 2011.

[33] Cicciotti M, Xenos DP, Bouaswaig AEF, Thornhill NF, Martinez-Botas R. Onlineperformance monitoring of industrial compressors using meanline modelling.In: Proceedings of ASME Turbo Expo 2014, GT2014-25088, June 16–20, 2014,Düsseldorf, Germany; 2014.

[34] Tirnovan R, Giurgea S, Miraoui A, Cirrincione M. Surrogate modelling ofcompressor characteristics for fuel-cell applications. Appl Energy2008;85:394–403.

[35] Twohig D. Utility optimization: driving economic performance through theutilization of automation technologies. Automation Island, InTech; 2011.<www.isa.org> [accessed 28.09.13].

[36] Jenícek T, Kralík J. Optimised control of generalized compressor station. In:27th Annual meeting pipeline simulation interest group (PSIG), October 18–20, Albuquerque, New Mexico; 1995.

[37] Luo X, Zhang B, Chen Y, Mo S. Operational planning optimization of steampower plants considering equipment failure in petrochemical complex. ApplEnergy 2013;11:1247–64.

[38] Varbanov PS, Doyle S, Smith R. Modeling and optimization of utility systems.Trans IChemE, Part A, Chem Eng Res Des 2004;82(A5):561–78.

[39] Cicciotti M, Xenos DP, Bouaswaig AEF, Thornhill NF, Martinez-Botas R. Physicalmodelling of industrial multistage centrifugal compressors for monitoring andsimulation. J Mech Eng Sci 2014.

[40] Mansour M, Ellis JE. Methodology of on-line optimisation applied to achemical reactor. Appl Math Model 2008;32:170–84.

[41] Martinez Prata D, Schwaab M, Luis Lima E, Carlos Pinto J. Simultaneous robustdata reconciliation and gross error detection through particle swarmoptimization for an industrial polypropylene reactor. Chem Eng Sci2010;65:4943–54.

[42] Kopanos GM, Xenos DP, Cicciotti M, Pistikopoulos EN, Thonrhill NF.Operational planning of networks of compressor stations: The air separationplant case. Appl Energy 2015.

[43] Kim M, Yoon SH, Domanski PA, Payne WV. Design of a steady-state detector forfault detection and diagnosis of a residential air conditioner. Int J Refrig2008;31:790–9.

[44] Xenos DP, Cicciotti M, Bouaswaig AEF, Thornhill NF. Preprocessing of raw datafor developing steady-state data-driven models for optimizing compressorstations. In: 10th International conference on control, Control 2014,Loughborough, UK; 2014.

[45] Rosipal R, Kramer N. Overview and recent advances in partial least squares. In:Subspace, latent structure and feature selection: statistical and optimizationperspectives workshop (SLSFS 2005) revised selected papers. Lecture notes incomputer science, vol. 3940. Berlin, Germany: Springer-Verlag; 2006. p. 3451.

[46] Wikipedia. Root-mean-square deviation. <en.wikipedia.org/wiki/Root-mean-square_deviation> [accessed 25.04.14].

[47] Matlab Linear Regression. <mathworks.co.uk/help/matlab/data_analysis/linear-regression.html> [accessed 25.04.14].

[48] Matlab Optimization Toolbox. mathworks.co.uk/help/optim/ug/fmincon.html> [accessed 25.04.14].

[49] Brooks DG, Carroll SS, Verdini WA. Characterizing the domain of a regressionmodel. Am Stat 1988;42(3):187–90.

[50] Mitra S, Grossmann IE, Pinto JM, Arora N. Optimal production planning undertime-sensitive electricity prices for continuous power-intensive processes.Comput Chem Eng 2012;38:171–84.

[51] Kahrs O, Marquardt W. The validity of hybrid models and its application inprocess optimization. Chem Eng Process 2007;46:1054–66.

[52] Matlab statistics toolbox. <mathworks.co.uk/help/stats/linearmodel.fit.html>[accessed 25.04.14].