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Computation of energy efficient driving speeds in conveying systems

Modell-basierte Optimierungsverfahren zur Berechnung energie effizienter Bewegungsprofile in Förderanwendungen
Stefan Windmann, Oliver Niggemann and Heiko Stichweh

Abstract

This article addresses the automatic optimization of driving speeds in conveying systems. Electric drives in existing conveying systems are usually accelerated and decelerated according to predetermined movement profiles. Such an approach is inflexible for conveying applications with changing constraints and, in many cases, not optimal with respect to energy efficiency. In the present work, a method for automatic computation of energy efficient movement profiles is proposed. The proposed method is based on accurate models for electric drives and several types of conveying applications such as roll conveyors, belt conveyors and vertical conveyors. Furthermore, joint energy efficiency optimization for two drives, which are attached to an intermediate circuit, is investigated. Thereby, additional constraints on the energy flow between the drives are imposed in order to reduce load peaks and energy feedback into the grid. The resulting optimization problem is a mixed integer quadratic program (MIQP), which can be solved in a few milliseconds. Experimental results show that energy losses of electric drives are cut down by using the obtained non-trivial movement profiles instead of standard trapezoid movement profiles. The additional constraints on the energy flow between two drives lead to further significant improvements with respect to the overall energy losses.

Zusammenfassung

Der Beitrag adressiert die energie-effiziente Ansteuerung der elektrischen Antriebe in Förderanwendungen. Elektrische Antriebe in Förderanwendungen werden in der Regel entsprechend vordefinierter Bewegungsprofile beschleunigt und abgebremst. Ein solcher Ansatz ist für Förderanwendungen mit wechselnden Randbedingungen unflexibel und in vielen Fällen nicht energie effizient. In der vorgestellten Arbeit wird daher eine Methode zur automatischen Optimierung der Bewegungsprofile vorgeschlagen. Die vorgeschlagene Methode basiert auf genauen Modellen der elektrischen Antriebe und verschiedener Arten von Förderanwendungen wie Rollenförderern, Bandförderern und Vertikalförderern. Weiterhin wird die gemeinsame Optimierung für zwei Antriebe untersucht, die an einen Zwischenkreis angeschlossen sind. Dabei werden Randbedingungen bzgl. des Energieflusses zwischen den beiden Antrieben berücksichtigt, die es ermöglichen, Lastspitzen und Rückspeisungen in das Netz zu reduzieren. Insgesamt ergibt sich ein gemischt-ganzzahliges Optimierungsproblem, welches in wenigen Millisekunden gelöst werden kann. Experimentelle Untersuchungen zeigen, dass die Verwendung der optimierten, nicht-trivialen Bewegungsprofile anstelle trapezförmiger Bewegungsprofile zu einer signifikanten Reduktion des Energieverbrauchs führt. Insbesondere durch die zusätzlichen Randbedingungen bzgl. des Energieflusses zwischen den beiden Antrieben ergeben sich deutliche Einsparungen.

Appendix A Logic and magnitudes in MIQP

Logic and magnitudes are expressed in MIQP as follows [23]:

1. Alternative restriction groups/equivalences

(31)ifx1C1s=1,ifx2C2s=0

with continuous variables x1, x2, continuous bounds C1, C2 and 0-1-variable s can be written as

(32)x1Mx(1s)+C1,x2Mxs+C2,

where Mx denotes a big constant, which should not constrain the range of x. In this work, values of

(33)Mx=xmaxxmax

are used, where xmin and xmax denote known limits for the range of x.

2. Propositional formulas with atomic formulas P1,P2,,Pk can be written as constraints with 0-1-variables y1,y2,,yk:

(34)P1:y1=1,¬P1:(1y1)=1
(35)P1P2Pk:y1+y2++yk1
(36)P1P2:y1=1andy1=1.

3. Magnitudes |x| of variables x are modeled by writing

(37)x=x(+)x(),|x|=x(+)+x()
(38)x(+)0,x()0
and adding the λ-factor λx|x|=λx(x(+)+x()) to the objective function, which asserts that either x(+) or x() is forced to be zero. In doing so, inefficient case-by-case analysis is avoided [1].

4. The signs sx of variables x can be defined as 0-1-variables [1]:

(39)x0sx=1.

Eq. (39) can be written as

(40)x()0sx=1,x(+)0sx=0

with x() and x(+) according to eq. (37).

5. The motor modes Q (Q=0: regenerative mode, Q=1: motoric mode) of electric drives are related to the signs of motor speeds and motor torques [25]:

(41)(SuSm)(¬Su¬Sm)Q
(42)(¬Su¬SmQ)(SuSmQ)(Su¬Sm¬Q)(¬SuSm¬Q)
with Boolean variables Su=1 and Sm=1 for positive signs of u and m, respectively. The Boolean eqs. (42) can be converted by means of (34)–(36) into the following set of MIQP constraints:

(43)su+smq1,susmq1su+sm+q1,susm+q1

with integer variables su, sm and q.

Appendix B Conveying applications

The three conveying applications considered in section 2.2 are detailed in this appendix. The operating principles of the three conveying applications are shown in Fig. 7.

Figure 7 Operating principles of the conveying applications: a) belt conveyor b) roll conveyor c) vertical conveyor.

Figure 7

Operating principles of the conveying applications: a) belt conveyor b) roll conveyor c) vertical conveyor.

The belt conveyor employs a drive roll with radius Rroll to transport the ware on a belt. In the roll conveyor, N transport rolls connected to the drive roll are used for transportation. The drive roll of the vertical conveyor moves the ware in vertical direction. In all three applications, the drive roll is powered by a drive. Velocity v of the conveyor and motor speed U of the drive are related by radius Rroll of the drive roll:

(44)U=60sminigearv2πRroll,

where igear denotes the gear factor. The drive has to realize torque

(45)Mm=mwRrollafor the horizontal conveyors a) and b)mwRroll(a+g)+mbeltRrollafor the vertical conveyor c)

to cause acceleration a of a ware with mass mw. The increased torque for vertical conveyors is due to gravity g=9.81ms2 and acceleration of the belt with mass mbelt, which is only significant for the vertical conveyor. The influence of gravity on the belt is neutralized due to the symmetry of parts, which are moved downwards and upwards, respectively. To accelerate the rolls in the respective conveying application, additional torque

(46)Mroll=J/Rroll

is required, where J denotes the overall inertia of the rolls in the respective application:

(47)J=Jrollfor the belt conveyor a) andthe horizontal conveyor c)NJroll+Jdrivefor the roll conveyor b)

with inertia Jroll of the drive roll for the belt conveyor and the vertical conveyor, and inertias Jroll and Jdrive of the transport rolls and the drive system of the roll conveyor, respectively.

Additionally, torque Mfriction is required to compensate for friction, which depends on the application:

(48)Mfriction=μfc+mwRrollμfgg+MωRrollvfor the belt conveyor a) andthe vertical conveyor c)mwg(2μmountRroll+μf)for the roll conveyor b)

with friction factors μfc, μfg, Mω, μmount and μf [13]. In total, torque

(49)M=(Mm+Mroll+Mfriction)/igear

is required. Transformation of (44) and (49) leads to the roll conveyor coefficients in Table 2.

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Received: 2017-9-7
Accepted: 2018-2-6
Published Online: 2018-4-6
Published in Print: 2018-4-25

© 2018 Walter de Gruyter GmbH, Berlin/Boston