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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


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.


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


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


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


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:


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

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]:


Eq. (39) can be written as


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]:

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:


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:


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


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


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


1. J. Bisschop, AIMMS – Optimization Modeling, Lulu, 2006. Search in Google Scholar

2. I. Boldea, Control Issues in Adjustable Speed Drives. Industrial Electronics Magazine, 2:32–50, 2008.10.1109/MIE.2008.928605 Search in Google Scholar

3. H. Chen, Y. Fang, and N. Sun, A Swing Constraint Guaranteed MPC Algorithm for Underactuated Overhead Cranes. IEEE/ASME Transactions on Mechatronics, 21(5):2543–2555, 2016.10.1109/TMECH.2016.2558202 Search in Google Scholar

4. K.Y. Chen, Sliding Mode Minimum-Energy Control for a Mechatronic Motor-Table System. IEEE/ASME Transactions on Mechatronics, 21(3):1487–1495, 2016.10.1109/TMECH.2015.2503123 Search in Google Scholar

5. W. Deprez, J. Lemmens, D. Vanhooydonck, W. Symens, K. Stockman, S. Dereyne, and J. Driesen, ISO Efficiency Contours as a Concept to Characterize Variable Speed Drive Efficiency. In XIX International Conference on Electrical Machines, 2010. Search in Google Scholar

6. A. Domahidi, E. Chu, and S. Boyd, ECOS: An SOCP Solver for Embedded Systems. In Proceedings European Control Conference, pp. 3071–3076, 2013. Search in Google Scholar

7. M. Duan and C.E. Okwudire, Energy-Efficient Controller Design for a Redundantly Actuated Hybrid Feed Drive With Application to Machining. IEEE/ASME Transactions on Mechatronics, 21(4):1822–1834, 2016.10.1109/TMECH.2015.2500165 Search in Google Scholar

8. H. Hu, M.C. Zhou, Z. Li, and Y. Tang, Deadlock-Free Control of Automated Manufacturing Systems With Flexible RRoute and Assembly Operations Using Petri Nets. IEEE Transactions on Industrial Informatics, 9(1):109–121, 2013.10.1109/TII.2012.2198661 Search in Google Scholar

9. R. Heydari and M. Farrokhi, Robust model predictive control of biped robots with adaptive on-line gait generation. International Journal of Control, Automation and Systems, 15(1):329–344, 2017.10.1007/s12555-014-0363-2 Search in Google Scholar

10. IBM, CPLEX Optimizer: High-Performance Mathematical Programming Solver, 2014. Search in Google Scholar

11. K. Inoue, K. Kotera, Y. Asano, and T. Kato, Optimal Torque and Rotating Speed Trajectories Minimizing Energy Loss of Induction Motor under both Torque and Speed Limits. In IEEE 10th International Conference on Power Electronics and Drive Systems (PEDS), 2013. Search in Google Scholar

12. J.H. Lee, Model Predictive Control: Review of the Three Decades of Development. International Journal of Control, Automation and Systems, 9, 2011. Search in Google Scholar

13. Lenze SE, Internal Report, 2014. Search in Google Scholar

14. H. Li and W. Yan, Model Predictive Stabilization of Constrained Underactuated Autonomous Underwater Vehicles with Guaranteed Feasibility and Stability. IEEE/ASME Transactions on Mechatronics, 2016. Search in Google Scholar

15. X. Luo, Data-driven predictive control for continuous-time linear parameter varying systems with application to wind turbine. International Journal of Control, Automation and Systems, 15(2):619–626, 2017.10.1007/s12555-015-0480-6 Search in Google Scholar

16. O.E.K. Onwunta and M.T.E. Kahn, Energy Efficiency and Reliability Improvement Strategies in Industrial Electric Motor-Driven Systems (EMDS). In Proceedings of the 8th Conference on the Industrial and Commercial Use of Energy (ICUE), 2011. Search in Google Scholar

17. D.F. Opila, X. Wang, R.B. Gillespie, J.A. Cook, and J.W. Grizzle, An Energy Management Control to Optimally Trade Off Fuel Economy and Drivability for Hybrid Vehicles. IEEE Transactions on Control Systems Technology, 20:1490–1505, 2012.10.1109/TCST.2011.2168820 Search in Google Scholar

18. P. Pisu and G. Rizzoni, A Comparative Study of Supervisory Control Strategies for Hybrid Electric Vehicles. IEEE Transactions on Control Systems Technology, 15:506–518, 2007.10.1109/TCST.2007.894649 Search in Google Scholar

19. L.B. Ristic and B.I. Jeftenic, Implementation of Fuzzy Control to Improve Energy Efficiency of Variable Speed Bulk Material Transportation. IEEE Transactions on Industrial Electronics, 59:2959–2969, 2012.10.1109/TIE.2011.2169639 Search in Google Scholar

20. D. Schroeder, Electrical Drives 1, Springer, 2009. Search in Google Scholar

21. A. Sciarretta and L. Guzzella, Control of Hybrid Electric Vehicles. IEEE Control Systems, 27:60–70, 2007.10.1109/MCS.2007.338280 Search in Google Scholar

22. A. Siegel, R. Schulz., K. Turek, T. Schmidt, and H. Zadek, Modeling the Energy Need of Storage and Retrieval Vehicles and Different Storage Operating Strategies for the Reduction of the Energy Need. Logistics Journal, 2013. Search in Google Scholar

23. L. Suhl and T. Mellouli, Optimierungssysteme, Springer, 2009. Search in Google Scholar

24. W.D. Solvang, H. Yu, and G. Sziebig, Developing automated and integrated flexible manufacturing systems. In Cognitive Infocommunications (CogInfoCom), 2014 5th IEEE Conference on, 2014. Search in Google Scholar

25. S. Windmann, O. Niggemann, and H. Stichweh, Energy efficiency optimization by automatic coordination of motor speeds in conveying systems. In IEEE International Conference on Industrial Technology (ICIT 2015), 2015. Search in Google Scholar

26. S. Zhang and X. Xia, Model and Energy Efficiency Optimization of Belt Conveyors. Applied Energy, 88:3061–3071, 2011.10.1016/j.apenergy.2011.03.015 Search in Google Scholar

27. D. Zuehlke, Smart Factory – Towards a factory of things. Annual Reviews in Control, 34:129–138, 2010.10.1016/j.arcontrol.2010.02.008 Search in Google Scholar

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