IN
DEGREE PROJECT MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS
STOCKHOLM SWEDEN 2020,
The Energy Efficiency Model of a DC Motor for the Control of HEVs
JIACHENG CAI
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF INDUSTRIAL ENGINEERING AND MANAGEMENT
Examensarbete MMK 2020:
TRITA-ITM-EX 2020:524
Energieffektivitetsmodellen för en likströmsmotor för styrning av HEV
Jiacheng Cai
Godkänt
2020-09-10 Examinator
Lei Feng Handledare
Tong Liu
Uppdragsgivare
Lei Feng
Kontaktperson
Fredrik Asplund
Sammanfattning
Denna avhandling studerar en likströmsmotor för en prototyp av ett elektriskt hybridfordon (HEV) för racing. Utvecklingen av optimeringsbaserade energihanteringsstrategier (EMS) kräver en precis kvasistatisk dynamisk modell av den drivande motorn, som inkluderar en en 2D-karta (effektivetetskarta) som beskriver hur verkningsgraden beror på moment och rotationshastighet.
Verkningsgraden hos likströmsmotorn varierar dock mycket beroende på arbetspunkt och verkningsgradskartan från databladen stämmer inte alltid med de olika applikationerna i verkligheten. Givet detta undersöker denna avhandling en fältprovsbaserad kvasistatisk modelleringsmetod för att uppskatta likströmsmotorns effektivitetskarta med endast flyttbara och begränsade testresurser.
Till att börja med är en testbänk designad, tillverkad, integrerad och konfigurerad med alla nödvändiga komponenter. Testbänken består av den motor som testas, en bromsmotor för att ge belastningsmoment, en servoförstärkare för vridmomentstyrning och mätning, samt en dator för datainsamling och strömförsörjning. Sedan utformas en fristående testplan som gör att så många olika testpunkter som möjligt kan täckas, baserat på bromsmotorn effektgräns. Därefter utförs experimenten successivt på testbänken där ingående elektrisk effekt och utgående mekanisk effekt mäts i jämviktsläget.
Flera olika metoder undersöks för att analysera den insamlade testdatan. Kvadratiskt medelvärde används för att minska variansen i testdatan. Ogiltiga outliers identifieras och filtreras ut baserat på hur mycket de avviker från medelvärdet. De godkända testpunkterna används för att bygga upp 2D-effektivitetskartan genom en fjärde gradens polynom regression. Därefter används tre olika metoder, linjära, kvadratiska och kubiska för att skapa kurvanpassningar genom polynomregression för att beskriva sambandet mellan ingångseffekt och utgångseffekt vid olika hastigheter. Resultaten visar att den kvadratiska metoden är det bästa alternativet eftersom det ger en mindre medelkvadratavvikelse och en hanterbar beräkningskomplexitet.
Avslutningsvis kan den kvasistatiska dynamiska modellen för en likströmsmotor, som inkluderar en 2D-effektivitetskarta med det hastighetsbaserade polynomuttrycket för ingångseffekt, skapas av en ny metod som förliter sig på mindre och enklare materiel än traditionella metoder. Denna metod kringår en stor del av den omständiga modulering som precis varvtalsstyrning kräver vilken även är väldigt beroende på högprecisionssensorer. Den formulerade 2D-effektivitetskartan kommer ge betydande stöd till framtida utveckling av modelbaserade energihanteringsstrategier
(EMS). Polynomuttrycket ger ett mer effektivt tillvägagångssätt för att uppskatta omedelbar energieffektivitet för en inbäddad systemapplikation.
Nyckelord
Effektivetetskarta, likströmsmotorn, testbänk, polynomuttryck, K-betyder
Master of Science Thesis MMK 2020:
TRITA-ITM-EX 2020:524
The Energy Efficiency Model of a DC Motor for the Control of HEVs
Jiacheng Cai
Approved
2020-09-10 Examiner Lei Feng Supervisor
Tong Liu
Commissioner
Lei Feng
Contact person
Fredrik Asplund
Abstract
This thesis studies a DC motor for a racing hybrid electric vehicle (HEV) prototype. The development of optimization-based energy management strategies (EMS) necessitates an accurate quasi-static model of the driving motor, which includes a 2D efficiency map with the torque output and rotating speed as the inputs. However, a DC motor's efficiency varies a lot at different operating points and the efficiency map from the technical manual does not match the various applications in reality. In view of this, this thesis investigates a field testing based quasi-static modeling method to construct the DC motor efficiency map with only portable and brief testing resources.
Firstly, a testbench is designed, manufactured, integrated, and configured with necessary accessories. The testbench consists of the motor under test, a braking motor to provide load torque, a servo-amplifier for torque control and sensing, a host computer for data acquisition, and power supplies. Then, a self-contained testing plan is designed by which as many as possible different testing points can be covered based on the braking motor's power limit. After that, the experiments are successively performed on the test bench, and the input electric power along with the output mechanical power at steady state are recorded.
Multiple data process methods are explored to analyze the collected testing data. Root mean square (RMS) is used to reduce the measuring variance. Invalid outliers are identified and filtered out based on the residuals. The qualified samples are employed to build up the 2D efficiency map by fourth-degree polynomial regression. Then, three methods, linear, quadratic, and cubic fittings are attempted separately to estimate the relationships between the input power and output torque at different speeds. The results show that the quadratic model is the best option which results in smaller root mean square error (RMSE) and fair computation complexity.
To conclude, the quasi-static dynamic model of a DC motor, which includes a 2D efficiency map and the speed-based polynomial expression of input power, can be properly established by a new method relying on less and simpler devices in contrast to those traditional methods. This method bypasses a bulk of tedious modulations on precise motor speed control which is heavily dependent on a high-precision sensor. The formulated 2D efficiency map will effectively support the future development of model-based EMS. The polynomial expression provides a more efficient approach to estimate instantaneous energy efficiency for an embedded system application.
Keywords
Efficiency map, DC motor, test bench, polynomial regression, K-means
FOREWORD
First of all, I must express my special thanks and gratitude to my supervisor, Tong Liu, for his enduring patient guidance, detailed explanation, and valuable advice during the thesis process.
Without him, I cannot finish the data processing phase and the thesis writing after I had to leave Sweden when my visa was expired. He gave me so much help through telephone contact during the coronavirus pandemic.
Second, I am very grateful to my examiner, Professor Lei Feng, for the precious opportunity to get involved in this project. He offered me the project topic timely after I was dropped from my previous topic hardly and gave me lots of support on the academic theories. His professionalism, erudite, and broad outlook has made me learn so much.
Besides, I really appreciate the support gave by Hans Johansson and Staffan Qvarnström, who help me get access to many required equipment and machined components. Hans Johansson provided the used testbench for other courses and gave lots of practical instructions. Staffan Qvarnström gave me the right to use and purchase many essential pieces of equipment. I cannot construct the testbench and conduct the experiments without their help.
Last but not least, I would also like to thank Seshgopalan, who gave me countless technical advice and accompanied me during the experiment process in summer vacation 2019. His rich practical experience has been beneficial.
Jiacheng Cai Ganzhou, China July 2020
NOMENCLATURE
Notations
Symbol Description
𝑃𝑃𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 Electric power of the tested motor (W)
𝑃𝑃𝑚𝑚𝑒𝑒𝑒𝑒ℎ Mechanical power of the tested motor (W)
𝐼𝐼𝑡𝑡𝑚𝑚 Current of the under tested motor (A) 𝑈𝑈𝑡𝑡𝑚𝑚 Voltage of the under tested motor (V) 𝑇𝑇 𝑒𝑒 Load torque for the tested motor (Nm) 𝜔𝜔𝑡𝑡𝑚𝑚 Angular Velocity of the tested motor (rad/s) 𝜂𝜂𝑡𝑡𝑚𝑚 Tested motor efficiency (1)
𝐼𝐼𝑎𝑎𝑒𝑒𝑡𝑡 Actual current of the braking motor (A)
𝐾𝐾𝑏𝑏𝑡𝑡 Torque constant of the braking motor (Nm/A)
𝐼𝐼𝐶𝐶𝐶𝐶𝐶𝐶 Command current given by servo-amplifier (A)
𝜔𝜔𝑎𝑎𝑒𝑒𝑡𝑡 Actual motor speed (RPM)
𝐼𝐼𝑅𝑅𝐶𝐶𝑅𝑅 Current processed by root-mean-square (A)
v Residual degrees of freedom (1) 𝑧𝑧𝑓𝑓𝑓𝑓 Forecasted value (1)
𝑧𝑧𝑓𝑓 Observed value (1)
𝑧𝑧̅ Mean of the observed value (1)
e Residual value (1)
𝑣𝑣𝑏𝑏 Efficiency values of the benchmark (1) 𝑣𝑣𝑟𝑟 Efficiency values of the reduced dataset (1)
𝜀𝜀 Absolute error
𝑃𝑃𝑓𝑓𝑖𝑖 Input power of the tested motor (W)
𝑅𝑅𝐵𝐵𝑓𝑓𝑖𝑖𝑡𝑡 Internal regenerative resistance (Ω)
𝑈𝑈𝑣𝑣𝑒𝑒𝑐𝑐𝑐𝑐 Output voltage of the voltage changeable power supply (V)
𝐼𝐼𝑣𝑣𝑒𝑒𝑐𝑐𝑐𝑐 Output current of the voltage changeable power supply (A)
Abbreviations
HEV Hybrid Electric Vehicle
ITRL KTH Integrated Transport Research Lab
GHG Greenhouse Gas
EV Electric Vehicle
ICE Internal Combustion Engine
EM Electric Motor
EMS Energy Management Strategy
ECMS Equivalent Consumption Minimization Strategy
DP Dynamic Programming
IM Induction Motor
ORMEL96 Oak Ridge Motor Efficiency and Load 1996 PMSM Permanent Magnet Synchronous Machine
FM Frequency Modulation
VCPS Voltage Changeable Power Supply
RMS Root-Mean-Square
R-square Coefficient of Determination
SSE Sum-of-Squared Errors
SST Sum-of-Squares Total
SSR Sum-of-Squares due to Regression RMSE Root-Mean-Square Error
ADVISOR Advanced Vehicle Simulator
TABLE OF CONTENTS
SAMMANFATTNING (SWEDISH) 1
ABSTRACT 4
FOREWORD 7
NOMENCLATURE 9
TABLE OF CONTENTS 11
1 INTRODUCTION 14
1.1 Background 14
1.2 Problem Statement 16
1.3 Purpose 19
1.4 Methodology 19
1.5 Delimitations and Limitations 20
1.5.1 Delimitations 20
1.5.2 Limitations 20
1.6 Ethical Considerations 21
1.7 Deposition 21
2 FRAME OF REFERENCE 22
2.1 The Quasi-Static Model of DC Motor 22
2.2 Review of Torque Measurement Methods 23 2.3 Selected Devices and Software for Testbench Development 23
2.3.1 Under-Test Motor Specification 24
2.3.2 Braking Motor Specification 24
2.3.3 Braking Motor Servo-amplifier and Software 24
2.4 Experimental Principle 26
2.5 Applied Data Processing Methods 26
2.5.1 Root Mean Square Signal Processing 26
2.5.2 Polynomial Approximation 27
2.5.3 K-Means Classification Method 27
2.5.4 Linear and Quadratic Regression Methods 28
2.5.5 Fit Evaluation Methods 29
2.6 Summary 29
3 IMPLEMENTATION 31
3.1 Experiment Preparation 31
3.2 Test Bench Construction 31
3.2.1 Test Bench Installation Problem 31 3.2.2 Supplementary Parts Manufacture 32
3.2.3 Test Bench Integration 33
3.3 Experimental Procedures 35
4 RESULTS 37
4.1 Representative Value for Single Point 37
4.2 Motor Speed-Torque Curve 38
4.3 Scatter Plot 38
4.4 Approximation 39
4.4.1 Outliers Removing 39
4.4.2 Approximation Results 40
4.4.3 Efficiency Map Approximation 41
4.4.4 Accuracy and Density of Measurement Points 42 4.4.5 Validation of the Efficiency Map Approximation 43 4.4.6 Approximation of Input Power versus Torque 43
5 DISCUSSIONS 50
5.2 Discussion on Research Questions 50
6 CONCLUSION AND FUTURE WORK 52
6.1 Conclusion 52
6.2 Recommendation 52
6.3 Future work 53
7 REFERENCES 54
APPENDIX A: DC MOTOR DATASHEET 57
APPENDIX B: BLDC MOTOR DATASHEET 58
1 INTRODUCTION
1.1 Background
Facing the challenges of climate change, strict regulations and customer preferences, sustainable development and environmentally friendly ideas have assumed unprecedented importance than ever before (Larminie & Lowry, 2012). The United Nations Intergovernmental Panel on Climate Change (Intergovernmental Panel on Climate Change, 2011) Fourth Assessment Report (AR4) concluded that over 90 % of the observed increase in global average temperature since the mid- 20th century is due to the observed increase in anthropogenic greenhouse gas (GHG) concentrations. The transportation system generally contributes a considerable part of total GHG emissions. In 2007, around 39 % of total emissions were caused by transportation globally (Çağatay Bayindir et al., 2011). As the common industrial product used in the transportation industry, vehicles made up most of the total transportation emissions and energy consumption (Shindell et al., 2011). Even the developed countries which promote the green and sustainable economy cannot deny the fact that the expansion of the vehicle fleet and the resulting mobility lead to a serial of substantial environmental problems. For instance, the U.S. transportation system was responsible for 27 % of total U.S. GHG emissions in 2008, second in sectoral emissions behind the electric power industry. In particular, highway vehicles dominated the U.S. transportation sector’s energy consumption and CO2 emissions. They accounted for 80 % of the sector’s energy use and emitted 78 % of total transportation CO2 emissions in 2007 (Greene et al., 2010). In Germany, over 20 % of the energy consumption and emissions in 2010 were owing to the transport, the majority of which were due to road traffic (Helms et al., 2010). So, improving tank-to-wheel efficiency of the vehicle has a substantial impact on global carbon footprint and emission reduction given to over 1.5 billion motor vehicles running on the road today worldwide (Kodjak, 2015).
Given the target set by IPCC that a reduction of at least 50% in global CO2 emissions, compared with the levels in 2000, should be achieved by 2050, Hybrid electric vehicle (HEV) has been identified as a promising technology to address these environmental challenges (Millo et al., 2014).
Thanks to the built-in energy buffers (flywheel, battery, and/or capacitor), HEV has an additional degree of freedom in terms of the power supply on the powertrain, which allows for more flexible operation. The hybrid powertrain has unique advantages over conventional vehicle (CV) because it provides the functionalities such as conducting regenerative brake, shutting down the internal combustion engine (ICE) at low speed or low torque demand scenarios, recharging the energy buffer by the redundant power from the ICE (Murgovski et al., 2012). The hybrid driving technique may significantly improve the fuel economy of the HEV since urban driving usually contains many low-speed driving regions and frequent speed up/down, all of which lead to a vast amount of fuel waste to CV. Comparing to the pure electric vehicle (EV), HEV is not entirely dependent on the charging stations along with the road network because its hybrid driveline ensures similar endurance mileage as CV so that the range anxiety to EV can be relieved. Besides, HEV can accept a smaller onboard battery pack compared with EV, which saves the total gross mass and thus the driving power at the same driving scenario. From the perspective of energy conservation, EV’s higher energy efficiency than CV is only because the electric motor (EM) generally enjoys a higher operating efficiency than the ICE, while HEV can, additionally, making use of the energy management strategy (EMS) to seek for a balance that both EM and ICE can operate with close- to peak efficiencies but without compromising the vehicular performance.
To sum up, HEV can offer drivers a similar range as CV powered byfossil fuels but also can lead to environmental benefits as EV (Kelly et al., 2012). The actual market performance proves the superiority of HEV, whose forecasted sales number is 3.3 million in 2025 (Pillot, 2016) compared
with less than 2 million in 2016, which indicates a continuously growing trend in the global vehicle market.
As the essential element to improve HEV’s great advantages on energy conservation and emission reduction, the EMS varies a lot according to different HEV powertrain characteristics and driving conditions. That is why the research on EMS is always a hot topic in the HEV field. Since 2012, an R&D team in KTH has been working on the development of a hybrid electric racing car. This HEV is aiming to compete in Shell Eco-marathon (Shell, 2020) where the HEV is required to finish a driving task by its highest possible fuel efficiency. After several years of design and reform, the powertrain architecture is almost determined. The current research emphasis is mainly concentrated on the update and improvement of the EMS based on this fixed HEV. Due to the conflict between the exploding volume of software in range and complexity and the limited development period and cost, the traditional EMS development method cannot meet the requirements of the new automotive controller (Ao et al., 2008). So, the model-based V-model development methodology has been adopted by many reputable automotive companies to shorten the development process and cost. In this case, the under-development EMS is a model-based system.
Most of the model-based EMS demand reliable efficiency maps of each propelling component on the powertrain to achieve the real-time optimized torque split control. For example, the equivalent consumption minimization strategy (ECMS) is a method to split the powertrain torque demand to ICE and EM to realize a minimal sum of transient fuel consumption and weighted electricity consumption. For this target, both the accurate and reliable efficiency maps of the ICE and EM must be known a priori (Sciarretta et al., 2004). It has been used to find the best instantaneous engine state and torque allocation for an HEV, and this controller can improve the vehicle’s fuel efficiency by up to 50% (Liu et al., 2018). Besides, dynamic programming (DP) is an algorithm to obtain global optimization results, which can be served as a benchmark to improve an existing online algorithm. As a popular model-based EMS, DP relies on the accurate efficiency maps of both ICE and EM. For example, based on the DP results of an HEV model, a lookup table based real-time control strategy is developed from a rule-based control algorithm, which results in a reduction of 66 % of fuel consumption (Wang & Lukic, 2012). Lin et al. (2003) investigated the problem of predicting the best fuel economy of a parallel hybrid truck over a driving cycle. A rule- based control strategy was improved by the DP algorithm through optimizing the operation points on the ICE efficiency map. In addition, Schouten et al. (2003) present a fuzzy-logic-based power controller for a parallel hybrid vehicle, and the efficiency maps of the powertrain components have been used to design this controller.
Except for the dynamic model of the HEV body, the quasi-static models of EM and ICE are also essential to realize optimal energy management meanwhile satisfy the power demand on the powertrain. These quasi-static models are characterized by the 2D efficiency map, which covers various operational conditions, mainly determined by the torque output and the spinning speed. In other words, the development of an appropriate EMS for a unique HEV highly relies on a reliable and complete 2D EM efficiency map. Furthermore, EM’s quasi-static model can be also presented by the form of integrated relation grids between motor input power and output torque at different speeds. On the basis of these relation grids, real-time electric energy consumption can be conveniently estimated according to the torque demand by an online controller. For example, Khodabakhshian et al. (2017) researched a real ICE characterized by the fuel rate map from the measurement. Then, the efficiency map-based relation grids, which state the ICE fuel rate as the functions of output torque at different speeds, have been regressed to rapidly estimate the fuel rate for an onboard model predictive controller.
However, the provided datasheets and prior information of the EM are not capable of building such an accurate quasi-static model or plotting a reliable efficiency map, while the accessible experimental resources are limited in the current situation to test EM’s efficiency. Normally, the test of EM efficiency requires dedicated testing equipment, such as dynamometer, proceeded in
the professional organizations through the complicated method to ensure accurate testing results.
A typical dynamometer could measure the speed and torque of the EM under test so that the output power and efficiency of the tested EM can simply be calculated (Agamloh, 2011). However, the high cost and complicated configuration of deploying such a dedicated EM dynamometer make this conventional method unfeasible to this project given the available experimental environment and project budget. Therefore, a new method based on simple instruments will be put forward in this paper to explore the EM’s efficiency map.
1.2 Problem Statement
The selected HEV, named “Elba”, has a unique parallel powertrain, including three propelling components, an ICE, a brushless DC (BLDC) motor, and a DC motor like Figure 1 shows (KTH ECO CARS, 2020). Both the ICE and the EMs have a significant impact on the powertrain overall energy efficiency, but the DC motor plays a more straightforward role than the BLDC motor does because of its higher efficiency. When there is a positive torque demand from the electric path, the DC motor is always the preferred EM, and the BLDC motor is not used unless the DC motor is saturated. For this reason, this paper selects the DC motor as the research object for efficiency exploration. The BLDC will be explored by the following researchers, and the method applied to this DC motor can also be valid to the BLDC motor. The no-load speed of this DC motor is 4620 rpm with a stall torque of 7.37 Nm, and the nominal power is 200 W.
Figure 1. Elba powertrain architecture
At present, there are several commonly used methods for estimating the motor’s efficiency. Most of them could be categorized into three groups, namely, analytical methods, software methods, and field experiments by dedicated devices such as dynamometer.
Firstly, analytical methods do not rely on the real-time measurement results from specific experiments. In contrast, they estimate the motor’s efficiency based on the collection of guidelines or prior knowledge. Analytical methods can be directly employed to extract or estimate the efficiency value from the datasheet of the motor. Otherwise, when the datasheet is unavailable or unreliable, another alternative can be used as a substitute. US Department of Energy (1997) derived several lookup tables based on empirical rules. The average efficiencies for standard motors at various load points can be found in these lookup tables based on the motor size and load level in percentage. In this case, neither the complete datasheet nor the sufficient prior knowledge about this DC motor is available. Therefore, analytical methods are invalid to measure the DC motor to be studied.
Secondly, the efficiency of the Induction Motor (IM) can be approximated through software methods, which model the equivalent circuit of the motor based on physical laws and simulate the real operating conditions. For instance, Oak Ridge National Laboratory has developed ORMEL96 (Oak Ridge Motor Efficiency and Load 1996), a software program that uses an equivalent circuit method to estimate the load and efficiency of an in-service motor (Wallace et al., 2001). This software only needs nameplate data and motor speed to compute the efficiency, whose accuracy can be improved if the user inputs the stator resistance. However, the ORMEL96 is valid only to AC induction motor since it is programmed only for the IM’s equivalent circuit. Nevertheless, the resultant errors of these methods, easily affected by the testers’ experience, can be as much as 10%
or even more in efficiency estimation (Lu et al., 2007). Therefore, the software methods cannot be used in this project since the motor under test in this paper is a DC motor, which demands a highly accurate efficiency map.
Thirdly, field experiments can be designed dedicatedly for each under test motor based on versatile dynamometer or specifically configured devices. These methods collect real operating parameters such as speed, torque, current, voltage, etc. at several sampling points from the experiments.
However, the field experiments cannot cover all the operating points within the working range, and only a limited number of experiments can be conducted due to the constrained project schedule and budget. So, the measured points should be carefully selected to represent a possibly large operating region, and thereby the unmeasured points could be estimated by approximation methods. For example, the efficiency map of the traction system of a commercial EV is deduced by experimental data acquired during an on-road test drive (Depature et al., 2014). The efficiency intermediate values are determined by a linear interpolation based on the acquired operating points.
Besides, Novak et al. (2017) present a method for calculation of the Permanent Magnet Synchronous Machine (PMSM) efficiency map. Then, field experiments are proceeded to measure the efficiency of operating points under different speeds and torques. The measured efficiency map, which is interpolated with a third-order polynomial, is used to verify the described method.
Specifically, polynomial interpolation is a method to estimate values between known data points by the polynomials that the interpolating function passes through all the data points. It may work well for smooth data set when the second and higher derivatives are small (Old Dominion University, 2009). But it typically produces a rather jagged result if the data points are not closely spaced. On the other hand, polynomial regression can be used to find a surface such that the square distance from each data point to the surface is minimized. The fitting surface may not pass through any data point. Nevertheless, the advantages of polynomial regression including the broad range of functions can be fit, flexible, and useful where a model must be developed empirically, etc (Pant, 2019). So, different orders of polynomial regression could be used to estimate the efficiency map based on the measured points.
In general, the methods of field experiments can be classified into two groups, one relies on the dynamometer and another relies on customized testbench to proceed tests. A dynamometer is defined as a device for applying torque to the rotating member of the under-test machine, and it is equipped with means for indicating torque and speed of the under-test machine (Powerlink, 2020).
Following the testing procedures for DC machines according to the IEEE Standard 113-1985 (IEEE Power Engineering Society, 1984), credible efficiency ratings usually can be obtained by dynamometer. Specifically, the input power is measured from the readings of the input voltage and current. When the dynamometer is used and therefore the torque and speed are measured, the output shaft power 𝑃𝑃𝑐𝑐ℎ𝑎𝑎𝑓𝑓𝑡𝑡 is obtained from Equation (1) as follows according to the standard.
These test data can be transmitted to a data acquisition system, and then the efficiency of the test point can be simply derived.
𝑃𝑃𝑐𝑐ℎ𝑎𝑎𝑓𝑓𝑡𝑡 = 𝑇𝑇 ∙ 𝑛𝑛
𝑘𝑘 (1)
Where 𝑃𝑃𝑐𝑐ℎ𝑎𝑎𝑓𝑓𝑡𝑡 is in W, 𝑇𝑇 is the load torque in Nm, 𝑛𝑛 is the rotation speed in rpm, and 𝑘𝑘 is a constant equal to 9.549, which converts the unit of speed value from rpm to rad/s.
According to Killedar (2012), who has experience of 32 years in the dynamometer field, since the advanced dynamometers, controls, and measuring instruments for engine or motor testing have become more complex and expensive, a dedicated space called “Test Cell” is in request to accommodate these sophisticated instruments and controls. These devices require dedicated space and trained electricians to operate. However, there is not enough space, budget, or skilled dynamometer operators available during the thesis work. The dynamometer-based method cannot be applied to this project considering the complicated devices, high costs, and safety concerns.
Other than the dynamometer, a customized testbench also could be functional to a specified EM, which has been implemented effectively in many cases due to its higher flexibility.
For example, Buechner et al. (2013) investigated the nonlinear modeling and identification of a DC motor. Two test benches are constructed to measure the speed, torque, and friction. One of them is called motor testbed that consists of the analyzed DC motor, an analog incremental rotary encoder made by Heidenhain, a shaft fixing to the optional rotation lock, a dual-range torque sensor made by Kistler, and the testbed loader, which is another DC motor. All these components are arranged aligned with the rotatory axis. The speed and shaft torque can be measured from these sensors so that the output power can be calculated. Although this motor testbed is not used for the efficiency measurement in the paper, the component arrangement of this motor testbed is quite valuable since these works are related to motor speed and torque measurement, which are also required in this thesis.
Another example is an experimental setup built to research the efficiency of an IM-based electric vehicle powertrain with variable DC-link bus voltage (Prabhakar et al., 2016). The experimental setup of powertrain configuration consists of a constant DC source, a DC-DC converter, a three- phase voltage source inverter, the under-test IM, an eddy current braker to provide and measure braking torque, an incremental encoder to measure the motor speed. For the implementation of a closed-loop control strategy, the voltage and current signals are sensed by a voltage transducer (LV25-P) and current sensors (LTS6-np). The measured data are monitored and recorded by dSPACE as a platform for implementing real-time control. By varying the motor speed and braking torque, the IM based powertrain has been tested at different operating points. And the efficiencies for all the powertrain parts, including the efficiency of DC-DC converter, inverter, motor, and whole system, have been calculated. The author presents thorough mathematical expressions of the input power, output power, and efficiency for different parts of the powertrain, which are useful references for efficiency analysis. But it should be noted that the braking torque is sensed using an eddy current braker, which is not available at the lab at present.
The last example is an autonomous motor testing system based on a customized testbench developed by Haines et al. (2019). This system can produce efficiency maps covering the complete power drive system. In this testbench configuration, two identical three-phase EMs, one acts as the tested motor and another one is the braking motor, were coupled to a torque meter and driven by two controllers. An encoder is attached to the shaft to measure the rotary speed. This testing system, built by LabVIEW, contains a top sequencer layer that defines the planned measurement points and trajectory, and a lower testing implementation layer. The top layer also sets the temperature, the steady-state boundary, and the conditional consequences. For instance, to maintain constant experimental temperature, the sequenced measurement point would be interrupted if required to apply either a high or low power operating point to heat or cool the system, respectively. In terms of the unstable or unreachable measurement point when the motor control is saturated, the test sequencer will get rid of these planned operating areas, which is helpful for edge detection. The standard deviations and mean values of operating parameters, including the motor speed, torque, and temperature, from both motors’ control systems were used to determine
for the efficiency map measurement based on several IEEE standards, such as IEC 60034-2-1 (The International Electrotechnical Commission, 2014), IEEE 1812-2014 (IEEE, 2015), could be studied for collecting reachable and steady operating points and planning measurement trajectory in this thesis.
In general, all these three examples of specialized testbench required torque sensor to be functional, which is common in most of the customized EM test benches. However, unfortunately, there is no torque sensor available in the workshop for this thesis partly due to the high cost and little use in other on-going projects at KTH. At present, the available experimental resources only include several power supplies with limited upper bounds, several multimeters, a BLDC motor which could be used to provide the braking torque, a controller for the BLDC motor as well as some simple machining equipment. Therefore, there is a technical gap between the demand for accurate motor efficiency testing and deficient experimental resources. Therefore, a new method should be developed to complete the motor efficiency testing assignment only by such limited resources.
In terms of the map-based quasi-static model, which is used to enable rapid power estimation for online controller, multiple relation grids are supposed to estimate the electric input power as the functions of the output torque at different speeds. However, there is no direct speed control of the under-test DC motor, because the available controller is unreliable and hard to use. The absence of direct speed control leads to the scattering of the test points, all of which have different speeds.
So, a challenge to regress the relation grids at different speeds has appeared. Therefore, a unique data processing method needs to be proposed to approximate the quasi-static model based on the scattered data points while maintaining the quality of regressions.
1.3 Research Questions
To fulfill the technical gap mentioned above, this thesis presents a novel and practical method for estimating the efficiency map of a DC motor based on a testbench comprising of some simple and plain devices, and transforming the map to a quasi-static model. This method comprises four steps:
1. Build a new testbench based on available resources.
2. Efficiency measuring on limited amount of available operating points.
3. Verify and process the collected data to plot a 2D efficiency map.
4. Regressing a quasi-static model based on the map for rapid energy consumption estimation.
In fact, these four steps also state the tasks of this thesis work. The ultimate purpose of all the tasks is to acquire the efficiency map and its corresponding quasi-static model for EMS development and implementation. To be clear, this thesis ought to answer the following research questions:
1. How to build a motor testbench, which could maintain steady state in different working conditions within the operating area, based on limited experimental resources?
2. How to select or distribute the testing points within the reachable operating area, and sweep the testing points while maintaining as much measurement consistency as possible?
3. After the operating data have been collected, how to construct the 2D efficiency map of the under-test motor?
4. Compare the different approximation methods and research which method, linear or quadratic one, is more suitable to transform the efficiency map to a quasi-static model of this DC motor?
1.4 Methodology
The whole project consists of three main research phases from the research preparation to the field experiment, and then to the data processing. The detailed information is described below.
The first research phase is literature review. The target is to study the state of the art of related fields, including the applied theories, methods, and experimental devices used in the thesis work.
The datasheets or technical documents of the available equipments and software will be reviewed to get a sufficient understanding of the current equipments, and the missing parts that need to be fixed. The common method of EM testing will be researched. The working principle of the dynamometer, especially the electrical dynamometer, will be studied since it is similar to the braking motor available in the laboratory. Different motor torque measurement methods or techniques will be evaluated. Then, the advantages and disadvantages of different hardware and computing methods will be examined and compared. After that, the proper forms of testbench configuration and the sensing technique could be preliminarily determined considering the obtainable resources at KTH.
In the second phase, the real field experiment will be performed. Regardless of the obstructions like the limited equipment, technical support, and time shortage, the field experiment is a more reliable and credible method than the simulation to measure motor efficiency. By this method, the characteristics of the motor could be revealed via analysing the experimental data. The types of braking device, sensor, controller, and software for the experiment will be selected so that the test could proceed in an efficient way. The limitations of the available experimental environment, such as the maximum current of the power supply, the maximum speed for the braking motor will be determined. The precise speed control in the experiment will be compromised with the feasible experiment implementation supported by limited resources. Next, the detailed test procedures will be formulated in a flowchart that describes the process of field experiment and data recording.
The final phase is data processing. The collected data will be processed in several stages. Firstly, polynomial regression will be utilized to approximate a preliminary function relationship between motor speed, torque, and efficiency. Then, outliers within the collected data will be identified and removed by the residual values in the preliminary approximation. After that, the rest test points will be used to approximate the final function relationship and the efficiency map can be derived.
Next, K-means will be proposed to classify the scattered data into multiple groups according to the speeds. At last, the linear and quadratic approximation can be applied respectively between the motor input power and the output torque in each classified group. To generalize and draw the conclusion, the goodness of the regressions by the linear and quadratic method will be compared and thus trying to answer the research question.
1.5 Delimitations and Limitations
1.5.1 Delimitations
The scope of the presented project is limited in the efficiency measurement of a DC motor used in an HEV prototype, as well as the transformation from the measured efficiency map to a quasi- static model that presents input power in the form of relation grids.
It includes the design of the experiment, test bench building, motor testing, and data processing.
The project’s scope does not cover the usage of this efficiency map for concrete EMS development.
1.5.2 Limitations
As mentioned in Section 1.2, the available devices in the laboratory may not be able to satisfy the demands of work as expected fully. It is worth noting that the capability of the test bench limits the experimental range, which would affect the efficiency map coverage and approximation. Since there are limitations from the maximum current (5 A) of power supply for the tested motor and the fastest rotational speed (3500 rpm) of the braking motor, the presented experiment can neither cover all the operational field nor measure the whole efficiency map of the tested motor on the
current test bench. In other words, only part of the operational field will be tested, and part of the efficiency map will be approximated.
Besides, the tested motor controller is supposed to work for the closed-loop control of speed, but the encoder cannot cooperate with the unreliable controller software, so the precise speed control plan has been abandoned. By an alternative approach, the resulting scattered data will be processed by K-means and the final results show it has a similar effect to the closed-loop control of the tested motor.
Moreover, the vibration on the test bench and the incompact connections among components would cause noise signals and introduce variances into the recorded signal. These factors might affect the overall experiment results if no counteractions are employed. Taking root-mean-square (RMS) values over a certain period of time could effectively reduce the variance and obtain a generalized test value for each operating point.
The parameters of braking and tested motors may vary as the EM continuously works and the heat from dissipated energy increases the surrounding temperature. Especially the temperature around the braking motor goes up noticeably if the needed load is increased. As the motor parameters changing, part of the experiments may no longer remain consistent according to the single variable control principle. Since there is no temperature sensor available in the laboratory, precise temperature regulation cannot be realized. To mitigate the inconstant temperature problem, the high load tests are implemented intervallically and outliers will be excluded before the efficiency map approximation.
1.6 Ethical Considerations
I will make sure that plagiarism does not exist and state the referenced content in this thesis work.
The research on HEV is beneficial to environmental improvement and energy conservation, but the reduced fuel demand may damage the petrochemical industry profit to some extent. Besides, most of the experiments in this thesis will be conducted in the summer vacation or after class time so that the potential dangers and noises would not affect others in the school.
1.7 Deposition
This thesis is organized as follows. In Chapter 2, the quasi-static model of the DC motor is built up. The technical specifications of the hardware and software, which are used to construct the testbench are introduced. Some significant methods for data processing are introduced as well.
Chapter 3 describes the design procedures of the testbench, and then the experiment procedures are explained in detail. The subsequent data processing and all the relevant results are demonstrated in Chapter 4. Chapter 5 analyzes the reasons for these results, answers the research questions, and concludes this thesis. At last, Chapter 6 proposes recommendations and suggestions for future work.
2 FRAME OF REFERENCE
This chapter introduces the applied modeling method of the tested motor for HEV. Then, the hardware and software to be used are presented, together with their pros and cons are discussed.
At last, the data processing methods are studied.
2.1 The Quasi-Static Model of DC Motor
The main objective of modeling for EMS is to reproduce the energy flows within the powertrain and the vehicle,in order to obtain an accurate estimation of fuel consumption and battery state of charge, based on the control inputs and the road load. Considering the driving cycle is normally prescribed in the EMS optimization, Elba is assumed to follow the driving cycle exactly. The driving cycle can be subdivided into small intervals, during each of which a representative operating point with constant speed, torque, and acceleration is applied to approach the actual situation.
A complete HEV dynamical model, including the models of all powertrain elements, is used to support the EMS development and evaluate its performance in simulation. With regard to the over fast dynamics of the EM but a relatively longer sampling time of the periodic control, a quasi- static EM model where fast dynamics are neglected can simplify the EMS complexity without obviously compromising its performance on estimating the total energy consumption during each time interval (Onori et al., 2016). Thus, the under tested motor is quasi-static modeled using an efficiency map in this thesis since only the EM features concerning the energy consumption in steady state are studied.
The EM causality representation for quasi-static modeling is sketched in Figure 2. There are two input variables, the voltage 𝑈𝑈𝑓𝑓 and current 𝐼𝐼𝑓𝑓 at the DC link. The two output variables are the angular speed of the tested motor 𝜔𝜔𝑡𝑡𝑚𝑚 and the output torque, aka the load torque, 𝑇𝑇𝑒𝑒 (Guzzella &
Sciarretta, 2007).
Figure 2. The causality representation of an EM in quasi-static model
In such a case, the input electric power 𝑃𝑃𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 and output mechanical power can be written as Equation (2) (3):
𝑃𝑃𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 𝐼𝐼𝑡𝑡𝑚𝑚∙ 𝑈𝑈𝑡𝑡𝑚𝑚 (2)
𝑃𝑃𝑚𝑚𝑒𝑒𝑒𝑒ℎ = 𝑇𝑇𝑒𝑒∙ 𝜔𝜔𝑡𝑡𝑚𝑚 (3)
The EM efficiency at any operation point is defined as the corresponding output power 𝑃𝑃𝑚𝑚𝑒𝑒𝑒𝑒ℎ
divided by the input power 𝑃𝑃𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 as Equation (4) shows. Furthermore, an efficiency map, which can be expressed as a function of speed and torque, could present the efficiency of all the operation points within the measured area.
𝜂𝜂𝑡𝑡𝑚𝑚(𝜔𝜔𝑡𝑡𝑚𝑚, 𝑇𝑇𝑒𝑒) =𝑃𝑃𝑚𝑚𝑒𝑒𝑒𝑒ℎ
𝑃𝑃𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 × 100% = 𝑇𝑇𝑒𝑒∙ 𝜔𝜔𝑡𝑡𝑚𝑚
𝐼𝐼𝑡𝑡𝑚𝑚∙ 𝑈𝑈𝑡𝑡𝑚𝑚× 100% (4)
In this thesis, the tested motor only operates in motoring mode, so only the efficiency map for the first quadrant (positive speed and torque) will be studied. Alternatively, in generator mode, mechanical power becomes input, and electric power becomes the output.
2.2 Review of Torque Measurement Methods
As described in Equation (3), the load torque 𝑇𝑇𝑒𝑒 is one of the required variables to calculate the EM efficiency. There are generally five common torque sensing methods, including the load cell transducer, the rotary transformer, the analog frequency modulation (FM) wireless telemetry, digital telemetry, and obtaining from the loading device.
Firstly, the load cell transducer, aka reaction torque sensor, have two mounting flanges. One of the flanges is fixed to the ground or a rigid structural pedestal and another to the rotating shaft or rotary element. Shear forces will be generated between the two flanges when a torsional or twisting motion is applied. So, it is only utilized in measuring static torque.
For the next three methods, rotary transformer, analog FM wireless telemetry, and digital telemetry, they could measure dynamic torque by coupling between the EM and the load. Strain gauges are the key components for all these methods to measure the torque applied to a rotating shaft. The differences among them are the methods of signal transmission from the strain gauge and power supply. Rotary transformers have two rotating coils, one of which excites the torque sensor, while another takes back the data. This method is non-contact and accurate but less tolerant to extraneous loading conditions like bending moments and thrust loads (Sensorland, 2020).
The analog FM wireless telemetry uses a sensor with a built-in radio transmitter module, a battery, and a receiver. The signals from the strain gauges are amplified and modulated to a radio frequency signal by the transmitter. Then, this radio signal is received by a hoop antenna and decoded into analog voltage by the receiver. This method provides high tolerance of revolutions per minute (RPM) ratings and accurate with no-contact data transmission. However, it does have the drawback of needing a source of power on the rotating sensor, which makes it impractical for long term using and can be challenging to install and tune. As technology and electronics developed, digital telemetry became practical (Honeywell, 2020).
In terms of digital telemetry, a rotor electronics circuit board module is attached to the strain gauges. Signal conditioning and digitizing are done on the rotating sensor using this module. Then, a stationary signal-processing module with antenna receives the digital signals from the module, handling the communications with the rotating sensor. This type of sensor does have lower electrical noise and virtually no backlash, which provides improved resolution, stability, and accuracy. However, its great performance inevitably comes with higher costs, which constrains the application on more conditions.
In general, theses strain gauge-based torque sensors are limited by their high levels of maintenance, high cost and bulkiness regardless of the high accuracy. However, as mentioned in Section 1.2, there is no torque sensor available in this case. Alternatively, if obtaining the current going through the loading device’s winding becomes possible, the load torque can be calculated by multiplying the current by torque constant without the torque sensor. This indirect method eliminates the need for torque sensor so that it becomes the focused approach in this thesis and raises the request for the monitoring system of the loading device.
2.3 Selected Devices and Software for Testbench Development
Except for the under-test motor, the test bench uses a braking motor as the loading device, a controller for the braking motor, and a matched software on the host PC. Reviewing the technical
specifications of these devices and software is a prerequisite for the next work phases.
2.3.1 Under-Test Motor Specification
The research object, DC-motor used in Elba, is a type RE50 graphite brushed motor made by Maxon. The technical specifications of this DC-motor are shown in Table 1, referring to Appendix A.
Table 1. Technical Specifications of Tested Motor
Part number 370356
Normal Voltage 48 V
Rated Power 200 W
No-load speed 4900 rpm
No-load current 88.4 mA
Stall torque 7370 mNm
Nominal torque (max. continuous torque) 420 mNm Nominal current (max. continuous current) 4.58 A
Terminal resistance 0.608 Ω
Terminal inductance 0.423 mH
Torque constant 93.4 mNm/A
Speed constant 102 rpm/V
Speed/torque gradient 0.666 rpm/mNm
Max. efficiency 94%
Shaft diameter 8 mm
The under-test motor could operate in all four quadrants, including forward motoring, forward braking, reverse motoring and reverse braking, whereas only forward motoring operation is tested in this project since it is the only quadrant in which the tested motor operates as an actuator in the Elba’s powertrain system.
2.3.2 Braking Motor Specification
The braking motor available in the workshop at KTH is a synchronous servomotor, type 6SM37L- 4000. It is a BLDC motor for demanding servo applications such as industrial robots, machine tools, transfer lines, etc. It has a built-in hall effect sensor used to detect rotator position and speed.
Combined with matched SERVOSTAR 600 servo-amplifier, it is especially suited for tasks with high requirements of dynamics and stability. Moreover, the motor efficiency test under high load at high speed requires the stable load torque provided by the braking motor. Therefore, the provided servomotor is suitable for loading the tested motor in the experiment on most of the occasions. The technical specifications of this braking motor are shown in Table 2, referring to Appendix B.
Table 2. Technical Specifications of Braking Motor
Rated speed 4000 rpm
Torque constant 0.96 Nm/A
Rated torque at rated speed 1.2 Nm
Rated current 1.5 A
Rated power 0.5 kW
Winding resistance Phase-Phase 15.5 Ω
Winding inductance Phase-Phase 30 mH
2.3.3 Braking Motor Servo-amplifier and Software
The SERVOSTAR 600 servo-amplifier is a compact, fully digital drive-amplifier shown in Figure 3, which is designed to simplify installation, system setup, and system reliability. It has both analog and digital inputs/outputs, internal power regeneration options, and fault protections such as overvoltage, overspeed, and overtemperature. Multiple operational modes, including torque and velocity control from analog or digital command, enable the braking motor to output variable torque for the experiment.
Although it is relatively out of date for an electronic device, it can be conveniently configured and tuned by a user-friendly Windows based program called MOTIONLINK, as illustrated in Figure 4. MOTIONLINK provides PC oscilloscope and monitor modes so that the braking motor’s operating data could be recorded and analyzed later. The PC oscilloscope is used to evaluate the motor running condition whose data will be recorded into files first and then reviewed graphically on the computer screen. The monitor mode can also present the actual motor speed, setpoint speed, internal temperature, and effective current in real-time. Figure 5 shows the 𝐼𝐼𝐶𝐶𝐶𝐶𝐶𝐶, 𝐼𝐼𝑎𝑎𝑒𝑒𝑡𝑡, 𝜔𝜔𝑎𝑎𝑒𝑒𝑡𝑡 of the braking motor during the recording period, where 𝐼𝐼𝐶𝐶𝐶𝐶𝐶𝐶 is the command current given by servo- amplifier, 𝐼𝐼𝑎𝑎𝑒𝑒𝑡𝑡 is the actual current going through the braking motor winding and 𝜔𝜔𝑎𝑎𝑒𝑒𝑡𝑡 is the actual motor speed. The convenient built-in oscilloscope and monitor enable motor parameter recording for a period of time, which is crucial for checking whether the under-test motor is running at a specific operating point stability. In other words, MOTIONLINK, together with the servo- amplifier, also works as a data acquisition system.
Figure 3. SERVOSTAR 600 servo-amplifier
Figure 4. The main interface of MOTIONLINK
Figure 5. Oscilloscope interface of MOTIONLINK
The actual current 𝐼𝐼𝑎𝑎𝑒𝑒𝑡𝑡, going through the braking motor’s winding, could be monitored by the sensor in the servo-amplifier, and recorded in MOTIONLINK. With the known torque constant 𝐾𝐾𝑏𝑏𝑡𝑡, the load torque generated from the braking motor for the tested motor can be calculated as:
𝑇𝑇𝑒𝑒= 𝐼𝐼𝑎𝑎𝑒𝑒𝑡𝑡 ∙ 𝐾𝐾𝑏𝑏𝑡𝑡 (5)
2.4 Experimental Principle
As described in Section 2.1, the efficiency of the tested motor can be expressed as:
𝜂𝜂𝑡𝑡𝑚𝑚(𝜔𝜔𝑡𝑡𝑚𝑚, 𝑇𝑇𝑒𝑒) =𝑃𝑃𝑚𝑚𝑒𝑒𝑒𝑒ℎ
𝑃𝑃𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 × 100% = 𝑇𝑇𝑒𝑒∙ 𝜔𝜔𝑡𝑡𝑚𝑚
𝐼𝐼𝑡𝑡𝑚𝑚∙ 𝑈𝑈𝑡𝑡𝑚𝑚× 100% (6) where 𝑈𝑈𝑡𝑡𝑚𝑚, 𝐼𝐼𝑡𝑡𝑚𝑚 could be obtained from the readings of the voltage changeable power supply (VCPS) on display or the multimeter connected to it.
The speed of the tested motor 𝜔𝜔𝑡𝑡𝑚𝑚 is identical to the speed of braking motor since they are mechanically coupled rigidly. So, 𝜔𝜔𝑡𝑡𝑚𝑚 can be obtained from monitoring the braking motor speed 𝑉𝑉𝑎𝑎𝑒𝑒𝑡𝑡 by its build-in encoder through the servo-amplifier. But the unit of 𝜔𝜔𝑎𝑎𝑒𝑒𝑡𝑡 is rpm, which should be converted to rad/s by multiplying a coefficient (0.1047), i.e.:
𝜔𝜔𝑡𝑡𝑚𝑚 =𝜔𝜔𝑎𝑎𝑒𝑒𝑡𝑡∙ 𝜋𝜋
30 = 0.1047 ∙ 𝜔𝜔𝑎𝑎𝑒𝑒𝑡𝑡 (7)
To sum up, the efficiency of tested motor running at a specific operating point can be written as Equation (8) based on Equation (5)-(7).
𝜂𝜂𝑡𝑡𝑚𝑚 = 0.1047 ∙𝐼𝐼𝑎𝑎𝑒𝑒𝑡𝑡𝐾𝐾𝑏𝑏𝑡𝑡𝜔𝜔𝑎𝑎𝑒𝑒𝑡𝑡
𝐼𝐼𝑡𝑡𝑚𝑚𝑈𝑈𝑡𝑡𝑚𝑚 × 100% (8)
2.5 Applied Data Processing Methods
2.5.1 Root Mean Square Signal Processing The root mean square (RMS) is defined as:
𝑥𝑥𝑅𝑅𝐶𝐶𝑅𝑅 = �1
𝑛𝑛 � (𝑥𝑥𝑖𝑖 𝑓𝑓)2
𝑓𝑓 (9)
where 𝑛𝑛 is the number of samples through the amplifier over the sampling period.
This method could filter out the fluctuations of the collected data and summarize the representative values from the varying signals over a period (Law & Rennie, 2015). The RMS will be applied to the measured motor speed 𝜔𝜔𝑡𝑡𝑚𝑚 and current of braking motor 𝐼𝐼𝑎𝑎𝑒𝑒𝑡𝑡, respectively.
2.5.2 Polynomial Approximation
To generate the efficiency map of tested motor, a 3D surface approximation based on the fourth- degree polynomial with two independent variables was built to fit the filtered measured values (Achieser, 2013). The mathematical model of this approximation is shown in Equation (10).
𝑓𝑓(𝑥𝑥, 𝑦𝑦) = 𝑝𝑝00+ 𝑝𝑝10𝑥𝑥 + 𝑝𝑝01𝑦𝑦 + 𝑝𝑝20𝑥𝑥2+ 𝑝𝑝11𝑥𝑥𝑦𝑦 + 𝑝𝑝02𝑦𝑦2+ 𝑝𝑝30𝑥𝑥3+ 𝑝𝑝12𝑥𝑥𝑦𝑦2+ 𝑝𝑝03𝑦𝑦3+ 𝑝𝑝21𝑥𝑥2𝑦𝑦 + 𝑝𝑝40𝑥𝑥4 + 𝑝𝑝31𝑥𝑥3𝑦𝑦 + 𝑝𝑝22𝑥𝑥2𝑦𝑦2+ 𝑝𝑝13𝑥𝑥𝑦𝑦3+ 𝑝𝑝04𝑦𝑦4 (10) In this case, 𝑥𝑥, 𝑦𝑦, 𝑓𝑓(𝑥𝑥, 𝑦𝑦) correspond to the normalized speed, normalized torque, and efficiency of one operating point, respectively. Since the independent variables have different units and significantly different scales in this case, one represents the motor speed in rpm from 125 to 3496 while another represents the output torque in Nm from 0.02 to 0.43, the speed and torque are normalized first to have a mean of zero and a standard deviation of one. The normalization could maintain the relationship between variables with smaller values and not get conditioned by variables with a wider range of values, which would improve the approximation results noticeably.
For the measured dataset, the mathematical model can be written in matrix form as Equation (11) shows.
𝐅𝐅 = 𝐀𝐀 ∗ 𝐏𝐏 (11)
where 𝑚𝑚 is the number of measurement points.
2 2 3 2 3 2 4 3 2 2 3 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 3 2 3 2 4 3 2 2 3 4
1 1 A
=
m m m m m m m m m m m m m m m m m m m m
x y x x y y x x y y x y x x y x y x y y
x y x x y y x x y y x y x x y x y x y y
(12)
[
00 10 01 20 11 02 30 12 03 21 40 31 22 13 04]
P = p p p p p p p p p p p p p p p T (13)
1 1
( , ) ( , ) F
=
m m
f x y f x y
(14)
According to Equation (11), The parameter matrix 𝑃𝑃 can be resolved as:
𝐏𝐏 = 𝐀𝐀−1𝐅𝐅 (15)
2.5.3 K-Means Classification Method
K-means clustering was used to classify the scattered data points by different representative speeds.
Steinhaus (1956) firstly proposed and published the explicit idea of the K-means algorithm in the 1950s. Although K-means was put forward decades ago and many new clustering methods have been proposed since then, K-means is still widely used in many areas because it is relatively simple to implement and guarantees convergence.
Given a set of observations 𝐗𝐗 = {𝑥𝑥1, 𝑥𝑥2, ⋯ , 𝑥𝑥𝑓𝑓}, where each observation is a 𝑑𝑑-dimensional vector, K-means algorithm intends to partition the observations into a set of 𝑘𝑘 clusters, 𝐂𝐂 = {𝑐𝑐1, 𝑐𝑐2, … , 𝑐𝑐𝑘𝑘 }, such that the within-cluster sum of squares is minimized (Jain, 2010). Denote 𝜇𝜇𝑘𝑘 as the mean of points in the cluster 𝐶𝐶𝑘𝑘, the goal of K-means is to find:
2
argmin
1 µ= ∈ −
∑ ∑
n n
k n x c
c x (16)
The main steps of the K-means algorithm are presented below:
1. Randomly select 𝑘𝑘 observations as initial cluster centers and generate an initial partition by assigning each observation to its closest initial cluster center.
2. Compute new cluster centers.
3. Generate a new partition by assigning each observation to its new cluster centers.
4. Repeat step 2 and 3 until cluster centers are stabilized so that convergence has been reached.
The number of clusters 𝑘𝑘 is the most critical parameter, but there is no perfect mathematical criterion for choosing 𝑘𝑘. Normally, the K-means algorithm will run independently for different values of 𝑘𝑘 and the partition, which is generated by the minimum number of algorithm cycles.
2.5.4 Fit Evaluation Methods
The coefficient of determination (R-square) and the RMSE are two common indexes to evaluate the goodness of fit. To calculate the R-square, several relevant conceptions should be introduced first. The sum-of-squared errors (SSE) is the squared difference between the observed value and the predicted value as shown in Equation (17). The smaller the SSE, the better the prediction effect of the approximation. Another definition, the sum-of-squares total (SST), is the squared difference between the observed value and its mean value as expressed in Equation (18). For a regression model, the sum-of-squares due to regression (SSR) is the difference between the predicted value and the mean of observed value as shown in Equation (19). The rationale of these three terms is the total variability of the data set (SST) is the sum of the variability explained by the regression (SSR) and the unexplained variability stated as the error (SSE) as Equation (20) states.
2
1
( )
=
= ∑
n i−
fiSSE
iz z
(17)2
1
( )
=
= ∑
n i−
SST
iz z
(18)2
1
( )
=
= ∑
n fi−
SSR
iz z
(19)= +
SST SSR SSE (20)
where the 𝑧𝑧𝑓𝑓𝑓𝑓 is the forecasted value, 𝑧𝑧𝑓𝑓 is the observed value, and 𝑛𝑛 is the number of elements in the dataset.
The R-square is defined as the percentage of the explained SSR in the SST. The closer to 1 the R- square, the more data variability could be explained by the approximation model, which better fits the observations. Equation (21) shows how R-square is calculated.
2 = SSR = −1 SSE
R (21)
