# Improving the Autonomy of a Mid-Drive Motor Electric Bicycle Based on System Efficiency Maps and Its Performance

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. E-Bike Losses Characterization

#### 2.1.1. Charger

#### 2.1.2. Battery

#### 2.1.3. Controller, Motor, and Reduction Gears (CMRGs)

#### 2.1.4. Chain Drive

#### 2.1.5. Global System Efficiency

#### 2.2. Bicycle Dynamics

#### 2.3. Bicycle Performance

#### 2.4. Driving Strategy

#### 2.4.1. Maximization Function

#### 2.4.2. Chain Transmission Ratio and Assistance Level Selection Algorithm

- Identify the slope.
- Generate a row vector of possible system desired speeds $({v}_{j})$, for ${v}_{min}$:$\mathsf{\Delta}v$:${v}_{max}$ (m/s).$${v}_{j}=\left[\begin{array}{ccc}{v}_{min}& \dots & {v}_{max}\end{array}\right]$$
- Calculate a distance traveled $({d}_{j})$ row vector by integrating each position of ${v}_{j}$ vector by a differential of time.
- Calculate an RPM on the wheel $(RP{M}_{{w}_{j}})$ row vector by using Equation (19) at each position of the vector ${v}_{j}$.
- Generate an RPM on the crank spindle $(RP{M}_{cran{k}_{spindl{e}_{i,j}}})$ array in which each row represents the $RP{M}_{{w}_{j}}$ vector related to each gear ratio, through Equation (20).$$\begin{array}{cc}RP{M}_{cran{k}_{spindl{e}_{i,j}}}& =\left[\begin{array}{ccc}\frac{RP{M}_{{w}_{1}}}{T{R}_{chai{n}_{1}}}& \cdots & \frac{RP{M}_{{w}_{n}}}{T{R}_{chai{n}_{1}}}\\ \vdots & \ddots & \vdots \\ \frac{RP{M}_{{w}_{1}}}{T{R}_{chai{n}_{m}}}& \cdots & \frac{RP{M}_{{w}_{n}}}{T{R}_{chai{n}_{m}}}\end{array}\right]\hfill \\ & =\left[\begin{array}{ccc}RP{M}_{cran{k}_{spindl{e}_{1,1}}}& \cdots & RP{M}_{cran{k}_{spindl{e}_{1,n}}}\\ \vdots & \ddots & \vdots \\ RP{M}_{cran{k}_{spindl{e}_{m,1}}}& \cdots & RP{M}_{cran{k}_{spindl{e}_{m,n}}}\end{array}\right]\hfill \end{array}$$
- Generate a propulsion force (${F}_{{p}_{j}}$) row vector on the bicycle based on Equation (8), system parameters and considering a static equilibrium (no acceleration) system applied to each position of the vector ${v}_{j}$.
- Calculate a torque on the wheel (${T}_{{w}_{j}}$) row vector with each position of the vector ${F}_{{p}_{j}}$, Equation (13), and the wheel radius.
- Generate a torque on the crank spindle (${T}_{cran{k}_{spindl{e}_{i,j}}}$) array, in which each row represents the vector ${T}_{{w}_{j}}$ related to each chain transmission ratio and the chain efficiency by using Equations (14)–(16).$${T}_{cran{k}_{spindl{e}_{i,j}}}=\left[\begin{array}{ccc}{T}_{cran{k}_{spindl{e}_{1,1}}}& \cdots & {T}_{cran{k}_{spindl{e}_{1,n}}}\\ \vdots & \ddots & \vdots \\ {T}_{cran{k}_{spindl{e}_{m,1}}}& \cdots & {T}_{cran{k}_{spindl{e}_{m,n}}}\end{array}\right]$$
- Calculate a cyclist torque $({T}_{cyclis{t}_{i,j}})$ array with each position of array $RP{M}_{cran{k}_{spindl{e}_{i,j}}}$, cyclist power, and Equation (18).
- Calculate a torque delivered by CMRGs $({T}_{CMRG{s}_{i,j}})$ array by applying Equation (17) for each position of arrays ${T}_{cran{k}_{spindl{e}_{i,j}}}$ and ${T}_{cyclis{t}_{i,j}}$.
- Calculate a mechanical power of the CMRGs group $({P}_{me{c}_{CMRG{s}_{i,j}}})$ array by using Equation (5) for each position of arrays ${T}_{CMRG{s}_{i,j}}$ and $RP{M}_{cran{k}_{spindl{e}_{i,j}}}$.
- Calculate an efficiency in the CMRGs group $({\eta}_{CMRG{s}_{i,j}})$ array by applying, for each position of arrays ${T}_{CMRG{s}_{i,j}}$ and $RP{M}_{cran{k}_{spindl{e}_{i,j}}}$, the function expressed in Equation (21).
- Calculate a Watt-hours removed from the battery $\left(W{h}_{i,j}\right)$ array using, for each position of arrays ${P}_{me{c}_{CMRG{s}_{i,j}}}$ and ${\eta}_{CMRG{s}_{i,j}}$, Equations (5), (22), and (23).
- Calculate an energy consumption per unit of distance $(re{l}_{Wh/{d}_{i,j}})$ array by dividing each row of array $W{h}_{i,j}$ by vector ${d}_{j}$.$$re{l}_{\frac{Wh}{{d}_{i,j}}}=\left[\begin{array}{ccc}W{h}_{1,1}\xf7{d}_{1}& \cdots & W{h}_{1,n}\xf7{d}_{n}\\ \vdots & \ddots & \vdots \\ W{h}_{m,1}\xf7{d}_{1}& \cdots & W{h}_{m,n}\xf7{d}_{n}\end{array}\right]=\left[\begin{array}{ccc}re{l}_{\frac{Wh}{{d}_{1,1}}}& \cdots & re{l}_{\frac{Wh}{{d}_{1,n}}}\\ \vdots & \ddots & \vdots \\ re{l}_{\frac{Wh}{{d}_{m,1}}}& \cdots & re{l}_{\frac{Wh}{{d}_{m,n}}}\end{array}\right]$$
- Identify the minimum value in array $re{l}_{Wh/{d}_{i,j}}$ and its position (i, j). The ideal chain transmission ratio in which the system should operate is identified with the (i) position. The ideal crank spindle RPM is identified by finding the (i, j) position in array $RP{M}_{cran{k}_{spindl{e}_{i,j}}}$. The ideal value is selected using this value and the curves representing each assistance level.

#### 2.5. Study Case

## 3. Results

#### 3.1. Components Characterization Results

#### 3.1.1. Charger

#### 3.1.2. Battery

#### 3.1.3. CMRGs Group

#### 3.1.4. E-Bike Efficiency Map

#### 3.1.5. Wh/km Consumption Map

#### 3.2. Driving Strategies Results

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Terms of the controller, motor, and reduction gears (CMRGs) efficiency parametric equation.

Parametric Equation Coefficients | |||||

$C1$ | −5.2434 | $C8$ | −0.0026924 | $C15$ | 0 |

$C2$ | 6.8388 | $C9$ | 5.4134$\times {10}^{-5}$ | $C16$ | 4.6057 $\times {10}^{-5}$ |

$C3$ | −0.30491 | $C10$ | −3.1852 $\times {10}^{-7}$ | $C17$ | 0 |

$C4$ | 0.005517 | $C11$ | −0.011524 | $C18$ | 0 |

$C5$ | −3.6145$\times {10}^{-5}$ | $C12$ | −0.00017694 | $C19$ | 0 |

$C6$ | 0.86255 | $C13$ | 9.4452 $\times {10}^{-6}$ | $C20$ | 0 |

$C7$ | 0.045169 | $C14$ | −1.2101 $\times {10}^{-7}$ |

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**Figure 4.**(

**a**) Charger efficiency and power vs. battery state of charge (SoC). (

**b**) Battery efficiency vs. discharge current.

**Figure 7.**(

**a**) E-bike partial efficiency map with 34:21 transmission ratio (TR). (

**b**) E-bike global efficiency map with 34:40 TR. (

**c**) E-bike global efficiency map with 34:21 TR. (

**d**) E-bike global efficiency map with 34:11 TR.

**Figure 8.**(

**a**) Per unit of distance e-bike energy consumption map with 34:40 transmission ratio (TR). (

**b**) Per unit of distance e-energy consumption map with 34:21 TR. (

**c**) Per unit of distance e-bike energy consumption map with 34:11 TR.

Specifications | |
---|---|

Battery Type | Lithium-Ion |

Nominal Battery Voltage | 48 V |

Battery Electric Charge | 11.4 Ah |

Motor Placement | Mid-motor |

Motor Type | BLDC |

Reduction Gear Ratio | 1:21.9 |

Chain Wheel (number of teeth) | 34 |

Cassette (number of teeth) | 40/35/31/27/24/21/19/17/15/13/11 |

Controller Assistance Levels | 9 |

Wheel Diameter | 27.5 in |

C Ratio | 1/8 C | 1/4 C | 3/8 C | 1/2 C | 5/8 C | 3/4 C | 7/8 C | 1 C |

Battery Current (A) | 1.48 | 2.95 | 4.43 | 5.90 | 7.38 | 8.85 | 10.33 | 11.80 |

$SystemMass\left(kg\right)$ | ${C}_{r}$ | ${C}_{d}$ | $\rho \left(kg/{m}^{3}\right)$ | $A\left({m}^{2}\right)$ | ${v}_{wind}\left(m/s\right)$ | $CyclistPower\left(W\right)$ |
---|---|---|---|---|---|---|

100 | 0.0055 | 1.1 | 1.19 | 0.51 | 0 | 100 |

**Table 4.**Performance of strategies two and one under e-bike power alone (E stands for energy, $v$ stands for speed, ${\mathrm{TR}}_{\mathrm{chain}}$ stands for chain transmission ratio and AL stands for assistance level).

Slope (%) | Cadence 50 < RPM < 60 AL 4 | Configuration Used at Constant $v$ | Wh/km Optimization Strategy | Algorithm Output | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${E}_{elec}$ (Wh) | ${E}_{cyclist}$ (Wh) | Avg $v$ (km/h) | $T{R}_{chain}$ | AL | ${E}_{elec}$ (Wh) | ${E}_{cyclist}$ (Wh) | Avg $v$ (km/h) | $T{R}_{chain}$ | AL | |

0.0 | 72.0 | 0.0 | 24.1 | 34/11 | 4 | 72.0 | 0.0 | 24.1 | 34/11 | 4 |

1.8 | 112.3 | 0.0 | 15.6 | 34/17 | 4 | 106.1 | 0.0 | 15.6 | 34/13 | 3 |

3.5 | 172.8 | 0.0 | 9.9 | 34/27 | 4 | 164.6 | 0.0 | 10.6 | 34/19 | 3 |

5.2 | 243.4 | 0.0 | 9.7 | 34/27 | 4 | 236.6 | 0.0 | 10.3 | 34/19 | 3 |

7.0 | 314.9 | 0.0 | 8.4 | 34/31 | 4 | 308.2 | 0.0 | 9.2 | 34/21 | 3 |

8.8 | 386.9 | 0.0 | 6.5 | 34/40 | 4 | 380.6 | 0.0 | 9.0 | 34/21 | 3 |

10.5 | 458.9 | 0.0 | 6.4 | 34/40 | 4 | 450.7 | 0.0 | 7.8 | 34/24 | 3 |

12.3 | 534.2 | 0.0 | 7.2 | 34/35 | 4 | 516.0 | 0.0 | 7.7 | 34/24 | 3 |

**Table 5.**Performance of strategies two and one under e-bike and human power (E stands for energy, $v$ stands for speed, ${\mathrm{TR}}_{\mathrm{chain}}$ stands for chain transmission ratio and AL stands for assistance level).

Slope (%) | Cadence 50 < rpm < 60 AL 4 | Configuration Used at Constant $v$ | Wh/km Optimization Strategy | Algorithm Output | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${E}_{elec}$ (Wh) | ${E}_{cyclist}$ (Wh) | Avg $v$ (km/h) | $T{R}_{chain}$ | AL | ${E}_{elec}$ (Wh) | ${E}_{cyclist}$ (Wh) | Avg $v$ (km/h) | $T{R}_{chain}$ | AL | |

0.0 | 24.5 | 39.7 | 25.2 | 34/11 | 4 | 24.5 | 39.7 | 25.2 | 34/11 | 4 |

1.8 | 82.6 | 41.6 | 24.0 | 34/11 | 4 | 82.6 | 41.6 | 24.0 | 34/11 | 4 |

3.5 | 104.6 | 56.5 | 17.7 | 34/15 | 4 | 104.6 | 56.5 | 17.7 | 34/15 | 4 |

5.2 | 130.1 | 78.5 | 12.7 | 34/21 | 4 | 130.1 | 78.5 | 12.7 | 34/21 | 4 |

7.0 | 179.5 | 90.6 | 11.0 | 34/24 | 4 | 179.5 | 90.6 | 11.0 | 34/24 | 4 |

8.8 | 231.4 | 102.8 | 9.7 | 34/27 | 4 | 231.4 | 102.8 | 9.7 | 34/27 | 4 |

10.5 | 302.4 | 104.5 | 9.6 | 34/27 | 4 | 302.4 | 104.5 | 9.6 | 34/27 | 4 |

12.3 | 300.0 | 151.1 | 6.6 | 34/40 | 4 | 300.0 | 151.1 | 6.6 | 34/40 | 4 |

**Table 6.**Performance of strategies two and one (speed restriction modified) under e-bike and human power (E stands for energy, $v$ stands for speed, ${\mathrm{TR}}_{\mathrm{chain}}$ stands for chain transmission ratio and AL stands for assistance level).

Slope (%) | Cadence 50 < rpm < 60 AL 4 | Configuration Used at Constant $v$ | Wh/km Optimization Strategy Using 90% of Avg $v$ | Algorithm Output | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${E}_{elec}$ (Wh) | ${E}_{cyclist}$ (Wh) | Avg $v$ (km/h) | $T{R}_{chain}$ | AL | ${E}_{elec}$ (Wh) | ${E}_{cyclist}$ (Wh) | Avg $v$ (km/h) | $T{R}_{chain}$ | AL | |

0.0 | 24.5 | 39.7 | 25.2 | 34/11 | 4 | 17.0 | 43.3 | 23.2 | 34/17 | 6 |

1.8 | 82.6 | 41.6 | 24.0 | 34/11 | 4 | 73.3 | 45.8 | 22.0 | 34/15 | 5 |

3.5 | 104.6 | 56.5 | 17.7 | 34/15 | 4 | 85.21 | 62.8 | 16.0 | 34/13 | 3 |

5.2 | 130.1 | 78.5 | 12.7 | 34/21 | 4 | 116.6 | 82.5 | 12.2 | 34/17 | 3 |

7.0 | 179.5 | 90.6 | 11.0 | 34/24 | 4 | 164.0 | 100.8 | 10.0 | 34/27 | 4 |

8.8 | 231.4 | 102.8 | 9.7 | 34/27 | 4 | 215.1 | 113.0 | 8.9 | 34/15 | 2 |

10.5 | 302.4 | 104.5 | 9.6 | 34/27 | 4 | 290.3 | 108.5 | 9.3 | 34/21 | 3 |

12.3 | 300.0 | 151.1 | 6.6 | 34/40 | 4 | 259.7 | 170.8 | 5.9 | 34/35 | 3 |

**Table 7.**Performance of strategies three and one under e-bike power alone (E stands for energy, $v$ stands for speed, ${\mathrm{TR}}_{\mathrm{chain}}$ stands for chain transmission ratio and AL stands for assistance level).

Slope (%) | Cadence 70 < rpm < 90 AL 7 | Configuration Used at Constant $v$ | Wh/km Optimization Strategy | Algorithm Output | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${E}_{elec}$ (Wh) | ${E}_{cyclist}$ (Wh) | Avg $v$ (km/h) | $T{R}_{chain}$ | AL | ${E}_{elec}$ (Wh) | ${E}_{cyclist}$ (Wh) | Avg $v$ (km/h) | $T{R}_{chain}$ | AL | |

0.0 | 202.5 | 0.0 | 38.1 | 34/11 | 7 | 202.5 | 0.0 | 38.1 | 34/11 | 7 |

1.8 | 295.6 | 0.0 | 37.7 | 34/11 | 7 | 295.6 | 0.0 | 37.7 | 34/11 | 7 |

3.5 | 275.5 | 0.0 | 30.0 | 34/11 | 7 | 273.6 | 0.0 | 30.0 | 34/11 | 6 |

5.2 | 321.6 | 0.0 | 20.2 | 34/15 | 7 | 290.4 | 0.0 | 20.4 | 34/15 | 5 |

7.0 | 386.4 | 0.0 | 15.7 | 34/27 | 7 | 330.7 | 0.0 | 16.3 | 34/15 | 4 |

8.8 | 486.2 | 0.0 | 17.5 | 34/31 | 7 | 464.6 | 0.0 | 18.9 | 34/19 | 6 |

10.5 | 552.4 | 0.0 | 13.6 | 34/31 | 7 | 492.9 | 0.0 | 14.4 | 34/21 | 5 |

12.3 | 650.8 | 0.0 | 13.6 | 34/31 | 7 | 610.0 | 0.0 | 14.9 | 34/24 | 6 |

**Table 8.**Performance of strategies three and one under e-bike and human power (E stands for energy, $v$ stands for speed, ${\mathrm{TR}}_{\mathrm{chain}}$ stands for chain transmission ratio and AL stands for assistance level).

Slope (%) | Cadence 70 < rpm < 90 AL 7 | Configuration Used at Constant $v$ | Wh/km Optimization Strategy | Algorithm Output | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${E}_{elec}$ (Wh) | ${E}_{cyclist}$ (Wh) | Avg $v$ (km/h) | $T{R}_{chain}$ | AL | ${E}_{elec}$ (Wh) | ${E}_{cyclist}$ (Wh) | Avg $v$ (km/h) | $T{R}_{chain}$ | AL | |

0.0 | 158.4 | 26.0 | 38.5 | 34/11 | 7 | 158.4 | 26.0 | 38.5 | 34/11 | 7 |

1.8 | 181.4 | 30.7 | 32.6 | 34/13 | 7 | 176.2 | 30.2 | 33.1 | 34/11 | 6 |

3.5 | 223.7 | 35.4 | 28.2 | 34/15 | 7 | 198.2 | 35.8 | 27.9 | 34/11 | 5 |

5.2 | 253.9 | 44.7 | 22.4 | 34/19 | 7 | 218.9 | 44.4 | 22.5 | 34/11 | 4 |

7.0 | 295.2 | 56.2 | 17.8 | 34/24 | 7 | 247.7 | 54.6 | 18.3 | 34/11 | 4 |

8.8 | 359.0 | 63.4 | 15.8 | 34/27 | 7 | 313.9 | 60.7 | 16.5 | 34/15 | 4 |

10.5 | 445.0 | 63.9 | 15.6 | 34/27 | 7 | 406.6 | 57.5 | 17.4 | 34/17 | 5 |

12.3 | 505.9 | 73.4 | 13.6 | 34/31 | 7 | 447.4 | 68.3 | 14.6 | 34/21 | 5 |

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**MDPI and ACS Style**

Arango, I.; Lopez, C.; Ceren, A. Improving the Autonomy of a Mid-Drive Motor Electric Bicycle Based on System Efficiency Maps and Its Performance. *World Electr. Veh. J.* **2021**, *12*, 59.
https://doi.org/10.3390/wevj12020059

**AMA Style**

Arango I, Lopez C, Ceren A. Improving the Autonomy of a Mid-Drive Motor Electric Bicycle Based on System Efficiency Maps and Its Performance. *World Electric Vehicle Journal*. 2021; 12(2):59.
https://doi.org/10.3390/wevj12020059

**Chicago/Turabian Style**

Arango, Ivan, Carlos Lopez, and Alejandro Ceren. 2021. "Improving the Autonomy of a Mid-Drive Motor Electric Bicycle Based on System Efficiency Maps and Its Performance" *World Electric Vehicle Journal* 12, no. 2: 59.
https://doi.org/10.3390/wevj12020059