# Compressed Driving Cycles Using Markov Chains for Vehicle Powertrain Design

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State of the Art/Conventional Driving Cycles: Methodologies and Qualification

#### 2.1. Methods for Cycle Design Based on General Driving Segments

_{trip}are concatenated until the desired cycle time t

_{cycle}is reached. AUSTIN suggests three methods for concatenating the microtrips [7] (p. 116). In addition to the random selection of a microtrip from the available pool of trips (Random), that microtrip can also be combined so that the target values of the driving cycle, such as the Watson plot, are reached as well as possible (Best Incremental). A hybrid approach combines the two methods for chaining microtrips [7] (p. 117). From the set of driving cycle candidates generated, the one whose distribution of acceleration and speed is most similar to that of the real driving data set is selected [7] (p. 116).

#### 2.2. State-Based Design of Driving Cycles

_{i}depends on the speed at timestep t

_{i−1}with variable timestep size, allows the mapping of a driving cycle using a Markov process [6].

_{(i,j)}represents the probability of being in state s

_{j}at time t

_{(n+1)}, given that one is in state s

_{i}at time t

_{n}. More precisely, this is a 1st order Markov chain, where the next state depends only on the current state. A 2nd order Markov chain considers the last two states to select the next state.

_{ij}is then calculated as the quotient of the number n

_{ij}of detected state transitions from state s

_{i}to s

_{j}and the number n

_{ik}of all state transitions starting from state s

_{i}.

_{t}and an increase in speed of at least v over the duration t. A braking state is defined in the same way. The remaining driving points are classified as oscillating or idling states. For the optimal determination of the classification scheme (a, a

_{t}, v, t) different targets are classified in a comprehensive evaluation parameter. The values determined by DAI, EISINGER and NIEMEIER can be taken from [9]. A driving profile segmented according to this methodology is shown in the Figure 2 below.

#### 2.3. Limitations of the Previous Methodology

## 3. Enhanced Modal Cycle Construction (EMCC)

#### 3.1. Steps of Enhancement

_{i}is required at any time t

_{i}. The velocity v

_{(i+1)}of a next time t

_{(i+1)}is then calculated from the acceleration a

_{i}and the velocity v

_{i}of the past time. The calculation of a driving profile is therefore carried out iteratively for any points in time t

_{i}, i = 0 … n

_{1}), uniform (s

_{2}) and decreasing acceleration (s

_{3}), can be made:

#### 3.2. Steps of Cycle Construction

_{1}to S

_{12}, the relevant matrices contain the characteristic values required for the parametric calculation of a driving segment of the corresponding state. More details are shown below. In the third step, the transition matrix can be calculated with this, before a driving cycle is parametrically calculated in the final fourth step.

#### 3.2.1. Used Database

#### 3.2.2. Data Segmentation

**Step 1: Pre-Segmentation**

_{i}≥ 0.01 m/s

^{2}and the sum of the acceleration values in the driving points is greater than a

_{sum}≥ 0.1 m/s

^{2}. A deceleration sequence is defined in an analogous manner. A travel sequence is a deceleration sequence if the acceleration is less than a

_{i}≤ −0.005 m/s

^{2}for at least 3 s and the sum of the acceleration values in the individual travel points is less than a

_{sum}≤ −0.1 m/s

^{2}. Previous procedures define all points in a travel profile which have not been identified as acceleration or braking states as oscillating or idling states. Here, however, defined criteria are also to be set for these two driving states. To identify a cruising sequence, a driving sequence with a duration of 10 s is examined. The driving sequence is a cruising sequence if the deviation of the speed in the individual driving points from the average speed of the examined sequence does not exceed a defined tolerance window. This tolerance window is assumed as a function of the average speed of the examined sequence in order to avoid too strong a generalization. The permissible tolerance decreases with increasing speed. The tolerance range is between 25% at very low speeds to 2% at very high speeds. The identification of an idle state is only carried out on the basis of the driving speed. If the driving speed is lower than v

_{i}≤ 1 km/h for at least 3 s, an idle state is assumed for this sequence. The following figure shows a classified driving sequence according to the criteria mentioned above (Figure 9).

**Step 2: Unambiguousness**

- A cruising sequence is not interrupted by an acceleration sequence that is shorter than the current cruising sequence, e.g., in Figure 9. at time 1415 s.
- A cruising sequence is not interrupted by a braking sequence that is shorter than the current cruising sequence, e.g., in Figure 9. at time 1420 s.
- An acceleration sequence is not interrupted by a subsequent cruising sequence that is detected, e.g., in Figure 9. at time 1550 s.
- If the deceleration sequence is longer than the cruising sequence which has also been detected, the cruising sequence is interrupted at the starting point of the deceleration sequence. The braking sequence then dominates, e.g., in Figure 9. at time 1530 s.

#### 3.2.3. Data Analysis

_{1}–S

_{12}) are derived. Accordingly, after this step, a set of driving sequences is available for each driving state S

_{1}–S

_{12}.

_{seq}and initial acceleration a

_{start}of the corresponding state. In the second step, therefore, these descriptive characteristic values are identified. Once the variables have been identified for each state, these characteristic values can be assigned in the third step by means of a statistical analysis.

**Sequences of acceleration and deceleration:**

_{1}) and increasing deceleration (S

_{4}) are defined as follows:

_{2}or S

_{5}.

_{3}(acceleration) and S

_{6}(deceleration):

_{1}–S

_{6}, further characteristic values are derived from the set of corresponding sequences, which allow to construct such sequences parametrically. In the following, the further evaluation for the acceleration states S

_{1}–S

_{3}is shown. The evaluation of the deceleration states follows the same procedure and is not shown further here.

_{1}and S

_{3}, the acceleration in the next time step can be calculated using the scaling factor.

_{1}the acceleration value a

_{start}is required for the first time step of this state. Since the state’s steady acceleration (S

_{2}) and decreasing acceleration (S

_{3}) usually follow on another acceleration state, the initial acceleration value is taken over from the last state in the first time step. After Equation (3) the driving profile can then be calculated for the duration t

_{seq}.

_{seq}and the start acceleration a

_{start}for a speed range of the driving state S

_{1}(Figure 12).

_{1}, the statistical evaluation has shown that a constant value can be assumed for the sequence duration t

_{seq}. The start acceleration a

_{start}is selected in accordance with a frequency distribution, for the scaling factor scale

_{S1}the value is determined as a function of the start acceleration. The characteristic values of a state are summarized in a matrix relevant for this state and velocity range (Table 2).

_{1}and the current cycle speed is within the range corresponding to this matrix, then the first row of this matrix is selected with a probability of 32.2%. For the further acceleration states S

_{2}and S

_{3}, as well as the deceleration states S

_{4}–S

_{6}, an analogous data analysis is performed. In the following Table 3, the identified characteristic values for these states are summarized again.

**Sequences of Cruising:**

_{seq}the limits of the acceleration value a

_{(limit,pos/neg)}are needed. For each point in time of the cruising state, an acceleration value is then randomly selected which lies within these limits. As a result, the driving speed varies in a small window around the initial speed, which corresponds to the definition of a cruising sequence on which this paper is based. This simplified approximation of cruising sequences is considered appropriate for the objective here. However, the method shown by SCHWARZER and GHORBANI certainly offers higher accuracy.

_{7}–S

_{11}are defined, which are characterized by a different cruising duration. The limit values for the duration of a state vary for each speed range, but are set in such a way that after the cruising sequences have been divided into these driving states S

_{7}–S

_{11}, a fixed prescribed frequency distribution results. For example, 45% of the extracted cruising sequences of a speed range belong to the driving state S

_{7}. As a result of this state refinement, the range of the cruising duration t

_{seq}of one state decreases significantly.

_{7}–S

_{11}and speed range the statistical evaluation and derivation of the above identified characteristic values t

_{seq}, a

_{(limit,pos)}, a

_{(limit,neg)}is carried out according to a uniform procedure. The duration t

_{seq}of a cruising sequence is chosen quasi randomly according to the frequency distribution found in the real data (Figure 15).

_{(limit,pos/neg)}is the average value of a class formed for the duration of a cruising sequence. The following table shows a section of the relevant matrix for the driving state S

_{7}(<55 km/h) (Table 4).

^{2}and +0.05 m/s

^{2}(class 2). Depending on the speed range and cruising state, these characteristics vary.

_{1}–S

_{11}and the different speed ranges such a matrix exists with all relevant characteristic values and the frequency distributions to be taken into account. With the help of this matrix, segments of the driving states can be calculated parametrically.

**Sequences of Idle:**

_{12}, the duration of such a state is used as the only characteristic value. The speed and acceleration must be set to zero. In addition to traffic conditions, such as red lights or traffic jams, the duration of an idling sequence is also significantly influenced by the driver and his intentions. The data set used here was recorded with smartphones. Whether the vehicle was actually idling or whether the journey was simply not interrupted manually on the smartphone and the vehicle was already parked cannot be reconstructed from the data set. Therefore, the duration of an idle state will not be derived from the real driving data in this paper. The duration of the driving state S

_{12}is randomly chosen between one and 90 s. This value corresponds to the maximum circulation time of a traffic light and is considered useful here [15]. For the results in this paper, an equal distribution for values between 1 to 90 s is used. However, this can be adapted to specific applications, e.g., city cycles.

#### 3.2.4. Markov Chain and Transition Matrix

_{0}, X

_{1}, ...) with the states (S

_{1}, S

_{2}, ..., S

_{12}) by a row vector [10].

_{1}) is set as the first state. The state distribution for the time t = 0 s is thus obtained:

_{ij}. The entries of the transition matrix are calculated according to Equation (2). The following transition matrix results for the data set used here (Figure 16).

_{1}is followed with a probability of 99% by a further acceleration state S

_{2}or S

_{3}.

^{0}the next state distribution is calculated by a 1st order Markov chain.

_{1}is followed by driving state S

_{2}. With the help of the characteristic values of the 12 driving states and the Markov chain recorded in the relevant matrices, the driving cycle can finally be calculated.

#### 3.2.5. Calculation of Driving Cycle

_{1}of a driving cycle after the initial distribution, the following state vector results as an example.

_{4}(increasing deceleration) cannot be assumed at the current cycle speed close to zero, since otherwise a speed below zero results due to negative acceleration.

## 4. Results and Validation

_{(max,small)}= 160 km/h and a maximum initial (v = 0 km/h) acceleration a

_{(0,small)}= 2.0 m/s

^{2}is assumed here. Significantly higher values can be applied for the driving cycle of the luxury class. A maximum speed of v

_{(max,exe)}= 250 km/h and a maximum initial acceleration of a

_{(0,exe)}= 4.0 m/s

^{2}is permitted. The differences can also be easily seen in the generated cycles in (Figure 17).

#### 4.1. Valuation Based on a Characteristic Value

_{(avg,%)}; the idling portion n

_{LL}; the portion of acceleration states s

_{(acc,%)}; the portion of accelerations exceeding 70% of the maximum possible acceleration a

_{(max,70,%)}and the portion of decelerations exceeding 70% of the maximum deceleration a

_{(min,70,%)}. To calculate the CV, 400 driving cycles for each vehicle class (same input parameters) were constructed using the approach shown here. For each characteristic C

_{cycle,i}of a driving cycle, the minimum C

_{min,i,}and maximum value C

_{max,i}of the underlying 400 cycles are used to calculate the CV for the corresponding characteristic.

_{cycle.i}weighted in equal proportions into a scale between 1 and 10, where CV = 10 indicates a high vehicle load. The average cycle speed is then over 50% of the maximum speed. The idling component is then close to zero. In addition, high accelerations and decelerations often occur. The driving cycle will also be highly dynamic, as over 45% of the driving conditions correspond to a state of acceleration. These conditions clearly indicate a high vehicle load. Nevertheless, it should be noted that the CV value is a relative valuation according to other driving cycles.

_{(max,compact)}= 210 km/h and a maximum initial acceleration a

_{max}= 3.0 m/s

^{2}are assumed. The blue curve shows a driving cycle with a resulting CV value of CV = 5.9, the cycle shown in green has only a value of CV = 4.7. The difference can be seen on the one hand in the significantly increased dynamics in the urban area and on the other hand in the higher maximum and average cycle speed.

#### 4.2. Driveability Check

^{2}is called up by the driving cycle. The maximum possible acceleration calculated by a simplified longitudinal dynamics model is 1.7 m/s

^{2}at this point in time. For the acceleration process at second 470, here marked by a blue frame, the actual course shows that the test vehicle can hardly follow the target speed towards the end of the sequence. For this point in time, the driving cycle requires an acceleration of 1.7 m/s

^{2}. This value corresponds exactly to the maximum possible acceleration predicted from the vehicle data and the current vehicle speed via the longitudinal dynamics model. Since the actual speed here shows a slightly flatter curve, the maximum possible vehicle acceleration is actually called up. This is also consistent with the observations of the speedometer during the driving cycle on the chassis dynamometer.

#### 4.3. Representative Driving Cycles for Powertrain Design

_{max}= 2.0 m/s

^{2}in the initial state and a maximum speed of v

_{max}= 160 km/h is assumed for this vehicle class. After subdividing the driving cycle into sections A–F, each of these sections may be associated with a traffic condition. Sections A and B are generally assigned to urban traffic, where section A can be described as “light urban traffic”. The increased acceleration and braking usually occurs in “increased urban traffic” (Section B). Section C is referred to here as “dynamic interurban travel”. This section is characterized by frequent acceleration and braking. Section D is a general “stop and go” segment. This can occur both in city traffic and when driving on the motorway. The concluding sections E and F form two motorway sections. For section E, the maximum speed is reached, this section is associated with a “free motorway journey”. For the following section F, the cycle speed is limited to the recommended speed. Therefore, this section can be described as “restricted motorway travel”. In addition to these assigned areas, the cycle is characterized by the following characteristics. Overall, 43% of the times of this driving cycle are acceleration states, 6% of which have an acceleration that is at least 70% of the maximum possible acceleration. In total, 11% of the points in time are driven at a speed that is at least 80% of the maximum vehicle speed. If 99% of the maximum speed is reached, the same applies to the limit of maximum acceleration. This reaches 98%. These characteristic values result in an average speed of v

_{(avg,%)}= 0.34 (34% of the maximum speed) and in an idling ratio of 2%. These characteristic values result in the high CV for this driving cycle.

_{(max,WLTP)}= 131.7 km/h and the maximum acceleration a

_{(max,WLTP)}= 1.75 m/s

^{2}, it can generally be said that the WLTC is not well suited for the design and dimensioning of the powertrain. Even for the small car class, these limits are well below the possible accelerations and speeds of a typical vehicle in this class. The figure below shows a direct comparison of the WLTP with the driving cycle recommended in this paper for the small car class (Figure 21).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Modelling a driving cycle with a Markov chain [6].

**Figure 2.**Example of trip segmentation result by [9].

**Figure 3.**Modelling of a driving pulse [13].

S_{1} | S_{2} | S_{3} |
---|---|---|

${\mathrm{t}}_{\mathrm{seq}}$ | ${\mathrm{t}}_{\mathrm{seq}}$ | ${\mathrm{t}}_{\mathrm{seq}}$ |

${\mathrm{a}}_{\mathrm{start}}$ | ${\mathrm{scale}}_{\mathrm{S}3}$ | |

${\mathrm{scale}}_{\mathrm{S}1}$ |

Class | Relative Frequency | Initial Acceleration [m/s^{2}] | Duration [s] | Scaling Factor |
---|---|---|---|---|

1 | 0.32228 | 0.04125 | 3.5411 | 2628 |

2 | 0.27624 | 0.10625 | 3.5411 | 1886 |

3 | 0.15891 | 0.18047 | 3.5411 | 1673 |

… | … | … | … | … |

Driving State | Name | Number of Velocity Ranges | Characteristic Values | |
---|---|---|---|---|

Variable | Constant | |||

S1 | increasing acceleration | 3 | initial acceleration, scale (matched) | duration |

S2 | constant acceleration | 3 | duration | - |

S3 | decreasing acceleration | 3 | scale | duration |

S4 | increasing deceleration | 3 | initial deceleration, scale (matched) | duration |

S5 | constant deceleration | 3 | duration | - |

S6 | decreasing deceleration | 3 | scale | duration |

Class | Relative Frequency | Duration In s | Max. Acceleration in m/s^{2} | Max. Deceleration in m/s^{2} |
---|---|---|---|---|

1 | 0.092783505 | 1 | 0.001340254 | −0.002380952 |

2 | 0.06921944 | 2 | 0.050723404 | −0.034638298 |

3 | 0.082474227 | 3 | 0.030321429 | −0.105107143 |

… | … | … | … | … |

Cycle Time in s | Change City Rural at Time in s | Change Rural Highway at Time in s | Min. Velociy in km/h | Max. Deceleration in m/s^{2} | Max. Velocity/Acceleration |
---|---|---|---|---|---|

1800 | 600 | 1200 | 0.1 | −3 | vehicle specific |

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**MDPI and ACS Style**

Zähringer, M.; Kalt, S.; Lienkamp, M.
Compressed Driving Cycles Using Markov Chains for Vehicle Powertrain Design. *World Electr. Veh. J.* **2020**, *11*, 52.
https://doi.org/10.3390/wevj11030052

**AMA Style**

Zähringer M, Kalt S, Lienkamp M.
Compressed Driving Cycles Using Markov Chains for Vehicle Powertrain Design. *World Electric Vehicle Journal*. 2020; 11(3):52.
https://doi.org/10.3390/wevj11030052

**Chicago/Turabian Style**

Zähringer, Maximilian, Svenja Kalt, and Markus Lienkamp.
2020. "Compressed Driving Cycles Using Markov Chains for Vehicle Powertrain Design" *World Electric Vehicle Journal* 11, no. 3: 52.
https://doi.org/10.3390/wevj11030052