# Novel Multi-Vehicle Motion-Based Model of Trolleybus Grids towards Smarter Urban Mobility

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Conventional Analytical Approach

#### 2.1. Monitoring of Overloads and Overheating

^{2}raw copper wire should not carry more than 451 A [3]. Since the standard relates to a continuous operating condition, the vehicle average current absorption I

_{m}is to be used for monitoring possible overloads, which is given by

_{a}is the line current absorbed by the trolleybus and t

_{bs}is the mean time between stops (with dwelling times at stops included).

_{M}, which accounts for the assumed number of vehicles within the affected FS:

_{m}

_{,j}is the vehicle average current for the trolleybuses of the jth type and N

_{j}is the number of trolleybuses of jth type inside the FS.

_{M}in specific cases:

- Each bilaterally supplied FS is symmetrical in relation to the two TPSSs (i.e., the line feeders leaving the TPSSs are assumed to be of identical lengths (i.e., same voltage drop), and the TPSSs themselves equally share the power delivered to the FS. Therefore, each TPSS supplies half of the line current;
- The two catenary lines constituting the FSs with bidirectional traffic (two-way street or different outward and return routes) are assumed to be electrically in parallel (i.e., the corresponding positive and negative poles are theoretically connected at infinite points along the whole OCL length). This leads to a perfect halving of the line current at any position.

_{M}.

_{OCL}is governed by the following differential equation:

_{th}is the thermal time constant of the system, θ

_{OCL}

_{,0}is the initial temperature, C is the heat constant (°C mm

^{4}/A

^{2}), σ (A/mm

^{2}) is the electric current density, I

_{OCL}is the OCL pole current, and S

_{OCL}(mm

^{2}) is the cross-section of the OCL conductor. To verify the compliance with the aforementioned standard, the analysis is limited to the evaluation of the steady state temperature θ

_{OCL}

_{,ss}, obtained from Equation (3):

_{th}that has a much longer duration than current start-up transients. As a consequence, θ

_{OCL}

_{,ss}may be studied with reference to the vehicle average current absorption, which means that in the conventional approach, we resort to the overall average current I

_{M}. According to the FS configuration, I

_{OCL}in Equation (2) is expressed with the relative formula in Table 1.

#### 2.2. Probability-Based Calculation of the Line Voltage Drop

#### 2.2.1. Probabilistic Derivation of the Current Consumption at the Feeding Section Macro-Level

_{s}is used (i.e., the mean value of the current taken by the traction inverter during the starting period), and it is obtained as follows:

_{s}is the start-up time duration (until the maximum speed is reached).

_{s}

_{,j}is the start-up time duration for the trolleybuses of the jth type and t

_{bs}

_{,j}is the mean time between stops for the trolleybuses of the jth type. Ideally, given a set of an infinite number of similar vehicles, this factor represents the probability of each vehicle to be in the starting phase at each instant. Because the time between stops is always larger or equal to the start-up time duration, the calculated simultaneity factor always varies between zero and one.

_{s}

_{,j}is the starting current drawn by a trolleybus of the jth type.

_{c}

_{∞,m}= F

_{c}

_{∞}), the relationship between the simultaneity factor and the number of vehicles for different values of F

_{c}

_{∞}is depicted in Figure 1. Note that the horizontal axis has a logarithmic scale. The figure shows that the coincidence factor decreases with the increasing number of considered trolleybuses, as it becomes less probable that they are in their start-up phase simultaneously. For a very large number of vehicles, the coincidence factor F

_{c}is very close to the respective value taken by F

_{c}

_{∞}.

#### 2.2.2. Voltage Drop Evaluation with a Unilateral Power Supply

- (a)
- Uniformly distributed load

_{d}is the total current delivered by the TPSS at line end A.

_{l}is the OCL electrical resistance per unit length.

_{OCL}

_{,x}varies with quadratic law as a function of x, and it assumes the maximum value at line end B (see Figure 2) (i.e., for x = L), as expressed in the following formula:

- (b)
- Evenly spaced concentrated loads

_{1}, I

_{2}, …, I

_{N}at distances x

_{1}< x

_{2}< … < x

_{N}from the TPSS at point A. The maximum voltage drop at the x

_{N}points can be determined as follows by superposition of the effects:

_{u}) and bilaterally (k

_{b}) fed FSs. By increasing the number of vehicles, the plus factor decreases, as the load is shared among more line points.

_{d}takes the value of the total starting current I

_{S}. Note that for a very large number of vehicles, with k

_{u}being closer to 1, ΔV

_{OCL}

_{,max,u}assumes the result expressed by Equation (15) (i.e., we approach the condition of a uniformly distributed load).

#### 2.2.3. Voltage Drop Evaluation with a Bilateral Power Supply

_{OCL}= V

_{OCL}

_{,A}= V

_{OCL}

_{,B}.

- (a)
- Uniformly distributed load

_{l}) is given by Equation (10). According to assumption 1 in Section 2.1, each TPSS delivers a current equal to I

_{d}/2 = i

_{l}·L/2. At a distance x from the TPSS, the OCL current is given by

_{OCL}

_{,x}varies with quadratic law as a function of x, but this time it takes the maximum value at the midpoint of the OCL section (i.e., for x = L/2). The following result is achieved:

- (b)
- Evenly spaced concentrated loads

_{d}= I

_{S}. Similar to the previous case, for a very large number of vehicles, ΔV

_{OCL}

_{,max,b}, assumes the result expressed by Equation (22).

_{l}as twice the pole resistance per unit length.

## 3. Literature on the Simulation Tools

#### 3.1. Fortran Language-Based Model

#### 3.2. Variable Resistor-Based Catenary Modeling Using Simulink

## 4. Proposed Multi-Vehicle Motion-Based Model of the Catenary in Simulink

#### 4.1. Modular Catenary Modeling

_{hs}) are arranged symmetrically with respect to the central current source [25]. The “H” configuration derives from the choice of preserving the symmetry of the system to be able to apply changes more easily when extending the system itself. Although the accuracy of the introduced model might appear to be a shortcoming when compared to the variable resistor-based approach, such a discrepancy is compensated for by virtue of the theoretical possibility of increasing the spatial resolution at will. Nevertheless, for the purpose of this manuscript, a spatial discretization of 20 m is deemed sufficiently accurate.

#### 4.2. Electrical Parameter Evaluation

_{Gen}is a column vector including the currents supplied by the generators between the positive and negative OCL poles inside each referenced subsystem, n is the number of SR blocks, N represents the number of vehicles considered, M is a sparse matrix whose only elements m

_{ij}(with i = 1, …, n and j = 1, …, N), being equal to one, exist by virtue of the matching of the trolleybus positions and the referenced subsystems, and I

_{T}is a column vector containing all the trolleybus currents set as inputs to the model. The current delivered by the generic ith generator within the ith SR block is defined as follows:

_{j}) is inside the ith SR block (SRB

_{i}), then the element m

_{ij}equals one, and the current drawn by T

_{j}(I

_{T}

_{,j}) is assigned to I

_{Gen}

_{,i}; otherwise, m

_{ij}equals zero.

#### 4.2.1. Voltage at the Trolleybus Location

_{OCL}

_{,i}at the ith SR block (i.e., the voltage V

_{mes}

_{,i}measured by the ith voltmeter depicted in Figure 6) to the line voltage V

_{T}

_{,j}at the location of the jth trolleybus, provided that the jth vehicle is within the ith SR block itself. The voltages seen by the trolleybuses are collected in a vector computed as follows:

_{j}) can be expressed as

_{ij}and m

_{ji}change over time, and hence, the position of the respective switched-on generator varies temporally, as well as the V

_{OCL}

_{,i}value assumed by the voltage V

_{T}

_{,j}seen by the jth vehicle.

#### 4.2.2. OCL Temperature

_{OCL}

_{,i}is the OCL positive-pole current at the ith SR block (i.e., the current I

_{mes}

_{+,i}measured by the ith upper ammeter displayed in Figure 6).

## 5. Case Study of Feeding Sections in Bologna and Discussion of Results

- FS “Sant’Isaia-Carducci” (FS 1): a bilaterally supplied FS (two feeding TPSSs (i.e., TPSS “Sant’Isaia” and TPSS “Carducci”) from which the FS designation was derived) with a basic topology (Figure 7a) (i.e., there exists an approximate symmetry of the double-bifilar line between the supply points). Rounded to the nearest multiple of 20 (i.e., the assumed span length in meters), the total bifilar length was 6240 m;
- FS “Marconi-Carducci” (FS 2): a bilaterally supplied FS with a complex structure (Figure 7b) due to the simultaneous presence of a double-bifilar line (towards TPSS “Carducci”) and a single two-wire loop (in the direction of TPSS “Marconi”), as well as the power support via several reinforcing feeders. Assuming the approximation made in the case above was valid, the total bifilar length was 4540 m.

_{th}at the substation terminals of 830 V DC (hypothesizing 615 V on the AC side of the rectifier) and an equivalent series resistance of 68 mΩ in steady state conditions [27]. Based on the knowledge gained on Bologna’s trolleybus infrastructure, each TPSS is supposed to be grounded through a 0.5 Ω resistor.

- When analyzing the line current and temperature distributions, the total average current I
_{M}in the FS was set as the overall load absorption (refer to Section 2.1) (i.e., for the sake of simplicity, each current source modeling the trolleybus drew (when due) a constant current I_{T}_{,j}= I_{M}/N); - Because of the structure of the conventional approach, we knew that it provided only one current and one temperature value for each FS, which both related to a worst-case scenario. Therefore, for the purpose of comparison, it was sensible to pick the maximum line current and temperature that the simulation gave along the whole FS;
- Similar to what has been emphasized about the current and temperature, the line voltage profile was evaluated by imposing the total starting current I
_{S}in the FS as whole load absorption (see Section 2.2). Each vehicle thus drew a constant current I_{T}_{,j}= I_{S}/N; - The maximum voltage drop along the catenary section belonging to a certain FS was given by the difference between the maximum and the minimum voltage value.

#### 5.1. Feeding Section “Sant’Isaia-Carducci”

#### 5.1.1. First Simulation Type: OCL Morphology Analysis of FS 1

_{OCL}(i.e., position along the single bifilar line to which the trolleybus was connected). The following remarks are provided:

- A detail of the voltage trend emphasizes the 20-m spatial discretization;
- The horizontal axis coordinates of the voltage peaks correspond to the positions of the line feeders leaving from the TPSSs, as it should be. One may observe how the S’Isaia’s feeder connection point with the catenary was not in the exact extremity of the FS but slightly displaced towards the center;
- The voltage V
_{T}is characterized by a undulatory behavior, owing to the presence of the equipotential bonding system set across the double-bifilar line. The slight increase in voltage in correspondence of such equipotential connections is attributable to the reduction in the catenary electrical resistance. Moreover, the effect of the double-bifilar OCL was that the trolleybus movement along one of the bifilar lines affected the current distribution in its counterpart, and hence a sort of symmetry in the undulatory phenomenon exists between the left and right halves of the plot.

#### 5.1.2. Second Simulation Type: Method Comparison for FS 1

_{OCL}

_{,min}> 500 V, I

_{OCL}< 451 A, and θ

_{OCL}

_{,ss}< 80 °C). Since the conventional method disregarded the voltage drop for the feeders and rectifier (41 V), the MVMB model resulted in a noticeably higher total voltage variation. On the other hand, the small mismatch (7.8 V) for the OCL voltage variation between the two approaches was attributable to the TPSS “S’Isaia” feeder, which was connected about 300 m (refer to Figure 7a) after the beginning of the FS, leading to a virtually shorter OCL stretch length. The latter phenomenon is apparent in Figure 9, specifically in the first and last 300 m of the OCL section, where the null value of the current absorption (see Figure 9b) due to the absence of vehicles in that time instant did not produce any additional voltage drop or overtemperature. As for the current, the result obtained with the simulation was reasonably higher, since our Simulink model did not present the ideal double-bifilar line described in hypothesis 2 in Section 2.1. Indeed, it was not possible to achieve the perfect halving of the contact line current at any position that would occur in the case of OCLs, ideally in parallel. As one may see, the results of the two methods did not differ markedly. This means that the conventional method assumption of symmetry of the FS between the TPSS feeders could be adopted.

#### 5.2. Feeding Section “Marconi-Carducci”

#### 5.2.1. First Simulation Type: OCL Morphology Analysis of FS 2

- The voltage V
_{T}was characterized by an undulatory behavior towards the TPSS “Carducci” at the plot extremity, owing to the presence of the equipotential bonding system set across the double-bifilar line. A symmetry in the undulatory phenomenon existed between the left and right plot extremities; - The single two-wire ring presence was marked by the larger valleys belonging to the line section highlighted in red. Within these areas, no voltage equalizers were found.

#### 5.2.2. Second Simulation Type: Method Comparison for FS 2

## 6. Modeling Techniques Comparison

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 6.**“H” configuration of the OCL span model, showing the Simulink representation (

**a**) and circuit schematic drawing (

**b**).

**Figure 7.**Bologna’s FSs selected for the case study. The trolleybus routes (oriented by arrows) of the FS “S’Isaia-Carducci” (

**a**) and FS “Marconi-Carducci” (

**b**) are plotted together with TPSSs and supply points (connection points of line feeders with the OCL in green and of reinforcing feeders in yellow).

**Figure 9.**Distributions of OCL voltage (

**a**), pole currents (

**b**), and steady state temperature (

**c**) along FS 1. Trolleybus positions are marked by red arrows.

**Figure 11.**Distributions of OCL voltage (

**a**), pole currents (

**b**), and steady state temperature (

**c**) along FS 2. Trolleybus positions are marked by red arrows.

OCL Pole Current | OCL Structure | ||
---|---|---|---|

Single-Bifilar | Double-Bifilar | ||

Power supply configuration | Unilateral | I_{M} | I_{M}/2 |

Bilateral | I_{M}/2 | I_{M}/4 |

Label | Description | Parameters |
---|---|---|

R_{OCL}_{,hs} | OCL half-span (10 m) pole resistance | 2.475 (mΩ) |

S_{OCL} | OCL nominal cross-section | 100 (mm^{2}) |

C | OCL heat constant | 2.5 (°C mm^{4}/A^{2}) |

τ_{th} | OCL thermal time constant | 600 (s) |

θ_{OCL}_{,0} | OCL initial temperature | 30 (°C) |

Label | Description | Parameters |
---|---|---|

r_{F}_{,1S} | TPSS “S’Isaia” feeder pole resistance per unit length | 0.0250 mΩ/m |

r_{F}_{,1C} | TPSS “Carducci” feeder pole resistance per unit length | 0.0250 mΩ/m |

Label | Description | Conventional Method Parameters | MVMB Model Parameters |
---|---|---|---|

V_{OCL}_{,max} | OCL maximum voltage | 750 V | 789 V |

V_{OCL}_{,min} | OCL minimum voltage | 613 V | 659.8 V |

ΔV_{OCL} | OCL voltage variation (V_{OCL}_{,max} − V_{OCL}_{,min}) | 137 V | 129.2 V |

ΔV_{tot} | Total voltage variation (V_{th} − V_{OCL}_{,min}) | 137 V | 170.8 V |

I_{OCL} | OCL positive-pole maximum current | 187.5 A | 201 A |

θ_{OCL}_{,ss} | OCL positive-pole maximum steady state temperature | 39 °C | 40 °C |

Label | Description | Parameters |
---|---|---|

r_{F}_{,2M} | TPSS “Marconi” feeder pole resistance per unit length | 0.0188 mΩ/m |

r_{RF1}_{,2M} | TPSS “Marconi” reinforcing feeder 1 pole resistance per unit length | 0.0601 mΩ/m |

r_{RF2}_{,2M} | TPSS “Marconi” reinforcing feeder 2 pole resistance per unit length | 0.2338 mΩ/m |

r_{F}_{,2C} | TPSS “Carducci” feeder pole resistance per unit length | 0.0150 mΩ/m |

r_{RF}_{,2C} | TPSS “Carducci” reinforcing feeder pole resistance per unit length | 0.0375 mΩ/m |

Label | Description | Conventional Method Parameters | MVMB Model Parameters |
---|---|---|---|

V_{OCL}_{,max} | OCL maximum voltage | 750 V | 777.9 V |

V_{OCL}_{,min} | OCL minimum voltage | 603 V | 686.7 V |

ΔV_{OCL} | OCL voltage variation (V_{OCL}_{,max} − V_{OCL}_{,min}) | 147 V | 91.2 V |

ΔV_{tot} | Total voltage variation (V_{th} − V_{OCL}_{,min}) | 147 V | 143.3 V |

I_{OCL} | OCL positive-pole maximum current | 416 A | 383 A |

θ_{OCL}_{,ss} | OCL positive-pole maximum steady state temperature | 73 °C | 61.6 °C |

Modeling Features | ||||
---|---|---|---|---|

Variable Number of Vehicles | Complex FS Morphology | User-Friendliness | ||

Modeling methods | Conventional analytical approach | ✓ | ✕ | ✕ |

Train-sim | ✓ | ✓ | ✕ | |

Variable resistor imitation | ✕ | ✕ | ✓ | |

MVMB model | ✓ | ✓ | ✓ |

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**MDPI and ACS Style**

Barbone, R.; Mandrioli, R.; Ricco, M.; Paternost, R.F.; Cirimele, V.; Grandi, G.
Novel Multi-Vehicle Motion-Based Model of Trolleybus Grids towards Smarter Urban Mobility. *Electronics* **2022**, *11*, 915.
https://doi.org/10.3390/electronics11060915

**AMA Style**

Barbone R, Mandrioli R, Ricco M, Paternost RF, Cirimele V, Grandi G.
Novel Multi-Vehicle Motion-Based Model of Trolleybus Grids towards Smarter Urban Mobility. *Electronics*. 2022; 11(6):915.
https://doi.org/10.3390/electronics11060915

**Chicago/Turabian Style**

Barbone, Riccardo, Riccardo Mandrioli, Mattia Ricco, Rudolf Francesco Paternost, Vincenzo Cirimele, and Gabriele Grandi.
2022. "Novel Multi-Vehicle Motion-Based Model of Trolleybus Grids towards Smarter Urban Mobility" *Electronics* 11, no. 6: 915.
https://doi.org/10.3390/electronics11060915