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Electrical power companies are using more underground cables rather than overhead lines to distribute power to their customers. In practice, cables are generally installed in some compact ductbanks. Since the cost of underground cables is very expensive, using the entire space of a ductbank is extremely important. But such usage is limited due to the overheating of cables. Overheating is generally caused by overload, which means the carrying current exceeds the ampacity of a cable. The ampacity of a cable depends on not only the material and design of a cable but also the distance between different cables. Thus the configuration of cables determines the total ampacity value and the potential use of a ductbank. In this paper, the best configuration based on ampacity is achieved for a three-row, five-column ductbank that is buried at a depth of one meter below the earth’s surface. Both balanced and unbalanced scenarios are considered, and all cables have two available types to be selected.

Underground cables have more advantages than overhead lines since cables offer better protection and are not as unsightly in appearance in urban areas. In practice, cables are generally installed in some compact ductbanks in order to provide easier installation of multiple cables in a concrete space [

cables. Overheating is the most significant factor in decreasing cable service life [

However, a cable’s ampacity value is decided not only by its own characteristics but also by neighboring cables. The heat generated by one cable can influence the maximum value of the current of one nearby. This influence is called the mutual heating effect [

Although some researchers have studied cable configuration optimization [

It is known that a cable’s ampacity is based on the highest allowable temperature that cable can hold without overheating, and it is influenced by the mutual heating effect of nearby cables. To properly design a cable system and optimize cable configuration, calculating the ampacity value of various cables with different cross-sections and sizes is extremely important. Typically, an underground cable consists of four layers, including a conductor layer, insulation layer, shield layer, and jacket layer [

Several publications proposed different methods to calculate cables’ parameters and their ampacities for both single and multiple cable configurations [

are then classified and summarized by Dr. George J. Anders [

In order to calculate the ampacity of cable i, a thermal circuit includes heating sources, and the thermal resistance of different layers is built based on the highest allowable cable temperature.

I i = [ Δ θ max − W d ( 0.5 T 1 + n ( T 2 + T 3 + T 4 ) − Δ θ i n t ) R T 1 + n R ( 1 + λ 1 ) T 2 + n R ( 1 + λ 1 + λ 2 ) ( T 3 + T 4 ) ] 0.5 (1)

Δ θ max = θ max − θ a m b (2)

Δ θ i j = W j ∗ T i j (3)

Δ θ i n t = ∑ j = 1 N Δ θ i j (4)

W j = n [ I j 2 R j ( 1 + λ 1 + λ 2 ) μ j + W d j ] (5)

T i j = ρ s 2π ln d ′ i j d i j (6)

where I i is the ampacity of cable i, I j is the carrying current of cable j, R i is the AC resistance of cable j, λ 1 is cable j’s shield loss factor, λ 1 is cable jacket loss factor, n is the conductors number, μ j is the loss factor, W d j is the dielectric loss of cable j, T_{1} , T_{2}, T_{3}, T_{4} are the thermal resistance of different layers including ductbank and soil, θ max is the highest temperature that allows the cable to operate without problems, θ a m b is the ambient temperature, and Δ θ i n t is the reduction factor of conductor temperature. All these parameters depend only on the material and design of the cable and of the surrounding conditions [

From the Equations (1)-(6) shown in Section 2.1, it can be noticed that to find the ampacity of cable i, carrying currents of all other cables should be pre-known, given the mutual heating effect. So if these equations are applied to all cables, a set of mutually interconnected equations is obtained. However, a set of interrelated equations is challenging to solve, and frequently, the iteration method can be used. But it is time-consuming, and not convergent in some conditions. So a more efficient method is the optimization method, which is recommended by Dr. Moutassem [

The optimization problem for multiple cables installed in a ductbank for a specific configuration can be summarized as follows:

The objective function is

I 1 + I 2 + ⋯ + I n (7)

The constraint for cable 1 is:

1 d 1 I 1 2 + c 12 d 1 I 2 2 + ⋯ + c 1 n d 1 I n 2 ≤ 1 (8)

Similarly, the constraints for the other cables are:

c i 1 d i I 1 2 + c i 2 d i I 2 2 + ⋯ + 1 d i I i 2 + ⋯ + c i n d i I n 2 ≤ 1 (9)

Using MATLAB, the constraints can be acquired in a matrix form.

c ∗ ( I . 2 ) ′ . / d ′ (10)

where all elements in matrices c and d are calculated based on Equations (13)-(14) in Appendix B and matrix c has one on its diagonal terms.

The procedure for finding the total ampacity value for a specific configuration of cables in a ductbank is completed. The next step is to find the configuration that leads to the maximum total ampacity value and minimum total ampacity value. The method applied in this paper includes three steps. Firstly, assume all ducts have their own cables with some initial guess as to current values. Secondly, randomly choose some of these cables to have current equal to zero, which means these ducts don’t have cables installed in them. Thirdly, find the best and worst configuration that produces the maximum total ampacity value and minimum total ampacity value. But since, in this program, the types of cables should be selected automatically, one more step is added that introduces additional ducts for different cable types selection. The steps of configuration optimization of cables in a ductbank are shown in

In this paper, a three-row, five-column ductbank is selected. It is buried at a depth of one meter below the earth’s surface. The distance between two ducts in the same row is 0.3 meter, and the distance between each row is 0.5 meter, which is shown in

For two cables per phase, the second type of cable is selected, and the maximum ampacity of each cable is 655 A, as shown in

cable has more influence on those nearby. Thus the total ampacity is smallest. The difference between these two values proves that configuration optimization for cables in a ductbank is very important.

Normally configuration in

optimization is followed. These two results of ampacity values are compared. For configuration in

The temperature limitation of type two cable is 90˚C and the resulting temperatures of all cables are below 90˚C . Moreover, the temperatures of the cables are symmetric in

For three cables per phase, the second type of cable is selected, and the maximum ampacity of every cable is 566 A, which is shown in

Cable # | Temperature ˚C |
---|---|

1a | 86.4 |

2a | 89.8 |

1b | 89.8 |

2b | 86.4 |

1c | 84.9 |

2c | 84.9 |

Cable # | Temperature ˚C |
---|---|

1a | 81.1 |

2a | 87.4 |

3a | 89.9 |

1b | 89.4 |

2b | 84.5 |

3b | 89.7 |

1c | 83.6 |

2c | 89.8 |

3c | 87.1 |

For unbalanced condition, a particular example: I b = 1.05 I a and I c = 1.1 I a is studied in this section.

For two unbalanced cables per phase, the best configuration is shown in

Cable # | Ampacity A | Temperature ˚C |
---|---|---|

1b | 651 | 86.9 |

1a | 620 | 89.4 |

2a | 620 | 88.7 |

1c | 682 | 87.9 |

2b | 651 | 87.2 |

2c | 682 | 86.7 |

For three unbalanced cables per phase, where c is the highest loaded phase, the best configuration based on ampacity is shown in

If the highest loaded phase is changed from phase c to phase b and then to phase a, a general pattern for the unbalanced condition is observed, where H means highest loaded phase; L means lowest loaded phase; M means medium loaded phase. It is noticed that the best configuration for balanced condition and unbalanced condition based on ampacity is different according to

Installing cables in ductbanks occurs more and more frequently nowadays since their installation is easy. Use of the full potential of a ductbank is extremely important for reasons of economy. This paper proposes using the optimization method to find the best and worst configuration for cables in a ductbank based on ampacity. The best and worst configurations are decided for both balanced and unbalanced scenarios, which are different from common sense without optimization. For an unbalanced condition, a particular example is studied and

Cable # | Ampacity A | Temperature ˚C |
---|---|---|

1c | 594 | 82.8 |

2c | 594 | 89.7 |

1a | 540 | 86.6 |

3c | 594 | 86.1 |

1b | 567 | 88.3 |

2a | 540 | 87.7 |

2b | 567 | 86.9 |

3a | 540 | 89.3 |

3b | 567 | 87.4 |

then extended to a general pattern for unbalanced cables in a ductbank. In the future, the impacts of optimization during abnormal condition will be discussed. The study will include different faulted phases and loading conditions. Based on the results and conclusion, best case and cable configuration under abnormal condition will be presented.

The authors would like to thank all the members of the Clemson University Power Research Association (CUEPRA) for their guidance, data and financial support for this research.

Sun, B. and Makram, E. (2018) Configuration Optimization of Cables in Ductbank Based on Their Ampacity. Journal of Power and Energy Engineering, 6, 1-15. https://doi.org/10.4236/jpee.2018.64001

In order to solve the optimization problem, several solver programs have been built, such as Cplex and Gurobi. However, these programs require considerable amounts of time to build optimization models. To build a model quickly, efficient modeling programs and languages are needed. Yalmip is one of the most powerful and convenient toolboxes for mathematical optimization model building [

Yalmip is a free MATLAB toolbox for modeling optimization problems. It solves the optimization problem in combination with external solvers. The toolbox simplifies model building of optimization in general and focuses on control-oriented optimization problems in particular [

In order to write ampacity calculation equations in an optimization form, the Equations (2)-(6) in Section 2.1 are combined into Equation (1) for cable (1), and the following equation is obtained [

I 1 = [ Δ θ max , 1 − W d , 1 ( 0.5 T 1 , 1 + n 1 ( T 2 , 1 + T 3 , 1 + T 4 , 1 ) ) R 1 T 1 , 1 + n 1 R 1 ( 1 + λ 1 , 1 ) T 2 , 1 + n 1 R 1 ( 1 + λ 1 , 1 + λ 2 , 1 ) ( T 3 , 1 + T 4 , 1 ) − n 2 [ I 2 2 R 2 ( 1 + λ 1 , 2 + λ 2 , 2 ) μ 2 + W d 2 ] ∗ T 12 + ⋯ + n n [ I n 2 R n ( 1 + λ 1 , n + λ 2 , n ) μ n + W d n ] ∗ T 1 n R 1 T 1 , 1 + n 1 R 1 ( 1 + λ 1 , 1 ) T 2 , 1 + n 1 R 1 ( 1 + λ 1 , 1 + λ 2 , 1 ) ( T 3 , 1 + T 4 , 1 ) ] 0.5 (11)

T i j = ρ s 2 π ln ( d ′ i j / d i j ) (12)

For all other cables, similar result equations can be obtained as well.

Let

c i j = n j [ R j ( 1 + λ 1 , j + λ 2 , j ) μ j ] ∗ T i j R i T 1 , i + n i R i ( 1 + λ 1 , i ) T 2 , i + n i R i ( 1 + λ 1 , i + λ 2 , i ) ( T 3 , i + T 4 , i ) (13)

d i = [ Δ θ max , i − W d , i ( 0.5 T 1 , i + n i ( T 2 , i + T 3 , i + T 4 , i ) − ( ρ s / 2π ) [ ∑ N j W d , j ln ( d ′ i j / d i j ) ] ) R T 1 + n R ( 1 + λ 1 ) T 2 + n R ( 1 + λ 1 + λ 2 ) ( T 3 + T 4 ) ] 0.5 (14)

so that the ampacity calculation can be solved as an optimization problem.

Parameters | Value | Parameters | Value |
---|---|---|---|

N | 3 | ρ s | 1 |

R | 0.079e−3 | W d | 0 |

λ 1 | 0 | T 1 | 0.341 |

λ 2 | 0 | T 2 | 0 |

θ a m b | 20 | T 3 | 0.095 |

θ max | 90 | T 4 | 0.637 |

μ | 1 | D e | 72.9 |

Parameters | Value | Parameters | Value |
---|---|---|---|

N | 1 | ρ s | 1 |

R | 0.0763e−3 | W d | 0 |

λ 1 | 0 | T 1 | 0.341 |

λ 2 | 0 | T 2 | 0 |

θ a m b | 20 | T 3 | 0.095 |

θ max | 90 | T 4 | 0.751 |

μ | 1 | D e | 35.8 |