This Neher–McGrath paper describes a method of esti mating the steady-state temperature of electrical power cables for commonly encountered co nfigurations. By estimating the temperature of the cables, cables’ safe long-term c urrent-carrying capacity (termed “ampacity”) is determined. The paper described two-dimensional highly symmetric simplified calculations which have formed the basis for many cable applicat ion guidelines and regulations. Complex geometries, or configurations that require three-di mensional analysis of heat flow, may require more complex tools such as finite element analysis.

The calculation of the temperature rise and load capability of cable systems

ANSI standards for power equipment, and a vast store of technical literature, describe various methods by which thermal ratings may be adjusted if actual weather conditions are known or if the "overload" is to be applied for a limited period of time. These methods have been given various names including dynamic thermal rating, on-line rating, and dynamic ratings to describe the process of adjusting thermal ratings of power equipment for actual weather conditions and actual electrical load patterns. This paper discusses in detail a project undertaken by the Electric Power Research Institute (RP 3022-7) as part of its research on Flexible AC Transmission. This project avoids dependence on temperature measurement, instead, calculating critical equipment component temperatures based solely on real-time weather and electrical current. Inexpensive, commercially available weather stations, digital data loggers, and IBM-compatible PC computers are combined with sophisticated thermal algorithms to yield a portable, flexible, instrumentation method which can rate several transmission circuits simultaneously, including underground cable, overhead lines, power transformers, current transformers, switches, bus, line traps, and circuit breakers. Useable increases of 5% to 15% in the thermal capacity of transmission equipment circuits result.

Real-time monitoring and dynamic thermal rating of power transmission circuits

Expressions for computing the external thermal resistance (T4) of buried cables, in both the IEEE and the IEC standards, are applicable to a limited number of installation geometries. In this paper, a method for the computation of T4 using the finite element approach is presented. With this method, a parametric study on how cable ampacity is affected by different configurations of the backfills is performed. The obtained results are compared with those of the IEC and IEEE standards (Neher-McGrath) and published extensions by El-Kady and Horrocks. Important differences can be observed for nonstandardized situations.

Effects of Backfilling on Cable Ampacity Analyzed With the Finite Element Method

This paper defines a mathematical model which simulates the growth of a power system and determines the least cost expansion plan for a system of distribution substations. A new approach employing linear and integer programming is used to optimize the system's substation capacities subject to the constraints of cost, load, voltage, and reserve requirements. The model has been successfully applied to a 1600 square mile urban area served by 70 distribution substations.

An Interactive Procedure for Sizing and Timing Distribution Substations Using Optimization Techniques

The flow of water from soil to plant roots is controlled by the properties of the narrow region of soil close to the roots, the rhizosphere. In particular, the hydraulic properties of the rhizosphere are altered by mucilage, a polymeric gel exuded by the roots. In this paper we present experimental results and a conceptual model of water flow in unsaturated soils mixed with mucilage. A central hypothesis of the model is that the different drying/wetting rate of mucilage compared to the bulk soil results in nonequilibrium relations between water content and water potential in the rhizosphere. We coupled this nonequilibrium relation with the Richards equation and obtained a constitutive equation for water flow in soil and mucilage. To test the model assumptions, we measured the water retention curve and the saturated hydraulic conductivity of sandy soil mixed with mucilage from chia seeds. Additionally, we used neutron radiography to image water content in a layer of soil mixed with mucilage during drying and wetting cycles. The radiographs demonstrated the occurrence of nonequilibrium water dynamics in the soil-mucilage mixture. The experiments were simulated by numerically solving the nonequilibrium model. Our study provides conceptual and experimental evidences that mucilage has a strong impact on soil water dynamics. During drying, mucilage maintains a greater soil water content for an extended time, while during irrigation it delays the soil rewetting. We postulate that mucilage exudation by roots attenuates plant water stress by modulating water content dynamics in the rhizosphere.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2013WR014756

Nonequilibrium water dynamics in the rhizosphere: How mucilage affects water flow in soils

The transient temperature rise of a cable system under a constant load, that is, the manner in which the temperature of the conductor rises with time after the constant load is applied, is a matter of considerable interest to cable engineers, since it may serve as a basis for short time ratings, and for the determination of the temperature variations with a varying load. Although these determinations are most readily made by means of an analogue computer nevertheless there is need for rnathematical methods of determining this basic constant-load-temperature transient. This paper presents a simplified mathematical procedure which it is believed suffers only from the actual procedure, apart from theory, being extremely intricate and laborious. In this respect only is it limited to a degree. While a portion of the procedure is admittedly somewhat laborious, the computations re quired in this part involving the application of vector algebra are such that they may be done by an engineering assistant who need not necessarily be familiar with cable engineering.

A simplified mathematical procedure for determining the transient temperature rise of cable systems

It has been customary in calculating the temperature rise of cables in duct to take account of the nature of the load cycle, since this has the effect of reducing the temperature rise from that which would result if the load were maintained continuously at a constant value. With the advent of pipe-cable systems and the increased use of directly buried cables in the United States, the problem naturally arises as to how to calculate the effect of the load cycle in these cases as well. This matter is discussed at some length in an AlEE conunittee report (ibid., vol. 70, pt. I, 1951, pages 45-52). While this report presents the results of many calculations of the effect of various loss cycles on a number of typical cable systems, it does not offer a simplified procedure for making such calculations. This paper presents such a procedure. The procedure involves merely the determination of the internal thermal resistance of the cable system in conventional manner, and adding to it the effective external thermal resistance as given in a table as a function of cable diameter and loss factor. By the use of simple nomographs, the procedure is made to include the effects of changing the earth constants, depth of burial, or frequency of the loss cycle. A rigorous straightforward method of solution of the temperature rise under sinusoidal loss cycles is also given for use where additional accuracy is required.

J. H. Neher

Papers

The calculation of the temperature rise and load capability of cable systems

A simplified mathematical procedure for determining the transient temperature rise of cable systems

Procedures for calculating the temperature rise of pipe cable and buried cables for sinusoidal and rectangular loss cycles [includes discussion]

Oil Flow and Pressure Calculations for Self-Contained Oil-Filled Cable Systems [includes discussion]

Power distribution system parameters [includes discussion]