WHITEPAPER CABLE CONDUCTOR SIZING

Transcription

1 WHITEPAPER CABLE CONDUCTOR SIZING FOR MINIMUM LIFE CYCLE COST Bruno De Wachter, Walter Hulshorst, Rodolfo di Stefano July 2011 ECI Available from

2 Document Issue Control Sheet Document Title: Publication No: White Paper Cable Conductor Sizing for Minimum Life Cycle Cost Cu0105 Issue: 01 Release: 06/07/2011 Author(s): Reviewer(s): Bruno De Wachter, Walter Hulshorst, Rodolfo di Stefano David Chapman, Hans De Keulenaer, Stefan Fassbinder Document History Issue Date Purpose Prepared Approved 1 06/07/2011 Initial publication 2 3 Disclaimer While this publication has been prepared with care, European Copper Institute and other contributors provide no warranty with regards to the content and shall not be liable for any direct, incidental or consequential damages that may result from the use of the information or the data contained. Copyright European Copper Institute. Reproduction is authorised providing the material is unabridged and the source is acknowledged. Page i

3 CONTENTS Summary... 1 Optimum is several times larger than standard... 2 Cable sizing according to the standards... 2 Economical cable sizing: the basics... 4 Searching for the cross section with lowest LCC... 5 Recalculating the cost to a present value... 5 Taking the actual loading of the cable into account... 5 Isolating cable characteristics from operational and financial values... 5 Optimal cross section depending on the cable price... 6 An example of a calculation... 6 Table with optimal values... 7 Searching for the current density with lowest LCC... 7 Calculating the losses based on the current density... 7 The cost of energy losses per ton of conductor material... 8 Choosing an optimal current density for a particular economic lifetime... 8 Graphical representation... 8 Taking the scrap value into account... 9 Table with optimal values Additional advantages of larger cross sections Improved power quality Increased flexibility Round-up Annex Table with a few optimal conductor cross sections Table with a few optimal current densities Page ii

4 SUMMARY Energy prices are high and expected to rise. All CO 2 emissions are being scrutinized by regulators as well as by public opinion. As a result, energy management has become a key factor in almost every business. To get the most out of each kilowatt-hour, appliances must be carefully evaluated for their energy efficiency. It is an often overlooked fact that electrical energy gets lost in both end-use and in the supply system (cables, busbars, transformers, etc.). Every cable has resistance, so part of the electrical energy that it carries is dissipated as heat and is lost. Such energy losses can be reduced by increasing the cross section of the copper conductor in a cable or busbar. Obviously, the conductor size cannot be increased endlessly. The objective should be the economic and/or environmental optimum. What is the optimal cross section necessary to maximize the Return on Investment (ROI) and minimize the Net Present Value (NPV) and/or the Life Cycle Cost (LCC)? This paper will demonstrate that the maximizing of the ROI results in a cross section that is far larger than which technical standards prescribe. Those standards are based entirely on safety and certain power quality aspects. This means there is room for improvement a great deal of improvement in fact. We will calculate the conductor cross section for minimum LCC and maximum RoI. Two different models are used in this presentation: 1) The first model (see page 4) is more detailed and calculates the most economical cross section for a specific cable connection. 2) The second model (see page 6) is ideal for obtaining an approximation of the ROI with copper conductors and for developing a company-wide policy for cable sizing, independent of the particular rated current of a single connection. It calculates the optimal current density. Calculating the environmental optimum using a Life Cycle Analysis (LCA) is not within the scope of this whitepaper. However, it is worth mentioning that this environmental optimum lies at an even larger cable cross section than the economic one. Indeed, the environmental impact of energy losses quickly mounts to high levels compared to the environmental impact of copper a 100% recyclable material. Page 1

5 OPTIMUM IS SEVERAL TIMES LARGER THAN STANDARD The following examples show the order of magnitude of the gap between the technical minimum standard and the economic optimum. Take for instance a cable with a rated current of 100 A and a nominal voltage of 230 V. According to the minimum technical standard, this cable should have a minimum cross section of 25 mm 2 to avoid excessive heat production. The economic optimum depends on market and operational conditions. Assume an electricity price of 100/MWh, a cable price of 0.30/(mm 2 x m), a life time of 10 years, and an interest rate of 7.5%. Furthermore, suppose an average loading of 65% over 3,700 hours per year (42% of the time). Those figures result in an optimal cable section of mm 2, a cross section that is nearly three times the standard. Assuming an average loading of only 40% during a mere 1,400 hours per year (16% of the time) for the same cable, electricity prices, and lifetime of 10 years, the optimal cable section would still be mm 2, or nearly twice the safety standard. The following chapters will show how those calculations are made. CABLE SIZING ACCORDING TO THE STANDARDS The international technical standards for cable sizing take safety and certain power quality aspects into account, but NOT energy efficiency. According to these technical standards, the minimum cross section of a cable is defined by the most stringent of three restrictions: 1) The thermal impact of the maximum rated current 2) The voltage drop created by the maximum rated current 3) The electro-dynamic impact of the strongest short circuit current The first restriction is defined in the technical standard IEC (Electrical Installations for buildings): The heat production in the cable should be restricted to avoid the creation of hot spots that could affect the insulation quality or which could be dangerously hot to touch. By choosing a sufficient cross section for the cable, its electrical resistance will remain low, as will the production of heat. The second restriction is stipulated in the standard DIN : : The maximum rated current will create a voltage drop in the cable. This should not be higher than 3% of the nominal voltage to ensure the proper functioning of all appliances. By choosing a sufficient cross section for the cable, its electrical resistance will remain low, as will the voltage drop. The third restriction is defined in the technical standard IEC (Short-circuit currents in three phase AC systems): The short-circuit current I sc, multiplied by the time-current curve of the circuit breaker, should not cross the time-current characteristic expressing the electro-dynamic strength of the cable. The latter is proportional to the cable cross-section. Page 2

6 For shorter cables and/or higher voltages, the first restriction (heat production) will be the most stringent in most cases. For longer cables and/or lower voltages, the second restriction (voltage drop) that will be the most stringent in most cases. This whitepaper will demonstrate that a fourth criterion should be taken into account when choosing the conductor cross section: 4) The cost of the total energy losses in the cable over its economic lifetime should not be higher than the investment cost of the cable. This whitepaper will show that this fourth criterion will be the most stringent in the large majority of the cases. Page 3

7 ECONOMICAL CABLE SIZING: THE BASICS The power losses in a cable at a given moment in time can be calculated using the following formula: P loss = I 2 x (ρ/a) x l With: I = the current in the conductor (depending on the load) ρ = the specific electrical resistance of the conductor A = the cross section of the conductor l = the length of the cable This should be multiplied by the total time of operation over the lifetime of a cable to obtain the total lifetime energy losses E l : E losses = (I 2 x (ρ/a) x l) x t life We see that the energy losses are inversely proportional to the cross section of the conductor. The investment cost of the cable, on the contrary, increases close to linear with its cross section. The economic cable cross section will be the point where the sum of the investment cost and the cost of the losses goes through a minimum. C Total = C Investment + C Losses Page 4

8 SEARCHING FOR THE CROSS SECTION WITH LOWEST LCC Even though the basic principle is simple, calculating the cable section S leading to the lowest life cycle cost C Total introduces a few complexities. C Total = C Investment [S] + C Losses [1/S] RECALCULATING THE COST TO A PRESENT VALUE The initial step in this process is to recalculate the cost of the losses to reveal the operational cost at the date of installation. An interest rate (i) should be set, as well as the economic lifetime of the cable (n). The cost of the losses should then be multiplied by the following capitalization factor: N (i,n) = ((1 + i) n 1) / (i (1 + i) n ) Moreover, the average electricity tariff (T) over the economic lifetime of the cable should be estimated. This leads to the formula for the cost of the energy losses: C losses = (I 2 x (ρ/a) x L) x t econ life x T ariff electr ( /kwh) x N(i,n) TAKING THE ACTUAL LOADING OF THE CABLE INTO ACCOUNT A second point is that the cable will not be loaded at its rated power continuously. This means the time t is not the complete economic lifetime of the cable, but only the time the cable will be loaded: t operational hours econ life. Moreover, the current I will not be the rated current, but the average of the currents that really flow through the cable, a figure that depends on the average relative loading P loading = P load /P rated. I real = (I real /I rated ) x I rated = (P load /P rated ) x I rated = P loading x I rated with P loading a figure varying between 0 and 1 This leads to the following equation: C Losses = (P loading ) 2 (I rated 2 x (ρ/a) x l) x thours x T ariff [ /kwh] x N (i,n) ISOLATING CABLE CHARACTERISTICS FROM OPERATIONAL AND FINANCIAL VALUES Now we can put all the factors which are not cable characteristics into one operational and financial value F. F = (P loading ) 2 x t hours x T ariff [ /kwh] x N (i, n) The following table gives a few average values for F per sector. Industry Interest N (year) Energy Price (euro/mwh) Loading T (hours) F (euro/w) Iron 7,5% % ,62 Non Ferrous 7,5% % ,62 Paper 7,5% % ,62 Chemical 7,5% % ,88 Datahotel 7,5% % ,32 Office 7,5% % ,43 Page 5

9 In general industrial conditions, F will vary between 0.5 and 20 /W. The average value of F across all European industry sectors is 4.24 /W. For a long-term investment in a cable that will be constantly loaded close to its rated power, F can rise above 50 /W. The LCC formula now becomes: C T = C I + C L C T = C I[S] + 1/A x I rated 2 x ρ x length x F OPTIMAL CROSS SECTION DEPENDING ON THE CABLE PRICE The specific resistance of a copper conductor is µω*cm at a typical operating temperature of 105 C (1.720 µω*cm at 20 C) or mω*mm. Consequently, the equation for one meter of cable (1,000 mm) now becomes: C T = C c x A + 1/A x I r 2 x x F With: C c the cable price in euro per mm 2 cable cross section and per meter cable length A the cable cross section expressed in mm 2 I r the rated current in Ampère F the operational and financial value in /Watt This equation gives the total Life Cycle Cost of 1 meter of cable in euro. The optimal cross section A of a cable is the point where the curve of this equation goes through its minimum. It can be proved mathematically that this minimum will always lie at the point where the first part and the second part of the sum are equal. This occurs when: C c x A = 1/A x I r 2 x x F Or: A 2 = I r 2 x x F / Cc We can calculate the optimal cross section A from: A = I r x x (F/C c ) 1/2 With I r the rated current of the connection, F a financial and operational value varying per sector, and C c the cable price per meter and per mm 2 cross section. AN EXAMPLE OF A CALCULATION Assume a cable in the Iron Sector that will carry a rated current of 200 A. The average loading in this sector is 65%, while the average operating time is 3,700 hours per year. If the additional assumptions are an interest rate of 7.5% and an economic lifetime of 10 years, the F factor will be 7.62 euro/w. At the current copper price, the cost of one meter of cable (3 phases + neutral) can be estimated to be approximately 0.30 per square millimeter of conductor cross section: 0.30 /(mm 2 x m), or for each of the four conductors. Page 6

10 The most economical conductor cross section for this cable will be: A = 200 x x (7.62/0.30) 1/2 = mm 2 Note that according to the technical standard, the minimum conductor cross section for this cable should only be 53 mm 2. TABLE WITH OPTIMAL VALUES There is a table in the annex providing the optimal cross section for a number of representative sectors, depending on the rated current I r. SEARCHING FOR THE CURRENT DENSITY WITH LOWEST LCC The calculations executed above are relevant for anyone interest in the most economical cross section for a single connection. However, to develop a general, company-wide policy for cable sizing, the optimum should be expressed in terms that are independent of the particular rated current of a single connection. This can be accomplished by setting a general, company-wide figure for the ratio between the current and the cross section. This figure is also called the current density (A/mm 2 ). CALCULATING THE LOSSES BASED ON THE CURRENT DENSITY How can the optimal current density for a company be set? Remember that the goal is to minimize the Total Present Value of the cables throughout the entire company. This Total Present Value will be composed of an investment cost paying for the material plus the cost of losses that are induced in this material over its economic lifetime, recalculated to the present time. Those losses per amount of conductor material are related to the current density: P losses = I r 2 x ρ/a x L If J = I r /A is the current density, then it follows that: P losses = J 2 x A 2 x ρ/a x L = J 2 x A x L x ρ = J 2 x V olume x ρ In other words, the power loss per volume is: P losses / V olume = J 2 x ρ Or with the density of copper being 8.94 ton/m 3 and the value of ρ being µω*cm at an average operating temperature of 105 C (1.720 µω*cm at 20 C), we get: P losses [kw/ ton] = (J[A/mm 2 ]) 2 x 2.3 The power loss per ton of copper conductor is proportional to the square of the current density. Page 7

11 THE COST OF ENERGY LOSSES PER TON OF CONDUCTOR MATERIAL If the cable is fully loaded all the time, we must multiply P losses by 8,760 hours (one year) to get the annual energy losses. However, while not impossible, this is almost never the case. We therefore need a correction factor, the Hour-Loss-Equivalent (HLE). This figure is calculated by taking the average over the year of the (actual power/rated power) 2. To give an idea of this figure, the HLE in the electricity distribution grid in France is 3,050 hours. In most industrial sectors, this figure will probably be lower. A good estimate for HLE is HLE = 0.65 x W annual [TWh] / P total [MW] = 0.65 x HPE with W annual the total annual energy consumption of the site (= Work), P total the total power of the site, and HPE the Hour-Power-Equivalent. The annual cost of energy losses C annual energy loss of one ton of copper conductor: C annual energy loss / ton = T ariff [ /kw] x HLE x J 2 x 2.3 With T ariff the electricity tariff, HLE the Hours of Load Equivalent, and D the current density. CHOOSING AN OPTIMAL CURRENT DENSITY FOR A PARTICULAR ECONOMIC LIFETIME In the optimal economic situation, the sum of the losses that are induced in one ton of copper conductor over its economic lifetime should be equal to the investment cost, namely the cost of one ton of copper. If: The cost of 1 ton of copper cable = the cost of the energy losses induced in this ton of copper over its lifetime Then: The economic optimal current density is achieved This means the optimal current density can be calculated using: J = (C 1ton / (T ariff [ /kwh] x HLE x 2.3 x N(i, n))) 0.5 GRAPHICAL REPRESENTATION A graphical representation of the current density can be derived from the above equation. Suppose a current density of 1.4 A/mm². Note that this density is consistent with the example in the former calculation model (page 6), where the optimal conductor cross section was calculated to be 144 mm 2 for a cable of 200 A rated current. Page 8

12 Furthermore, suppose an HPE of 3,700 hours, an electricity price of 0.1 /kwh, and a copper price of 6,000 /ton. This copper price needs to be multiplied with an adjustment factor to get the actual cable price. A factor of 1.3 is a good estimate for this ratio, giving a cable price of 7,800 per ton of copper conductor used. INPUT DATA density (A/sq.mm) 1,4 loss (kw/ton) 4,508 HPE (hours) 3700 HLE/HPE 0,65 HLE/HPE x HPE 2405 electricity price ( /kwh) 0,1 yearly cost of loss ( ) 1084 cable price / ton copper 7800 Scrap value / new value 0 The graph that follows presents the curves for the cumulative annual losses, recalculated to the present time with interest rates of respectively 5% (purple), 7.5% (blue), and 10% (orange). The green curve represents the investment cost. Thus the economic lifetime of the cable will be 8.5 years, 10 years, and 12.5 years respectively. Varying the current density causes the economic lifetime to vary as well. The optimal current density can be found for a specific economic lifetime through iteration. Note that the payback period of 10 years for an interest rate of 7.5% is exactly the same as given in the example of the former model (page 6), meaning that both models are consistent. TAKING THE SCRAP VALUE INTO ACCOUNT There is still one element missing in the former example. When a copper cable is dismantled, the copper is not thrown away but recycled. The owner will receive a significant sum for the scrap. This scrap value should be recalculated according to current prices and subtracted from the investment value. Suppose for example a scrap value that is 50% of the value of new copper. INPUT DATA density (A/sq.mm) 1,4 loss (kw/ton) 4,508 HPE (hours) 3700 HLE/HPE 0,65 HLE/HPE x HPE 2405 electricity price ( /kwh) 0,1 yearly cost of loss ( ) 1084 cable price / ton copper 7800 Scrap value / new value 0,5 Page 9

13 In this case, the curves of the former example are as follows: The economic lifetime is reduced to approximately 5.5 years. To achieve a higher economic lifetime, a larger cable cross-section should be chosen. The higher the scrap value, the lower the current density should be chosen (= higher cross sections) to achieve a particular economic life-time of the cable. TABLE WITH OPTIMAL VALUES In the annex you can find a table providing the optimal current density depending on the loading (Hour Power Equivalent) and the chosen economic lifetime n. ADDITIONAL ADVANTAGES OF LARGER CROSS SECTIONS The advantages of a larger cable cross section are not limited to economic and ecological benefits. A larger cable cross section also has certain technical advantages, including increased power quality and the flexibility of the connection. IMPROVED POWER QUALITY When the cross section of a conductor is increased, the voltage drop over the line is reduced. This means that the voltage variation between a loaded and unloaded cable is reduced as well. As a result, power quality issues (harmonics, voltage dips, transients, etc.) will be less severe. Furthermore, since the line will be carrying a lower load, the risk of a power outage caused by an overload will be lower. INCREASED FLEXIBILITY The power of the load has to be estimated when a new connection is designed. This is not always an easy task. An increased cross section of the cable makes the connection more flexible in regards to a future increase in the load. As long as the cable remains within safety limits, the load increase will not require an immediate replacement of the cable. Page 10

14 ROUND-UP Technical standards prescribing the minimum cross section of cables take only safety and certain power quality arguments into account. However, the most economical cross-section is several times larger. This economical cross section follows from the minimum Net Present Value (NPV) of the cable, taking the energy losses over the lifetime of the cable into account. Calculating the NPV requires the rated current of the connection, as well as a few boundary values, such as the actual loading, the electricity tariff, the interest rate, and a chosen economic lifetime. A company-wide policy for choosing economic cable sections can be established by determining a preferred current density. This optimal current density will minimize the NPV of each ton of copper conductor, taking the energy losses over its lifetime into account. Calculating this current density requires the average loading of the cables, the electricity tariff, the interest rate, and a chosen economic lifetime. Once a company-wide current density is established, the optimal cross-section for each individual connection follows directly from the rated current. The optimal current density will be even lower if the scrap value of the copper is taken into account. This means the optimal cross-sections will differ even more from the standard. Note that economic and environmental advantages are not the only ones derived from using a larger cable cross section. It will also have a positive influence on power quality and increase flexibility regarding future load increases. Page 11

15 ANNEX TABLE WITH A FEW OPTIMAL CONDUCTOR CROSS SECTIONS For a cable price of 0.3 /(mm 2 x m) Economic cross section (mm2) I rated F = 2.43 (office) F = 7.62 (iron, paper) Standard cross section (mm2) , , TABLE WITH A FEW OPTIMAL CURRENT DENSITIES Optimal current density for lowest NPV of losses Economic HPE = 2000 HPE = 3000 HPE = 4000 Standard (approximately) n = 5 years n = 8 years n = 10 years n = 15 years i = 7.5% HLE/HPE = 0.65 Copper price = 6000 Cable price = 6000 x 1.3 = 7800 /ton of copper conductor Electricity cost = 0.1 /kwh Scrap value = 50% of the value of new copper Page 12