The limits of HVDC transmission

Energy Policy 61 (2013) 292–300

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The limits of HVDC transmission William F. Pickard n Department of Electrical and Systems Engineering, Washington University, Saint Louis, Missouri 63130, USA

H I G H L I G H T S







Renewable Energy is abundant, but not necessarily near population centers. Its transportation requires energy and can be a major systemic inefficiency. HVDC can be transmitted 10,000 km with 2% loss and near-optimal embodied energy. Such transmission meets the requirements of intergenerational equity.

art ic l e i nf o

a b s t r a c t

Article history: Received 15 May 2012 Accepted 18 March 2013 Available online 16 July 2013

Renewable energy is abundant, but not necessarily near the urban centers where it will be used. Therefore, it must be transported; and this transport entails a systemic energy penalty. In this paper simple qualitative calculations are introduced to show (i) that high-voltage direct-current (HVDC) power lines for megameter and greater distances are unlikely to achieve power capacities much beyond 2 GW, although they can be paralleled; (ii) that most sources and sinks of electric power are rather less than 10,000 km apart; (iii) that such long lines can be constructed to have transmission losses o 2%; and (iv) e that lines of such low loss in fact meet minimal standards of intergenerational equity. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Corona loss Ohmic loss Renewable energy

1. Introduction When I was very young, we lived about a mile away and slightly inland from a coal company that received on its dock, at the edge of Boston Harbor, shipments of Pennsylvania anthracite.1 Eventually, ton by ton, portions of this fossil marvel were trucked to our driveway, poured down a steel chute through a basement window, and deposited in our coal bin. Only then was it of any use for heating our home. And this was only the beginning: my mother had still to shovel it into the furnace, and I had to trundle the ashes out to the curb. What no one remarked upon at the time was that somewhere, somehow each ton of coal had been augmented by many megajoules of useful work extracting it from the ground, cleaning and comminuting it, and shipping it all the way to our basement. It never occurred to me that there was a hidden energy overhead on the heat that kept our home so snug. Anthracite in the ground of Eastern Pennsylvania was of no use to the citizens of Boston: it had first to be extracted from the ground and transported to that market, at the largely overlooked expenditure of no small amount of energy.

And, of course, the same is today true of the natural gas whose pipelines cover the Earth like a subterranean spider web. For example, the new Nord Stream natural gas pipeline under the Baltic from Russia to Germany is so large that its input compressor was predicted to need a month just to get the line's 1220 km length up to operating pressure (Kramer, 2011); the completed project is rated at 55  109 m3 y−1 of gas throughput with an endto-end pressure drop of 12 MPa, consistent with a significant pipe loss (Brierley, 2010). The accounting is rather more complex than this. For example, in the United States in 2011, the total supply of natural gas available was reckoned to be 24.55 tcf2 of which approximately 0.65 tcf were used for pipeline fuel and another 1.42 tcf for miscellaneous plant operations (Energy Information Administration, 2012); that is, slightly less than 92% of the gas produced was actually available for sale at the distal ends of the pipelines. The same is true also of electricity production and distribution: for example, just over 93% of a generating station's net electricity production actually shows up on customers' bills (Energy Information Administration, 2011). Some of the missing 7% can be attributed to transformer losses, but much is due to loss along the transmission lines themselves, normally due electrical

n

Tel.: +314 935 6104; fax: +314 935 3248. E-mail address: [email protected] 1 It is worth noting that this once-popular clean-burning low-ash fuel appears no longer to exist in commercially significant quantities (cf. Höök and Aleklett, 2010). 0301-4215/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enpol.2013.03.030

2 ‘tcf’ stands for ‘one trillion standard cubic feet’ but can be considered a surrogate energy measure equal roughly to 1 EJ.

W.F. Pickard / Energy Policy 61 (2013) 292–300

Nomenclature

Prated

a A ACEA

r R RC RL

AEEA CW(ξ)

d

d dmax dtol ET G h I(z) I

L Ltol Lmax

M N p P(z) Pgen Pohm

the scaling parameter a is introduced in Eq. (3) (m) cross-sectional area of a power line's conductor (m2) optimal cross-section predicted using classical economic analysis (m2) optimal cross-section predicted using embodied energy analysis (m2) a sum of capital invested in Treasury InflationProtected Securities (TIPS). A typical strategy is for the issuer to specify a set rate of return δ per period. The initial value of the security is CW(0). At the end of each time period ξ, the capital CW(ξ) is adjusted for inflation over the period just ended to an equivalent purchasing power CW(ξ+1), and a dividend δCW(ξ+1) is paid ($). 2d is the distance between the adjacent surfaces of the two bundle-pairs of a bipolar HVDC transmission line (m) end-to-end fractional voltage droop along a line: d ¼[I (RCL)]/Vgen (dimensionless) largest permissible d for a proposed transmission scheme (dimensionless) largest tolerable d in a proposed transmission scheme (dimensionless) energy embodied in operating an HVDC link over a life cycle of length T (J) G ¼As/L is the end-to-end conductance of one of the N-bundles (S) height of the lower surface of an N-bundle above the soil (m) the average current of a given cable conductor at a specific location along a transmission line (A) the ampacity of a particular physical cable in a bundle: that is, the maximum amount of current the single physical cable can carry without risking dangerous deviations in its material properties (A) end-to-end length of a transmission line (m) the longest practical length given a specific dtol (m) the greatest conductor distance between converter/ inverter stations (grid nodes) likely to be encountered in practice. It is not an absolute physical limit (m) mass of bipolar transmission line conductor: M ¼2LAρ (kg) the integral number of cables in an N-bundle (dimensionless) wholesale price of electric energy to a particular user ($ J−1) average power carried by an HVDC transmission line (W) generator power at the source end of a transmission line: Pgen ¼P(0) (W) ohmic loss over a section of transmission line (W)

resistance but possibly involving corona discharge if the line is of sufficiently high voltage. When an energy source (be it a gas well or a generator of electricity) is close to the end user, modest hydraulic or electrical resistance can perhaps be tolerated: the loss per 100 km of even 1% of the billable energy may seem insignificant when 100 km is the maximum extent of the service area; but the same rate of loss over 5000 km is 40% of the input energy and might even be

RI

R t T V(z)

Vgen Vrated u, v u0 x, y z γ δ κ k

λ Λ ξ Ξ ρ s seff

φ ω

293

Prated 4 Pgen is the nameplate power of the generating e (W) station radius of a single conductor of an N-bundle (m) radius of an N-bundle (m) conductor resistance per unit length (Ω m–1) total line resistance. For a balanced two-wire line, RL ¼ (2L)/(As) (Ω) typical resistance per unit length of a particular class of overhead transmission conductor. For example, a Hen class line (I ¼666 A, A ¼2.42  10−4 m2) has RI ¼116 μ Ω m–1; a Kiwi class line (I¼1607 A, A ¼11.0  10−4 m2) has RI ¼ 26.3 μ Ω m−1 (Ω m–1). either R or r (m) time (s) the useful lifetime of a transmission line conductor (s) the average voltage (with respect to earth) of a given cable conductor at a specific location along a transmission line (V) Vgen is the output voltage of the generating station: Vgen ¼ V(0), where Vgen and Vrated (V) Vrated is the nameplate voltage of the generator (V) bicylindrical coordinates defined in Fig. 1 (dimensionless) defines unique cylindrical surface parallel to the z-axis (dimensionless) the transverse Cartesian coordinates in a right-handed xyz rectangular coordinate system (m) a position coordinate along a transmission line (m) USA dollars per kilogram ($ kg−1) real instantaneous rate of return. For example, 1.5% per period translates to 0.015 period−1 (period−1). the relative spacing distance of the two bundle conductors of a bipolar HVDC line. κ¼d/R⪢1 (dimensionless) 2a constant of proportionality. For a balanced bipolar high-voltage dc transmission line, it is 2 (dimensionless) fractional ohmic dissipation (dimensionless) a smooth surface being assumed, the great circle distance between the source and the load (m) a member of the sequence of non-negative integers 0, 1,…, Ξ (dimensionless) a non-negative integer (dimensionless) mass density of the metal of the power line (kg m−3) electrical conductivity of the metal of the power line (S m−1) the effective conductance of a typical metallic conductor made of mixed steel and aluminum wires twisted together and having air spaces (S m−1) electric scalar potential of the conductors comprising an N-bundle (V) the Process Fuel Equivalent to produce conductor metallic conductor from ore. For aluminum, ω∼60  106 (Warner, 2008) (J kg–1)

described as “intolerable”. As the Twenty First Century proceeds and the world's unexploited legacy of fossil carbon-based fuel gradually diminishes, it is to be expected that transferring energy as chemical potential borne by fossil fuel will become less important whereas transmitting renewable energy as electricity will increase in importance and make issues of electrical transfer costs rather more significant (cf. Armaroli and Balzani, 2011). It is also possible that the real price per kilowatthour at the generator

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W.F. Pickard / Energy Policy 61 (2013) 292–300

y 2r 2R x = –d

x = +d

x

2

u = +u0 φ = +V

u = –u0 φ = –V y = –h soil

Fig. 1. Schematic representation of a bipolar HVDC transmission line. It consists (conceptually) of two parallel conductors of radius R (gold circles) whose centers are a distance h above the soil (assumed to be at earth potential, φ ¼0) and whose apposing faces are a distance 2d apart. The conductors may be: solid; hollow; stranded; or composed of a bundle of N parallel cables of radius r (lime circles), each located at the vertex of a regular polygon. Cartesian coordinates are shown. But, if w¼ u+iv , then the Cartesian system can be transformed into a bicylindrical one with the conformal mapping x−iy¼ a[ew+1]/[ew−1] (Moon and Spencer, 2003, pp. 64,89); and this enables simple quantitative modelling of the voltages and fields.

the transfer-loss question resolves itself simplistically into (i) what the loss mechanisms are along the transfer path, (ii) what can be done to suppress each mechanism, and (iii) an economic model by which one decides how much suppression can be afforded/ justified. For a variety of reasons, moving large (e.g., gigawatt) quantities of electric power over great distances is most economically handled by high-voltage direct-current (HVDC) lines, (cf. Hammons et al., 2012; Van Hertem and Ghandhari, 2010; Weigt et al., 2010). Further, it turns out that a configuration widely adopted is the bipolar one shown schematically in Fig. 1. The major electrical loss mechanisms are two: corona loss and ohmic loss. Corona loss begins when, to increase power transmission, the voltage is raised to the point at which high electric fields at the conductor surfaces ionize the surrounding air and produce line-to-line (or line-to-ground) current leakage. Therefore, for any bipolar configuration, the line is voltage-limited. This means that transmitting more power requires raising the current I. But the physics of line resistance are such that wasteful electric heating of the wires goes up as I2, thereby rapidly increasing power loss. Thus, for any bipolar configuration, I too is effectively limited. Of course, line power capacity can still be varied (i) by tweaking the bipolar configuration or (ii) by altering one's economic model. Corona loss and its suppression are discussed in Section 2. Ohmic loss and its suppression are discussed in Section 3. The question of economic model is rather more complex and will be postponed until Section 4, where its complexities will be described but no definitive answers provided.

will rise as well. That is, the Age of Fossil Fuel was characterized by a massive dowry of fossil fuel laid down (without direct cost to mankind) over a few hundred million years by happenstance biological and geological processes. It has yet to be demonstrated that purposeful human endeavor can capture renewable energy from sun and wind as easily as mankind could exploit this fossil legacy. And a simple comparison of population density maps with maps of insolation and mean wind velocity will reveal that the best sources of renewable energy are often far from the centers of human population where one would like to expend the energy. Taken in aggregate, the mega-system that ultimately must convey the world's energy from point to point can be likened to a Gordian knot: a knot that tangles together political imperatives (both local and trans-national), regulatory constraints, financial limitations, public acceptance, transport modes (electric lines, gas pipelines, barge routes, etc.), the electrophysical practicalities of power lines, and the frequent failure of the preceding complexities to satisfy the engineers' beloved postulate of time-invariance. In practice, it should prove exceedingly difficult to model such a snarl accurately, much less to optimize it globally. This is so especially because the system will be expected to function reliably for many decades, whereas many of the constraints applied to the initial design turn out to drift over time. Therefore this paper will focus narrowly and ask simple questions about the electrophysics and the embodied energy of the electrical transmission cable: the goal is to show the Policy Analyst how one might minimize the energy sunk into (i) smelting the aluminum for the cable and (ii) making up for resistive loss along the cable. Further, this paper will show that, even so, the loss along the longest anticipated lines should be reducible to acceptable levels. As a preliminary to discussing electrical transfer costs, it is desirable to state the general rule that (for a balanced bipolar dc transmission line) the average voltage V of a line conductor with respect to ground, when multiplied by the average current I carried by the wire, yields a product VI that is related to the net power P transferred along the line through the relationship

Corona loss occurs because (cf. Comber and Zaffanella, 1982), when the electric field at the surface of a bare wire becomes sufficiently large: (i) electric charge is injected into the air near the wire (producing acoustic noise and visible light and radiofrequency interference); (ii) this charge moves in the electric field thereby creating a leakage current; (iii) this current constitutes a parasitic load that cannot be directly billed to the customer, and also (iv) it wastes energy. At low line voltages, no corona occurs. As the voltage increases, an onset voltage is reached beyond which the electric field at the line conductor surface produces a corona discharge (e.g., Comber and Zaffanella, 1982; Denissov et al., 2005; Li et al., 2011; Zhou et al., 2012). Onset electric field is hard to predict in advance but is known to decrease with decreasing atmospheric pressure or increasing altitude (e.g., Denissov et al., 2005), high air pollution (e.g., Denissov et al., 2005), low humidity or very high humidity (e.g., Hu et al., 2011), and scarring of the line conductors. There seems, however, to be qualitative agreement that field strengths below about 1 MV m−1 suffice to avoid significant corona whereas field strengths above 3 MV m−1 may be risky (cf. Li et al., 2011). To provide a definite example that can be carried throughout this paper, it will be assumed that the bipolar configuration of Fig. 1 is employed. Naturally no simple electromagnetic solution exists for a real bipolar geometry. However, in the absence of corona and neglecting ground effects, the bicylindrical system analogous to Fig. 1 has a simple solution for the electric scalar potential (cf. Moon and Spencer, 2003, p. 89)

P ¼ ^kVI;

φðu; vÞ ¼ V

ð1Þ

where ^k is a constant dependent only upon the line geometry. In parallel with the calculation of the power ideally transferred, the engineer typically carries out a calculation of loss along the transfer pathway. With electricity, as with any form of energy,

2. Corona loss

u ; u0 4 0: u0

ð2Þ

This implies (i) that the surface of the left conductor is an equipotential φ ¼−V, (ii) that potential is antisymmetric about the y-axis, and (iii) that the maximum electric field occurs along the

W.F. Pickard / Energy Policy 61 (2013) 292–300

x-axis at the facing conductor-surfaces and is of magnitude V cosh u0 þ 1 Emax ¼ a u0

ð3Þ

where the constants a and u0 remain to be determined. It is readily shown that the bicylindrical mapping can be inverted to yield u0 ¼ arc cosh

dþR ¼ arc coshðκ þ 1Þ; R

ð4aÞ

where R may signify either R (for N 41) or r (for N ¼ 1), κ ¼ d=R⪢1;

ð4bÞ

and a ¼ R sinhu0 :

ð5Þ

Simple calculus then yields Emax ¼ ¼

V κþ2 V κþ2 ½1−ðκ þ 1Þ−2 −1=2 ¼ R u0 sinh u0 R κ þ 1 ln½κ þ 1 þ ðκ 2 þ 2κÞ1=2   V V 1 þ κ−1  1 : 1 þ Oðκ −2 Þ H R lnð2κ þ 2Þ R lnð2κ þ 2Þ

ð6Þ

Extensive numerical modelling results were amassed by Li et al. (2011) for: N ¼2, 4, 6, or 8 with emphasis on 6; h¼ 10, 14, 16, 18, 22 m with emphasis on 16; d+R 8 8, 13, 18, 23 m with emphasis on 8; R ¼0.35, 0.40, 0.45, 0.50 m with emphasis on 0.45; and 14.0 o r e o 20.5 mm with emphasis on 18.5 mm. e A. For single bare conductors, the measured corona onset voltage (COV) rises rapidly with increasing r; this is in satisfactory agreement with the numerical modelling results of Li et al. (2011), which offers a degree of validation for these methods. Over the same r-range, the predicted corona onset electric field (COEF) fell slowly. B. Whereas, for a bipolar six-bundle, the predicted COV rose less rapidly over the r-range than that of a single monopolar conductor. Concurrently, the predicted COEF dropped only 5% over the r-range (Li et al., 2011, Fig. 12). This supports the physical intuition that, over a wide range of conductor radii r, a bundle of conductors behaves electromagnetically somewhat like a conducting shell of radius R. This shell approximation for a bundle is further supported by numerical data that predict virtually no variation of COEF with N (Li et al., 2011, Fig. 10). That is, for “rule of thumb” calculations or “back of the envelope” estimations with N 4 5, Eq. (6) should suffice with e R¼ R. C. Validation of the assumed insensitivity to h that underlies Eq. (6) is likewise provided by Li et al., 2011, (Figs. 19 and 20). D. Validation of the relative insensitivity to d that is predicted by Eq. (6) is likewise provided by Li et al., 2011 (Figs. 17 and 18). The canonical case used by Li et al. (2011) had d ¼7.5 m, R¼ 0.45 m, r ¼0.0185 m, and N ¼6. For this six-bundle, a theoretical lower bound on the electric field should be that for R ¼R and is approximately {Emax/V}min∼0.62 m–1, whereas a theoretical upper bound on the electric field should be that for R¼ r and is approximately {Emax/V}max∼9.06 m–1. It is known experimentally that the onset voltage rises with N (e.g., Hu et al., 2011). And, theoretically, for the canonical case, the electric field near a bundle conductor on the x-axis should decrease from {Emax/V}max for N ¼ 1 to roughly {Emax/V}min as N increases to not quite πR/r∼76, at which point the conductors of the bundle will be almost touching. Eq. (6), though not an exact representation of physical reality, nevertheless suffices to suggest that increasing the COV will require increasing R while simultaneously increasing N to limit electric fields near the bare conductors of the bundles. Naturally, as the COV rises for the line, it becomes necessary to increase the

295

dimensions of the already gigantic valve halls, bypass switches, arrestors, wall bushings, etc. (cf. Hammons et al., 2012). This could be a disincentive toward going to higher voltages, not only because of materials cost but also because of the sheer difficulty of assembly.

3. Ohmic loss Ohmic loss occurs because the carriers of electric charge within the conductors encounter resistance to their motion: think of this as a sort of friction. Most major high-voltage direct-current (HVDC) power lines consist (at least conceptually) of two wires: at the generator end, one wire is raised to a voltage +V while the other is lowered to a voltage V; a current I then flows out of the positive terminal, down a wire of length L, through the load, and back to the negative terminal through another wire of length L. For the bipolar case (k¼ 2), the output power of the generator is described by P rated 4 P gen ¼ 2V gen I; e

ð7aÞ

where one requires that Vgen ¼ Vrated, the nameplate voltage of the generator. Obviously, I is determined by the load on the system. One normally assumes (i) that I ¼ Pgen/(2Vgen) and (ii) that I o I, the e rated ampacity of the bundle. However, due to bundle resistance there is an ohmic loss of power   2 2L 2L P gen ¼ P ohm ¼ I 2 ð7bÞ As As 4V 2gen and a fractional ohmic dissipation λ¼

P ohm L I 1 I ¼ ¼ ; As V gen G V gen P gen

ð7cÞ

where G ¼As/L is the end-to-end conductance of one of the conductor bundles, and where A (m2) is the cross-sectional area of conductor and s is the effective electrical conductivity of the cable cross-section. It is economically desirable to minimize λ, although practical methods of doing so are somewhat limited3: (i). The transmission distance L has a lower bound Λ that is set by the great circle distance between the generator and the load. The great circle distance between two points is easily determined using Web resources such as http://www.gpsvi sualizer.com/calculators#airport. Table 1 shows such distances for endpoints that seem relevant in planning a transnational supergrid. Allowing for geographically imposed detours is an imponderable that has here been treated using the arbitrary rule of thumb Lmax o 3/2Λ. The bounding inequality on an individual link of this supergrid then becomes Λ o L and

3 Λ  Lmax o 10; 000 km: 2

ð8Þ

(ii). Because ohmic line loss varies as the square of the line current, there is considerable incentive to deliver a specified power Pload at the minimum current consistent with keeping the line voltage V line oVgen below the corona onset voltage 3 Although, in this manuscript, cables are sized for current at rated power, the fact that λ∝I(Eq. (7c)) suggests that cable conductors might thriftily be sized for average current as long as the current during sustained bursts at rated power does not exceed the ampacity of the cable.

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W.F. Pickard / Energy Policy 61 (2013) 292–300

Table 1 Great circle distances Λ between generating sources in celebrated deserts and far distant urban centers (sinks). Rural solar source

Urban sink

Λ[km]

Alice springs (in the Australian outback) Timbuktu (in the Sahel of Mali) Palm springs (in the Mohave of California) Arica (in the Atacama of Chile) Jiayoguan (in the Gobi of China)

Bangkok (Thailand) Murmansk (in the Russian Arctic) Goose Bay (on the Atlantic coast of Canada) Caracas (Venezuela) Sapporo (Japan)

5526a 6885 4866 3223 3565

a Of course, crossing the Timor Sea and the subduction zone between Australia and Indonesia will be challenging. And some detour will be desirable to travel overland (where possible) up the Indonesian Archipelago and the Malay Peninsula.

Table 2 The cost metric γρ/s for several geochemically abundant, ductile, metals that are stable under atmospheric conditions. Compared to copper, the conductivity of aluminum is approximately 63% IACS (International Annealed Copper Standard). Metal

Price γ ($ kg−1)

Density ρ (kg m−3) d

Conductivity s (S m−1) e

γρ/s ($ S−1 m−2)

Aluminum Copper Steel Nickel Titanium Zinc

2.17 a 8.39 a 0.81b 18.75 a 8.8c 2.03 a

2.7  103 9.0  103 7.9  103 8.9  103 4.5  103 7.1  103

3.8  107 6.0  107 1.0  107 (Fe) 1.4  107 2.6  106 (0 1C) 1.7  107

1.5  10−4 13  10−4 6.4  10−4 120  10−4 15  10−4 8.5  10−4

a Cash prices at the London Metals Exchange as of 09 Mar., 2012. http://www. metalprices.com/FreeSite/# b Approximate composite carbon steel price for February 2012.http://www. worldsteelprices.com/ c Approximate price in first week of March 2012.http://www.steelonthenet. com/feeds/titanium.php d Values at ∼20 1C from CRC Handbook of Chemistry and Physics. http://www. hbcpnetbase.com/ e Values at ∼20 1C from CRC Handbook of Chemistry and Physics.http://www. hbcpnetbase.com/

(COV). A plot of maximum commonly used line voltage versus year (Hammons et al., 2012, Fig. 9) indicates that this has been stalled around 0.8 MV for nearly fifty years. Breakthroughs are of course possible, but can never be guaranteed4. Given the risk of corona and breakdown and the imperative for reliability, it seems prudent not to count on transmission voltages much beyond 1.0 MV. Illustrative calculations in this paper will be made using the well tested Vgen ¼ V750 ¼ 0.75 MV. (iii). λ∝1/G, and it might seem desirable to increase G. Observe, therefore, that each choice of line metal will have its own cost per siemens and that this will be cost per siemens ¼ ½γρALC½As=L ¼ γρL2 =s;

ð9Þ

where γ is the price of the metal and ρ is the mass density of the metal. L is fixed, and hence the useful material cost metric is γρ/s. This is tabulated in Table 2 for several good conductors that are stable under atmospheric conditions. By the γρ/s metric, aluminum is several-fold better than its rivals5. However, power line reality is not this simple, and issues of flexibility and tensile strength have led to stranded cable in which some of the strands are steel: in practice the effective

4 Witness the massive assault by Thomas Edison on batteries for motive force in automobiles. The challenge frustrated him in the decade before World War I (Sulzberger, 2004). And now, a century later, it is still frustrating manufacturers. 5 As discussed in Section 4.3, the initial refining of aluminum from ore is highly energy intensive. However, the line lasts a long time; and the aluminum in it is recyclable at much lower energy cost.

conductivity of a bundle conductor is seff ∼2.4  107 S m–1 (40% IACS), which could reflect adding a few more conductors to the bundle to compensate (a) for the lower conductance of steel and (b) the air spaces between the twisted strands. (iv). A is the sole remaining parameter that can be varied to control fractional loss: elevating A reduces λ. An ultimate lower limit to A is set by the level at which ohmic loss will cause the conductor temperature to rise above 660 1C, the melting point of aluminum6. A more realistic lower bound might be the level at which the loss will cause voltage at the load end of the line to collapse to brownout levels, say λmax∼⅛. With Vgen ¼V750, Eq. (7c) then leads to h i P gen L Amin 4 4 2:9  10–19 LP gen : ð10Þ 2 e λmax sef f 2V 750 e Multigigawatt lines that run a few thousand kilometers are now appearing (cf. Hammons et al., 2012). But here a conservative maximum for Pgen of 1.0 GW will be employed, yielding I¼ 667 A. Taking L ¼Lmax ¼7500 km, this gives Amin 4 22  10–4 m2 or a conductor radius of about7 26 mm. e other hand, demanding λ On the max o 0.01 would force wire e 2, yielding a conductor area into the range above 275  10−4 m rather too inflexible to string easily between transmission towers; in this instance, using instead a twelve-bundle configuration would yield conductors of a more reasonable radius, around 27 mm, and this is only slightly beyond the upper end of the radius-range of common bare line conductors. (v). The upshot of these calculations is that it seems technically feasible to build dedicated HVDC transmission lines of 1 GW capacity and negligible transmission loss ( o 1%) between e most likely nodes on the planet; these lines would originate at a center's nearest-neighbor resources of plentiful renewable electrical energy. However, to achieve so low a loss would involve using many conductors in parallel, the current rating (ampacity) of each such bundle being upwards of 10 kA.

6 Coincidentally, this is also roughly the temperature at which steel reinforcing cable, sometimes incorporated into a twisted aluminum conductor for strength, tends to undergo long term plastic deformation. 7 In actual commercial practice a conductor for this much current could well be composed of smaller diameter wires, some steel and some high conductivity aluminum, which have been twisted together and designated Aluminum Conductor Steel Reinforced (ACSR); this is a favored compromise between the tensile strength of steel and the conductivity of copper, and it has been popular for over a century (Thrash, 2012). A typical manufacturer will list a large number of such conductors. For example, The Southwire organization in the United States (Southwire, 2003) lists its Kiwi class as having 72 aluminum subconductors twisted together with 7 steel ones, which yield r ¼22.3 mm but an effective conductivity of 2.4  107 S m−1 (40% IACS); the ampacity of such a conductor is roughly 1607. Because the difficulty of bending a cylinder rises as r4, the maximum practical radius of a conductor is limited; and, judging from the usual commercial availability of ACSR conductor, that limit is around r ¼ 25 mm (A¼ 20  10−4 m2).

W.F. Pickard / Energy Policy 61 (2013) 292–300

(vi). By comparison, recent policy literature involving long distance transmission seems to accept as normal rather higher line losses, fractional ohmic losses, λ, being as high as 4 or 5% per megameter at rated load (Delucchi and Jacobson, 2011; Trieb et al., 2012). Such losses, prorated upwards for ultra-long multi-megameter lines, could quickly become unacceptable.

4. Economic justification 4.1. Introduction Presumably, a utility would endeavor to avoid corona loss because: (i) exceeding the COV means that the danger of a flashover to some nearby surface may be uncomfortably high; (ii) the public may complain about the audible sound; (iii) the public may complain about the air ions and/or the ozone being generated; and (iv) the corona is wasting electricity (PGandE, 2005). In this paper, it will be assumed that the operator of the transmission line will prudently avoid corona discharge under normal atmospheric conditions. But there will still be the question of choosing A 8. It is true of many construction projects that: (i) simply of preparing the site and launching the project is a dauntingly major expenditure whereas additional units of construction trend progressively cheaper9; and (ii) emplacing the various parts of the project is a construction expense may be relatively insensitive to the quality specified for those parts10. This suggests balancing (a) the cost of the aluminum in a conductor of effective area A against (b) the rate at which energy is dissipated in a conductor of that area. The conductor-cost of A will be just One−Time Conductor Cost ¼ C W ¼ A½2Lγρ:

ð11aÞ

The units of this cost is [$] whereas the monetized ohmic loss rate resulting from this particular choice of area will be Continuous Dissipation Cost ¼ p

2 2L 2 L P gen I gen ¼ p ; As As 2V 2gen

ð11bÞ

where p ($ J−1) is the wholesale cost of electrical energy, and a crude estimate of p is 72 $ MW h−1 ¼20 n$ J−1. The units of Continuous Dissipation Cost are, however ($ s−1) rather than ($). But how is one to compare the expense of increased conductor cross-section with the benefit of reduced dissipation when the two are not commensurable? It will be argued below that a totally satisfactory comparison may, in fact, not be possible. Instead, four quite different analyses will be given, none of which may seem all that satisfactory. But first, a simple numerical example will be given. Consider a typical Kiwi-class conductor11 of ampacity 1607 A and 8 An HVDC transmission line does not exist in splendid isolation but is part of a complex and costly system whose design is not entirely independent of the line. That is, the cost of the line itself must be considered along with the cost of the Balance of the System (BOS). The balance of system includes right of way costs, ACto-DC conversion costs at the source, DC-to-AC conversion costs at the terminus of the line, permitting costs, and many others besides. For simplicity of the analysis, these costs are assumed to be independent of the configuration of the line. In reality, this is not strictly so: for example, the pylons that support the N-bundles must become more robust as the weight of those bundles increases. 9 Thus, in deep mines, it is frequently the case that sinking a shaft and assembling the equipment and excavating the first 100 m3 of tunnel at working depth is very costly, while the incremental cost of each additional m3 of tunnel excavation trends ever lower. 10 Thus, the actual stringing an (N+1)-bundle of conductors might be only minutely more expensive than stringing an N-bundle, the cost increment of going from N to (N+1) being primarily in the extra conductor. 11 In the United States, bare overhead power lines are commonly made of stranded aluminum conductor with interspersed steel reinforcing strands. Such

297

resistance per kilometer of 26.3 mΩ. In our illustrative bipolar example of Pgen ¼1 GW, Vgen ¼750 kV and I¼667 A, the fractional generator-to-load voltage droop due to ohmic loss at full-load current is as displayed in Table 3; because the current is constant throughout the system, the fractional voltage drop d is numerically equal to the fractional loss of transmitted power. The message of this table is that, even though the current never reaches 50% of the ampacity, a 1000 km transmission link would be manifesting significant loss and 10,000 km link would be deep into brownout. That is, the efficient transfer of multi-gigawatt quantities of power over continent-spanning distances is technically feasible, but only if lines are overbuilt by the standards of sub-megameter transmission and utilize many-conductor N-bundles12. 4.2. Classical engineering analysis One traditional way of making a comparison between incommensurables is to imagine that, rather than ones investing an increment of capital, CW, into added aluminum to lower the loss of the transmission line, one had instead put it into inflation-indexed bonds of a known fixed rate of return δ. The latter choice would forthwith have left one with an undiminished liquid capital CW plus a secure continuous income: Continuous Income ¼ δC W ¼ δA½2Lγρ;

ð12Þ

whereas the former choice would have transformed CW into illiquid capital and yielded the prospective continuous savings given by Eq. (11b). Now add Eqs. (11b) and (12) to get a loss function: (11b) representing the ohmic line loss that cannot be avoided if the power is to be delivered; and (12) representing the secure return on the forgone conservative investment. Now differentiate this function to find that 2 dðlossÞ L P gen ¼ δ½2Lγρ–p 2 ; dA A s 2V 2gen

and a loss minimum at rffiffiffiffiffiffiffiffiffiffi P gen p ACEA ¼ ; 2V gen γδρs

ð13Þ

ð14Þ

In this model, it is clear that increasing the cross-sectional area of the line is a hugely profitable investment near A-0; and it remains marginally profitable until A-ACEA. As was to be expected, this measure is independent of the length of the line. Now let us see how this plays out practically in today's financial climate13. At present, the return on index-linked bonds is uncharacteristically low; but, normally, the annualized real-rate of return has seemed to run around 3%, so we shall take δ4 1  10−9 s–1. This then gives ACEA o 63  10−4 m2 and implies N ¼6e if Kiwi-sized e Such a configuration has many-fold the conductors are used. ampacity required to keep the line within thermal limits. The modest rate of return, when coupled with the illiquidity of the investment in the line, seems rather lower than would interest most investors. (footnote continued) lines come in a wide variety of sizes and styles, each with a code name, usually that of a bird [http://www.cable.alcan.com/NR/rdonlyres/65A29EE7-C5CA-44CF-AA DA-352EE2405F0D/0/UT0003.pdf]. The Kiwi is one of the largest sizes. 12 A Web search on “overhead transmission line” AND “N-bundle” yielded: 1980 hits for N ¼4; 137 hits for N ¼6; 127 hits for N ¼8; 3 hits for N ¼10; and 1 hit for N ¼12. One interpretation of these data is that the complexity of stringing the bundles for N 48 has not yet been overcome. 13 For simplicity, the calculations in the remainder of Section 4 will be made assuming pure aluminum solid conductor. Even though this would not be used in practice, the assumption does markedly facilitate laying out the concepts of the four analyses.

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W.F. Pickard / Energy Policy 61 (2013) 292–300

Table 3 Fractional line-voltage drop from generator to load along a single Kiwi conductor (ampacity ¼I ¼1607 A; A ¼11.0  10–4 m2) when Pgen ¼1 GW, RC ¼26.3 mΩ m–1, Vgen ¼ 750 kV, and I¼ 667 A. The fractional voltage droop is d ¼[I(RCL)]/Vgen. Clearly, line lengths beyond a thousand L (km) d (%)

1 2.34  10

10 −3

2.34  10

100 −2

2.34  10

−1

1000

10,000

2.34  101

2.34  101

kilometers will definitely require N-bundle lines. To hold d under 2% will require: N ¼2, 850 r L r 1700 km; ... ;N ¼4, 2550 r L r3400 km; ... ;N¼ 9, 6850 r L r 7700 km; ... ;N ¼12, 9400 rL r 10,250 km. To achieve a 2% loss over 10,000 km requires “overbuilding” the line to reduce resistive loss: to carry 667 A at rated loss requires a 12-bundle ampacity of 19,284 A.

4.3. Embodied energy analysis This viewpoint defines the relevant embodied energy14 ET over a life cycle of the line as the sum of (i) the primary energy required to mine and refine virgin aluminum for the line plus (ii) the total ohmic loss incurred over the lifetime T of the line:      ET ¼ energy per unit mass mass of line þ fT g P ohm ( 2 )   P gen 2L ¼ fωg 2LAρ þ fT g ; ð15Þ 4V 2gen As where ω (J kg−1) is the energy expenditure in mining and refining, that is the Process Fuel Equivalent of producing metallic conductor from ore; for aluminum, ω∼60  106 J kg−1 (Warner, 2008)15. Simple differentiation then predicts a unique optimal conductor cross-section of sffiffiffiffiffiffiffiffiffi P gen T ð16Þ AEEA ¼ 2V gen ωρs The lifetimes of transmission conductors can be relatively long; and there is anecdotal evidence (e.g., Schweiner et al., 2002) to suggest that the principal failure mode of lines is tower collapse rather than isolated and unprovoked conductor failure. The bare conductors themselves may last 70 y (e.g., Westermann et al., 2010); and here it will be assumed that T ¼2.0 Gs ¼65 y. For the parameters of our ongoing example with the Kiwi-sized conductor, this gives AEEA ¼110  10−4 m2 and implies N ¼10. 4.4. Take the money and run In reading a classical text on engineering economics (e.g., Grant and Ireson, 1970), one is struck by the understated reliance of their cost comparison techniques upon the assumption of a steady state: the purchasing power of money will endure unchanged; the real return on an investment can be counted upon to remain constant; and the economic model will not be disrupted by unforeseen players, unanticipated technologies, or acts of God. In the real world, this has seldom been the case: quantitative details of the future have always been unknowable, our reading of events yet to come disturbingly uncertain, and our entrepreneurial activities fiscally hazardous16. These risks may not scare off all 14 The term ‘embodied energy’ seems to have been coined in the 1970s and is frequently associated with the ecological economist Robert Costanza (1980). Since that time it has become a much used concept, and a Google search on it turns up in excess of 350,000 hits. As definitions of it go, that of Eq. (15) is relatively restrictive. However, possible added terms would not involve A, and therefore Eq. (16) would not change. 15 Warner's (2008) estimate was for a hypothetical ultra-efficient process. Present industrial practice could be closer to 210  106 J kg−1 (Rankine, 2012, pp. 4–7). 16 In finance, bubbles have a long and presumably well known history. There is a fascinating recent literature on them (e.g., Ferguson, 2010; Reinhart and Rogoff,

the venture capitalists; but they presumably do sharpen the remaining investors' desires to recoup their investments early on; and this argues for a high early net rate of return. This means that there may be a temptation to reduce the amount of aluminum in the line. Let I be the rated ampacity of a proposed line and dmax the maximum tolerable voltage droop over the line length Lmax. If then RI is the resistivity, Lmax ¼ dmax

V gen RI I

ð17Þ

If, for the standard example, with Vgen ¼750 kV and Pgen ¼1 GW, we accept I¼ 667 A and RI ¼116 μΩ m−1 (Hen class specifications, A¼2.42  10−4 m2) and set dmax ¼2%, it follows that Lmax ¼194 km. That is, a line that might suffice marginally for local area power transmission becomes quite unacceptable for long distance transmission; but, the capital markets being nervous, it might well not attract the extra capital needed to beef it up to suitable robustness. Under this economic model, it is not clear how a low-loss 7500 km line could be built prior to a severe energy crisis. Such a frugal outlook was recently described as follows: “Usually the cost of losses during the lifetime of a cable is not taken into account when selecting a cable size. In fact, the selection leads to the minimum admissible cross-sectional area, minimizing the initial investment cost of the cable, but not taking into consideration its whole life-cycle.” (Pezzini and Sumper, 2012, p. 15). 4.5. Veil of ignorance analysis This procedure takes its name from a sort of Gedanken experiment in which two parties, Present (who is going to size and pay for the conductor) and Future (who will have to live with the consequences of that choice), endeavor to negotiate an agreement. As the negotiation begins, a “veil of ignorance” is somehow imposed such that each party might with equal probability be assigned either the role of Present or the role of Future. If then, in the face of this ambiguity, both parties believe that a particular negotiated agreement is equitable, it probably is indeed equitable (cf. Rawls, 1971). As matters now stand over much of the world, there is still ample fossil fuel that can easily be shipped by well-developed conventional technology to electricity generating plants near the end user. There is little urgency and no immediate need to build long (4 500 km) low-loss lines. Nor is such need anticipated until some decades hence, at which time the end of the Age of Fossil Fuel arrives and, lacking the formerly unnecessary low-loss lines, Future must cope with energy starvation. The Author holds that a goodly majority of “blindfolded” negotiators would agree that enduring belt-tightening in the present is tolerable if it drastically lowers the chances of starvation in the future. This sort of concern for future generations is not unheard of in the energy field and indeed is a foundation stone of the International Atomic Energy Agency's safety standard on underground disposal of high level radioactive waste (IAEA, 1989, p. 7): “The burden on future generations shall be minimized by safely disposing of high level radioactive wastes at an appropriate time… Since the present generations benefit directly from their exploitation of nuclear energy, it is reasonable that they should bear the financial burden of waste disposal…. The safety of a high level waste repository in the post-sealing period shall not rely on active monitoring, surveillance or other institutional controls or remedial actions after the time when the control of the repository is (footnote continued) 2009; Wicker, 2008). Nevertheless, around the world, ostensibly preventable financial crises still occur.

W.F. Pickard / Energy Policy 61 (2013) 292–300

299

At its end, the Age of Fossil Fuel will probably have lasted about 300 years. And, for most of us who enjoyed its bounty, it was great while it lasted! But if we determine not to bequeath ample fossil fuel to future generations, ought we not to leave the knowledge, the technology, and the infrastructure, that might enable our descendents to avoid regression to pre-industrial poverty? It can be argued that now, sitting pretty close to peak oil (Maggio and Cacciola, 2012; Murray and King, 2012), we should adopt a Veil of Ignorance economic viewpoint and begin building for generations yet unborn. There are already enough losses in our electricity transmission system: capture losses at the generator or collector, AC↔DC conversion losses, transformer losses, storage losses, etc. Enough so that it would, at the very least, be inconsiderate to add an ohmic loss that would markedly increase the extant total. As a 2% loss for a long line would mandate an area A not hugely different for that obtained from an embodied energy analysis, it seems reasonable to go with the 2% analysis, as described for a realistic line in Table 3. The 2% loss-limit cited illustratively above does NOT imply that a line will necessarily be dangerously overloaded if the total current of the N-bundle is pushed above the nominal rated current Inom. The line will operate safely as long as I o NI. What has been shown e here is that multi-megameter transmission at low loss is entirely reasonable and is even desirable if one wishes to minimize the embodied energy consumed over the line's life-cycle. Operation above the nominal line power capacity may be a questionable decision from the viewpoint of embodied energy analysis; but it may, upon occasion, be exceedingly profitable to the line's operator.

gives d ∼42%, which seems undesirable. For d ∼2% over 7500 km, the allowable current per Kiwi-size conductor is only 76 A; hence, even a 12-bundle would carry only ∼900 A, and 667 A would require a 9-bundle. Consequently, even in a rosy prediction, the power capacity of a single energy-efficient long-line seems unlikely to exceed a power of twice the product of technically foreseeable generator voltage and maximum 12-bundle current calculated above, or 2(1.5  106) 900 ∼2.7 GW. And to have over 7500 km a 2% transmission loss with 750 kV, would imply a power capacity of 1.4 GW at 900 A. But if one were to opt for the security and simplicity of robustly mature technology, 1 GW @ 750 kV would seem to be a more cautious choice upon which to standardize. Third, a decision to rely upon 1-GW lines could have the collateral benefits of covering the service area of a mega-grid with a spider web of overhead HVDC lines18. But the connection rules of such an HVDC-web remain to be specified. Would each line be just an AC/DC source that transmits to a DC/AC load? Or might each segment of line begin and end at a generic node that could: AC/DC convert, DC/AC convert, store energy, or transfer power to another segment? The latter choice could make the spider web far more versatile and therefore far more fault tolerant of both natural disaster and sabotage. However, to enable a damaged smart grid to reconfigure itself to avoid a damaged region while simultaneously maintaining the low-loss flow of power between any two nodes would presumably require that many (most?) links were sized to be a part of an extremely long path; and this, though productive of resilience, could be construed as wasteful of material. In short, to put a robust HVDC backbone into either (i) North America above the Panama Canal or (ii) Europe west of the Urals plus MENA (the Middle East and North Africa), it might be wise to construct it of overbuilt links that could be rearranged to skirt natural disasters or spectacular acts of sabotage. Yes, construction costs would be higher. But perhaps such super-links could be regarded as defensive weapons that protect centers of production from catastrophic shut down. And they would make the grid backbones more robust and resilient.

5. Conclusions

References

relinquished.” (Emphasis added.) These principles are only simple intergenerational equity. And, extended to HVDC transmission, they call for enough aluminum in long lines to keep the resistive loss down to levels comparable to those now experienced by shorter lines, say 2%. 4.6. Summation

This paper began with the notion of estimating the practical limits of HVDC transmission via a single long bipolar line with bundle conductors. Using a restricted model that considered only line losses, it turned out that this single line, over thousands of kilometers, is limited in capacity to a few gigawatts, perhaps only 1 GW. First, the line to ground voltage is limited by the dangers of flashover. And if that can be avoided, there is still the issue, especially in foul weather, of corona loss with its environmental annoyances and its power dissipation. Lines up to and including 750 kV are tolerably well understood and backed by mature technology. But even 1000 kV lines for DC are somewhat visionary at present. And 1500 kV lines presumably require unspecified breakthroughs from unspecified directions. That is, seemingly, Pgen can be increased by no more than a factor of 2 from the voltage side. Second, today the largest overhead conductor ampacity readily available commercially is I ¼1607 A; and this conductor has a resistivity17 of 26.3 μΩ m−1. But 1607 A @ 750 kV over 7500 km 17 At 667 A (only 42% of ampacity), this resistivity implies a full load power loss of 12.1 W m−1 or 0.0121 GW per 1000 km for a Kiwi class cable. If instead, a line of the “correct” ampacity (Hen class) had been used, the loss would have been 0.0535 GW per 1000 km. The moral of this calculation is that traditional sizing of power line conductors can, over continent-spanning distances, lead to

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