Tecnología y electrificación de los muelles

( -) (

= )

(Da_k^pw E C_{ns k}^pw EC_s

E (2.21)

Where E(C_s)^pw_k and E(C_ns)^pw_k are the present worth expected value of the costs for buying the component during the system life in sinusoidal and non-sinusoidal operating conditions, respectively.

The actualization of the costs can be effected in a similar way as in the previous equations 11) and (L-12) of Appendix 2 - L. considering both the discount rate and the cost variation for buying the

component; the expected value of cost to be met for buying each component at year n in sinusoidal and non-sinusoidal regimes is linked to the expected value of the component life in these conditions,

respectively. To estimate these figures again, the cumulative damage theory can be applied, as in the case of deterministic methods. In such a case, we have to refer to the expected value of relative loss of life in the study period, E[∆R_L]. A more complete analytical formulation is reported in Appendix 2-L.

2.3. Methodology for Quantifying the Economic Impact of Other PQ Phenomena

2.3.1. Voltage and Current Unbalance

The sensitivity of electrical equipment to unbalance differs from one appliance to another. A short overview of the most common problems is given below:

Induction machines: The magnitude of the internally induced rotating magnetic field in induction machines (IMs) is proportional to the amplitude of the direct and/or inverse components. The rotational sense of the field of the inverse component is opposite to the field of the direct component. Hence, in the case of an unbalanced supply, the total rotating magnetic field becomes “elliptical” instead of circular and consequently could lead to three types of problems in IM operation. First, the machine cannot produce its full torque as the inversely rotating magnetic field of the negative-sequence system causes a negative braking torque that has to be subtracted from the base torque linked to the normal rotating magnetic field.

Secondly, the bearings may suffer mechanical damage because of induced torque components at double system frequency. Finally, the stator and, especially, the rotor are heated excessively, possibly leading to faster thermal aging. This heat is caused by induction of significant currents by the fast rotating (in the relative sense) inverse magnetic field, as seen by the rotor, and is also accompanied by vibrations. To be able to deal with this extra heating, the motor must be derated, which may require a machine of a larger power rating to be installed. In general, if voltage unbalance is permanently higher than 2%, the losses of fully loaded IM are likely to cause damage.

Synchronous generators: Synchronous generators are exposed to similar stress as IM when subjected to unbalance. However, they mainly suffer from excess heating. Special care must therefore be devoted to the design of stabilizing damper windings on the rotor, where the currents are induced by the indirect and homopolar components.

Capacity of transformers, cables, and lines: The capacity of transformers, cables, and lines is reduced due to negative-sequence components. The operational limit is in fact determined by the RMS rating of the total current, being partially made up of “useless” non-direct-sequence currents as well. This has to be considered when setting trigger points of protection devices, operating on the total current. The maximum capacity can be expressed by a derating factor, to be supplied by the manufacturer, which can be used to select a larger system, capable of handling the load.

Transformers: Transformers subject to negative-sequence voltages transform them in the same way as positive-sequence voltages. The behavior with respect to homopolar voltages depends on the primary and secondary connections and, more particularly, the presence of a neutral conductor. If, for instance, one side has a three-phase four-wire connection, neutral currents can flow. If at the other side the winding is delta-connected, the homopolar current is transformed into a circulating (and heat-causing) current in the delta winding. The associated homopolar magnetic flux passes through constructional parts of the transformer, causing parasitic losses in parts such as the tank, sometimes requiring an additional derating.

Electronic power converters: These are present in many modern devices such as adjustable-speed drives, PC power supplies, efficient lighting, etc., and the amount of power electronic converters is bound to increase further in the future. As a consequence of unbalanced supply, they can be faced with additional uncharacteristic harmonics, although, in general, the total harmonic distortion remains more or less constant. The design of passive filter banks dealing with these harmonics must take this phenomenon into account.

The table below specifies percent of extra losses from load unbalance as a function of neutral current resulting from unbalance to average phase current.

Table 2-1 Extra losses due to unbalance

% of additional losses from load unbalance Ratio of neutral current to

average phase current Transformers Low-voltage lines

0,5 6-8 40-50

2.3.1.1. Classification of Unbalance Costs

Economic losses due to voltage and current unbalance, i.e., the economic costs of unbalance (

K

_as), can be divided into two categories, namely technological losses (

K

_as^'' ) and electromagnetic losses (

K

^'_as).

The technological losses include losses resulting mostly from changes in the slip and torque of induction motors and consequential decrease in the output of motor-driven production equipment, a decrease in the induction motor’s maximum torque, reduced efficiency of single-phase electrical heating equipment, and reduced efficiency and lower quality of production due to changes in electric lighting. They also depend on the load type and should be calculated taking into account specific features of production processes.

The electromagnetic losses associated with voltage unbalance result mainly from an increase in active power losses, as well as increase in the active and reactive power demand, reduction of capacitors and synchronous machines’ reactive power with respect to the required value, accelerated aging of insulation, and reduced in-service time of light sources.

The annual costs of losses

K

_as^' due to unbalance can be expressed as the sum:

∑ ∑ ∑

Where

∆ K

_Pj = additional cost of power losses in the j element (equipment, load) in the considered facility due to voltage and current unbalance

K

Aj

∆

= additional cost of energy losses in the j element due to voltage and current unbalance

K

Rj = costs of restoration of the j element caused by aging of its insulation due to voltage and current unbalance

K

Q = cost of the reactive power reduction due to unbalance

K

0 = cost of light sources replenishment due to detrimental effects of voltage unbalance The overall unbalance costs (

K

_as) include costs of the negative-sequence unbalance (

K

_as2 =

K

_as^' ₂ +

''

K

as₂) and zero-sequence unbalance.

2.3.1.2. Additional Costs of Power Losses and Electric Energy Losses

Annual cost of additional losses in the j element (loads, the series and shunt transmission, and distribution equipment) is:

j pj

Pj

k P

K = ∆

₂

∆

(2.23)

Where:

∆ P

_2j = maximum additional losses in a year, caused by voltage unbalance (loads, shunt equipment, no-load losses in transformers, etc.) or current unbalance (transmission and distribution series equipment)

k

pj = unit cost of power losses at the level of the power system in which the j¹ element is connected

Annual costs of additional energy losses in the j element are:

j Aj

Aj

k A

K = ∆

₂

∆

(2.24)

Where:

∆ A

_2j = annual additional cost of energy losses in the j element caused by the voltage or current unbalance

k

Aj = unit cost of energy losses at the level of the power system in which the j element is connected

For practical purposes, ∆A₂ is often calculated from the formula:

j j

j

P

A

₂

= ∆

₂

∆ τ

(2.25)

Where

τ

_jis the annual duration of maximum losses

∆ P

_2j.

The relative power losses ₂ ²

100

N

*

P P P

∆

= ∆

∆

, where

∆ P

_N is the nominal losses, under permissible voltage unbalance conditions (i.e., the measured negative-sequence unbalance factor 2%) are negligible.

For example, according to [100] these losses are: (a) 6-kV and 10-kV induction motors with rated powers above 100 kW

∆P

₂^*

= 2 . 4 %

; (b) synchronous motors with rated powers above 100 kW ― 4.2%; (c) transformers in industrial networks 1 to 4%.

2.3.1.3. Costs of Equipment Restoration

The voltage or current unbalance causes additional heating of electrical equipment components that results in shortened in-service time of the equipment insulation. The equipment in-service life will also be shortened because of the intensification of ionizing processes caused by the voltage increase due to unbalance.

Under balanced supply conditions, the equipment in-service time equals

T

_S (in years). The operability costs of this equipment during time

T

_S, i.e., total annual costs of expanded reproduction and operating

1 The method of determining the losses ∆P₂ for various types of loads and transmission and distribution equipment is outside the scope of this report.

costs, are

K

_us, and discounted costs (constant over time) are

K

_uar. Under unbalanced conditions, the equipment in-service time equals

T

_a<

T

_S, total annual costs are

K

_ua, and annual discounted costs are

K

uar:

Where:

K

_i = investment expenditures associated with the equipment installation

p

S and

p

_a = coefficients of the equipment reproduction (fixed operating costs taken into account);

p

_S ≠

p

_a because of different depreciation periods in each case.

After elapse of time

T

_a, the equipment should be repaired or new identical equipment should be installed.

The expected cost of repair or installation of new equipment is

K

_m. Total annual costs incurred during the period (

T

_a-

T

_S) are:

Total discounted costs associated with premature replacement or necessary repair of equipment, the so-called annual costs of equipment restoration, are:

ni

In practical calculations,

p

_n≈

p

_S≈

p

_a can be assumed. As evident from the formula (2.28), the relative time of shortening the equipment life due to voltage and current unbalance has considerable influence on the costs K_R²:

As follows from research [69], under voltage unbalance conditions and the voltage unbalance factor of 2%, the average values of time

∆ T

^* are: (a) induction motors – 9.1%; (b) synchronous motors – 10.2%;

(c) distribution transformers – 2.3%; transmission transformers – 3.4%; converters – 3.4%; power capacitors – 20 to 25%.

2.3.1.4. Cost of Reduction of the Reactive Power Value

The reactive power of a capacitor bank is changing as a result of the voltage unbalance. Compared to the reactive power under the balanced supply voltage conditions, this power often decreases by ∆Q_K. The unbalance of currents in synchronous machines reduces their inductive reactive power by

∆ Q

_G.

2 The method of determining the time ∆T or ∆T^* for various types of loads is outside the scope of this report.

The deficit in reactive power

∆ Q = ∆ Q

_K

+ ∆ Q

_G should be replenished by means of additional compensation equipment, e.g. installation of additional capacitor banks. Total annual discounted costs associated with the installation and operation of additional capacitor banks are:

RQ pQ ZQ PA Qi Q

Q

p K K K K K

K = + + + +

(2.30)

where:

K

_Qi = investment expenditures for the equipment to compensate the reactive power

∆ Q p

Q = the expanded reproduction installment, including fixed operating costs

KPA = cost of power losses and energy losses in the compensation equipment

K

ZQ = costs of undependability caused by the compensation equipment unreliability

K

pQ = other costs associated with installation of the compensation equipment (positive or negative) concerning e.g. consequences of changes in the PQ parameters

K

RQ = costs of restoration of the compensation equipment resulting from the voltage unbalance effects

The costs

K

_Qiare essentially dependent on the compensation equipment, i.e. ,

K

_Qi

= f ( ∆ Q )

. Under the voltage unbalance conditions, the reactive power of a capacitor bank (Q_K) can be larger or smaller than its rated power (

Q

_KN) or the power under the balanced supply voltage conditions (

Q

_Ks).

The value of

∆ Q

_K

= Q

_KN

− Q

_Ks can therefore be positive or negative. Consequently, the costs

K

_Q can also be positive (additional loss) or negative (an extra profit).

For a synchronous machine operated under 2% voltage unbalance and the system p.u. negative-sequence reactance equal 0.24, the negative-sequence symmetrical component of machine currents is 8% [69], which for some types of machines (turbogenerators) is intolerable. This problem occurs particularly in industrial cogeneration plants with unbalanced load. In such cases, it is necessary to reduce the reactive power generated by synchronous machines. With voltage unbalance exceeding 3% at the terminals of a synchronous motor, both the motor current and the generated reactive power shall be reduced. Under voltage unbalance equal to 2%, the reduction of reactive power generated by a synchronous motor is 5 to 23% [69].

2.3.1.5. Costs of Replenishment of Light Sources

Voltage unbalance in lighting installations causes the voltage rise in one or sometimes in two phases. It results in shorter in-service time of lamps, increased active power demand, and in the case of discharge lamps, increased reactive power. A reduced voltage (in one or two phases) results in reduction of the luminous flux, reduction of power, and losses in lighting installation.

Total luminous flux (

Φ

_ns) of all light sources, supplied from different phases under the voltage unbalance conditions, may differ or differ insignificantly from the luminous flux (

Φ

_s) in the case of balanced voltages. If the flux under unbalanced conditions is lower, additional light sources with power

0

P

d

∆

and luminous flux output

∆Φ = Φ

_S

− Φ

_ns shall be installed in order to provide a luminous flux

required by standards. The annual discounted costs associated with this installation, i.e., annual costs of additional light sources, are:

(

_d

) ( )

_{d d i inne}

p

d

f P f p K K

K

₀

= ∆

₀

=

_Φ

∆Φ =

_{0 0}

+

(2.31)

Where:

K

_d0i = the cost of installation of additional light sources

0

p

d = coefficient of expanded reproduction (operating costs taken into account)

K

inne = other costs components (the cost of power and electric energy, the cost of power and electric energy losses, etc.)

The costs of replacement of light sources (

K

_w0)³ are the cost incurred in connection with premature replacement of incandescent or fluorescent lamps, or even entire luminaries, due to shortened in-service time. The number of light sources (lamps) to be prematurely replaced (

∆ L

) is associated with the considered facility and depends essentially on the form of voltage unbalance.

The annual discounted costs of replacement of lamps can be calculated in a similar way as costs of compensation equipment (capacitor bank) restoration because of their premature wear-out due to voltage unbalance, i.e.:

i w w s

w

p K

T

K T

_{0 0}

0 0 0

In document Estudio correlacional de variable que afectan la sostenibilidad en el transporte marítimo (página 30-37)