«MASTER THESIS Reactive Power Compensation Author: Jakub Kępka Supervisor: PhD. Zbigniew Leonowicz Wroclaw University of Technology 2 Reactive Power ...»
Faculty of Electrical Engineering
Reactive Power Compensation
PhD. Zbigniew Leonowicz
Wroclaw University of Technology 2
Reactive Power Compensation
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1. Acknowledgments Wroclaw University of Technology 5 Reactive Power Compensation
2. Introduction Over the last few years, the interest in reactive power compensation has been growing, mainly because of the way in which energy supplier charge a customer for reactive power. Moreover, the energy price is growing, what force the industry plants and individual customers to minimize energy consumption, including reactive power. The aim is to minimize reactive power flow in supplying and distribution systems, eliminate or minimize the charge for reactive power as well as aspire to active energy limitation, in result, reducing fare for electrical energy. In the matter of fact, the energy providers wants they customers to compensate reactive power. Energy suppliers determine the value of tgφ which has to be kept in order to avoid paying for reactive power.
There are few solutions, that allow handle the problem of reactive power compensation. One of them is reactive power compensator basing on power capacitors. This is the most popular compensating device, mainly because of economical reasons, they are relatively cheap comparing with i.e. active filters or compensation by means of electric motors. That is one of the reasons, for which Elektrotim company proposed the master`s thesis topic – “Design of automatic capacitor bank” They want to launch brand new product to their offer, that is Automatic Capacitor bank.
To begin with, the aim of the project was to design automatic detuned capacitor bank for reactive power compensation company with rated power of 200kVar, rated voltage of 400V and detuning factor p=7%. One out of few assumptions was to find supplier who offers low prices and average quality as well as the one, who offers very good quality of the power factor correction equipment in order to meet the requirements of Elektrotim company customers.
The first most important thing before design process get started is to familiarize oneself with standards. Then, knowing what are the requirements regarding capacitor banks in compliance with standards I could proceed to the market survey and compare the elements capacitor bank regarding price, features and quality. Next step is to perform all necessary calculations in order to buy the capacitor bank equipment with proper rating. After that, when all the elements will be ordered I design main and control circuits as well as equipment layout. As a last steps, technical documentation and test program has to be done.
Wroclaw University of Technology 6 Reactive Power Compensation
“Power is a measure of energy per unit time. Power therefore gives the rate of energy consumption or production. The units for power are generally watts (W). For example, the watt rating of an appliance gives the rate at which it uses energy. The total amount of energy consumed by this appliance is the wattage multiplied by the amount of time during which it was used; this energy can be expressed in units of watt-hours (or, more commonly, kilowatt-hours).
The power dissipated by a circuit element—whether an appliance or simply a wire—is given by . The term
the product of its resistance and the square of the current through it:
“dissipated” indicates that the electric energy is being converted to heat. This heat may be part of the appliance’s intended function (as in any electric heating device), or it may be considered a loss (as in the resistive heating of transmission lines); the physical process is the same. Another, more general way of calculating power is as the product of current and voltage: . For a ) to see that the formulas and resistive element, we can apply Ohm’s law ( amount to the same thing:”  (1)
3.2 Complex power “Applying the simple formula becomes more problematic when voltage and current are changing over time, as they do in a.c. systems. In the most concise but
notation, power, current, and voltage are all complex quantities, and the equation for power becomes  (2)
component is to be reversed. All this ought to make very little sense without a more detailed discussion of complex quantities and their representation by phasors. In the interest of developing a conceptual understanding of a.c. power, let us postpone the elegant mathematics and begin by considering power, voltage, and current straightforwardly as real quantities that vary in time. The fundamental and correct way to interpret the statement when I and V vary in time is as a statement of instantaneous conditions. Regardless of all the complexities to be encountered, it is always true that the instantaneous power is equal to the instantaneous product of current and voltage. In other words, at any instant, the power equals the voltage times the current at that instant. This is expressed by writing each variable as a function of time, 
where the t is the same throughout the equation (i.e., the same instant).
“However, instantaneous power as such is usually not very interesting to us. In power systems, we generally need to know about power transmitted or consumed on a time scale much greater than 1/60 of a second. Therefore, we need an expression for power as averaged over entire cycles of alternating current and voltage. Consider first the case of a purely resistive load. Voltage and current are in phase; they are oscillating simultaneously. The average power (the average product of voltage and current) can be obtained by taking the averages (rms values) of each and then multiplying them together . Thus,
Fig. 1 Power as the product of voltage and current, with voltage and current in phase, source  The energy that is being transferred back and forth belongs to the electric or magnetic fields within these loads and generators. Since instantaneous power is sometimes negative, the average power is clearly less than it was in the resistive case. But just how much less? Fortunately, this is very easy to determine: the average power is directly related to the amount of phase shift between voltage and current. Here we skip the mathematical derivation and simply state that the reduction
in average power due to the phase shift is given by the cosine of the angle of the shift:
which is identical because each rms value is related to the maximum value (amplitude) by a √. This equation is true for any kind of load. In the special case where there is only factor of resistance and no phase shift, we have φ=0 and cosφ=0, so there is no need to write down the cos f, and we get the formula from the previous page. In another special case where the load is purely reactive (having no resistance at all), the phase shift would be φ=90 and cosφ=0, meaning that power only oscillates back and forth, but is not dissipated (the average power is zero). The average power corresponds to the power actually transmitted or consumed by the load. It is also called real power, active power or true power, and is measured in watts.
There are other aspects of the transmitted power that we wish to specify. The product of current and voltage, regardless of their phase shift, is called the apparent power, denoted by the symbol S. Its magnitude is given by”  (6) “Although apparent and real power have the same units physically, they are expressed differently to maintain an obvious distinction. Thus, the units of apparent power are called volt-amperes (VA).” 
“Finally, we also specify what we might intuitively think of as the difference between apparent and real power, namely, reactive power. Reactive power is the component of power that oscillates back and forth through the lines, being exchanged between electric and magnetic fields and not getting dissipated. It is denoted by the symbol Q, and its magnitude is given by” 
Q I ·V · sinφ (7)
Again, note how the equation converges for the resistive case where Φ=0 and sin sinΦ=0, as there will be no reactive power at all. Reactive power is measured in VAR (also written Var or
VAr), for volt-ampere reactive. We can represent power as a vector in the complex plane:
namely, and arrow of length S (apparent power) that makes an angle f with the real axis. This is shown in figure below. The angle Φ is the same as the phase difference between voltage and current.” 
The reactive power of inductive and capacitive elements, QR and QL respectively, can be
(8) (9) where and are the maximum value of the energy stored in the magnetic field of the inductive elements of the circuit and electric field of the capacitive elements.  Basing on the law of conservation of energy, the input reactive power in the source – less circuit is equal to algebraic sum of reactive power of the inductive and capacitive elements included in a circuit, that is:  ∑ ∑ (10) Considering any electric circuit, one knows, that the generated reactive energy is equal to the consumed energy. According to this, that most of the loads in the industry are the loads that needs inductive reactive energy to operate. For this reason, the reactive power demand is much more than the generator is able to produce. Therefore, there are a devices that needs to be connected to the system in order to provide an extra source of inductive reactive power or devices which will absorb capacitive power. These type of devices are: capacitor banks, synchronous motors, and power electronic sources of reactive power. The cooperation of compensating devices with linear circuits causes the reactive component of the supplying current to decrease.
 Wroclaw University of Technology 12 Reactive Power Compensation
3.5 Power – time and frequency domain
There are many of theories about power in electrical circuits, but there are two groups that they are can be divided in. One of them is considering frequency domain while the second one is related to time domains. Before these two approaches are explained, there is a one definition that needs to be introduced, namely, distortion power. One needs to deal with distortion power, when the instantaneous values of the voltage and current at the circuit`s terminals do not fulfill the Ohm`s law. The distortion power can be determined basing on the active and apparent power (11) The distortion power of a linear electric circuit is very often referred to us a power of phase shift, because the phase shift between the voltage and current cause this power to appear.
3.5.1 Frequency domain
a) Budenau`s theory Power theory in the frequency domain was published in 1927 by Budenau. He introduced two equations for the power within nonlinear electric circuits supplied by sinusoidal voltage, with
periodical non sinusoidal current waveforms. Budenau defined the reactive power as :
He also introduced the power component, so called distortion power D, describing it as follows:
The load current contains active and reactive component of fundamental harmonic and a sum of higher order harmonics. The sum can be considered as the component of distortion of the current.
The formulas above says, that if the load current value and amplitude of active component are known, one can determine reactive component of load current .
Fryze`s theory The Fryze`s theory from 1932 assumes, that apparent power S of an electric circuit contains two components of the power, active component P and reactive component Q, described by the
following formulas :
(19) (20) (21) According to the theory, everything else than the active power is considered as reactive power, and should be removed from the circuit. Instantaneous current value of the receiver can be shown as a sum of active (ir) and reactive (iq) component of the load current 
4. Power factor and its influence on supplying source The power factor definition is correlated with sinusoidal current circuits. In the linear AC current circuit supplied by the sinusoidal voltage the power factor is referred to as cosφ, where φ is an angle of phase shift between the sinusoidal waveform of supplying voltage and sinusoidal current
waveform, that is  :