Power Factor in Electrical Power Systems with Non-Linear Loads
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1 Power Factor i Electrical Power Systems with No-Liear Loads By: Gozalo Sadoval, ARTECHE / INELAP S.A. de C.V. Abstract. Traditioal methods of Power Factor Correctio typically focus o displacemet power factor ad therefore do ot achieve the total eergy savigs available i facilities havig both liear ad o liear loads. Oly through Total Power Factor Correctio ca the savigs ad power quality be maximized. This paper provides a simplified explaatio of both Power Factor (PF) ad Total Power Factor (TPF) for various electrical systems havig both liear ad o liear loads. Electrical diagrams ad waveforms are combied with represetative methods for aalysis ad help to demostrate the applicatio of mathematical calculatios. The cocept of Total Power Factor is frequetly misuderstood yet it is a importat cosideratio for electrical istallatios which employ power electroics equipmet. A uderstadig of Total Power Factor, alog with the potetial problems, is a importat cosideratio whe applyig o liear loads such as VFD s, DC motor drives, Uiterruptible Power Supplies, ad other power or frequecy coverters icludig SCR cotrollers. The article explais the differece betwee Power Factor ad Total Power Factor, ad the ifluece of o liear, harmoic producig loads o overall power system quality, reliability ad eergy efficiecy. Power factor with liear loads Whe the loads coected to the system are liear ad the voltage is siusoidal, the power factor is calculated with the followig equatio: ( ϕ) pf = cos () Ufortuately, this formula has led to a misuderstadig of the power factor cocept. Power factor is the proportioal relatio of the active power (or workig power) to the apparet power (total power delivered by the utility or cosumed by the load). Usig this defiitio, the power factor must be calculated as: P pf = () S Whe the loads are liear ad the voltage is siusoidal, the active, reactive ad apparet power are calculated with the followig equatios: ( ϕ) P = VI cos (3)
2 ( ϕ) Q = VI si (4) S = VI (5) ad we ca easily see that the power factor is calculated accordig to the equatio (), which is the resultat of the vector relatioship betwee the active, reactive ad apparet power show i figure. Figure Figure A low power factor meas a that a low amout of the total power delivered or cosumed (S) is used as workig power (P) ad a cosiderable amout is reactive power (Q). the purpose of power factor correctio is to reduce the reactive compoet of the total power. This achieves a more efficiet use of the eergy because whe the power factor is improved the workig power is equal (or early equal) to the total power, ad reactive power is zero or egligible. The most commo way to correct power factor is by addig a capacitor bak coected i parallel with the power system. The capacitor bak (Q C ) supplies most of the reactive power eeded by the load ad a small amout is supplied by the utility (Q ), as show i figure. The origial agle ( ) reduced to a smaller value ( ) cos( ϕ ) > ( ϕ ). cos ϕ betwee the apparet ad active power is ϕ ad the power factor is improved because It is very importat to ote that the reductio i the agle obtaied by the power factor improvemet is a result of the vector relatioship betwee the active, reactive ad apparet power, but what we are really doig is reducig the reactive power, cosequetly the apparet power is also reduced ad the power factor is icreased. Power factor with o liear loads ad siusoidal voltage Whe the loads are o liear but the voltage is siusoidal, the curret has harmoics ad the active, reactive ad apparet power should ot be calculated usig the traditioal methods as demostrated by equatios (3), (4) ad (5). This meas that the equatio () ca ot be used to calculate the power factor whe o liear loads are cocered. The active power is the mea (or average) value of the istataeous power, so it ca be calculated as:
3 P = VI cos ( ϕ ) (6) The rms value of curret is a fuctio of the total harmoic curret distortio ad the rms value of the fudametal compoet of curret: I = I + THD (7) I The power factor, for o liear loads, ca be calculated usig equatios (5), (6) ad (7): pf = cos( ϕ ) (8) + THD I There are two terms ivolved i the calculatio of the power factor: cos( ϕ ) ad + THD. The term ( ) pf because it I cos ϕ is called displacemet power factor ( ) depeds of the phase agle betwee the voltage ad the fudametal compoet of the curret, ad it is similar to the power factor calculated with liear loads ad siusoidal voltage. The term + THDI is called distortio power factor ( pf dist ) because it depeds of the curret harmoic distortio. The power factor calculated as the product of the displacemet power factor ad the pf : distortio power factor is kow as Total Power Factor ( ) T Total Power Factor ( ) T pf, where pf T = fpdisp fpdist (9) If the reactive power of the loads icreases, the displacemet agle betwee the voltage ad the fudametal compoet of the curret also icreases ad the total power factor decreases. Likewise, if the total harmoic curret distortio icreases, the total power factor decreases. Oe ca see by equatio (8) that Total Power Factor will always be lower tha the displacemet power factor wheever harmoic distortio is preset. Total power factor correctio ca oly be achieved whe both displacemet power factor ad distortio power factor are corrected. This requires a two step process:. Reduce the displacemet agle betwee voltage ad curret.. Reduce the total harmoic curret distortio. If either of these steps is take without the other the total power factor will be icreased but it may ot be high eough to reach the miimum value required by the utility. Additioally, if oe step is take without the other, the Total Power Factor Correctio ad the correspodig efficiecies will ot be achieved. A vector relatioship, as show i figure 3, ca be obtaied from the active power (P), the fudametal reactive power ( Q ) ad the fudametal apparet power ( S ) F prior to F displacemet power factor improvemet. This relatioship allows us to visualize the effect disp
4 that a capacitor bak ( Q C F ) has o correctig the displacemet power factor (figure 4). Wheever capacitors are used, care should be take to avoid a resoat coditio betwee the capacitor bak ad the mai trasformer. Figure 3 Figure 4 The total power factor ca be improved by decreasig the harmoic curret distortio, which is accomplished by usig a filter istead of a capacitor bak. The capacitive part of the filter improves the displacemet power factor, while the combiatio of the reactor ad the capacitor bak decrease the total harmoic distortio of the curret. A twofold result is achieved, that is improvemet of the distortio power factor ad improvemet of the displacemet power factor. Impedace [ohms] Capacitive behavior Resistive behavior Iductive behavior Harmoic order Zreact Zcap Zfilter Figure 5 The Figure 5 shows the behavior of a harmoic passive filter as well as the behavior of its iductive ad capacitive parts. Below the tuig harmoic the behavior of the filter is like a capacitor ad above the tuig harmoic the behavior is like a iductor. At the tuig harmoic the behavior of the filter is like a resistor. We ca see that at the fudametal frequecy, the filter acts like a capacitor bak because its reactace is basically capacitive, so the filter improves the displacemet power factor.
5 At the tuig harmoic the filter is a very low impedace ad a great amout of curret at the tuig harmoic flows through it, decreasig the total harmoic curret distortio ad improvig the distortio power factor. The improvemet of both power factors (displacemet ad distortio) improves the total power factor. If a capacitor bak were used istead of a filter, the displacemet power factor would have bee improved. If there is o resoace, the distortio power factor does ot chage ad the total power factor icreases oly because the displacemet power factor also icreases, but the total power factor may ot be high eough to reach the miimum value required by the utility. If a resoat coditio is created betwee the capacitor bak ad the mai trasformer, the total harmoic curret distortio icreases so the distortio power factor degrades ad the fial result is a low total power factor eve with a capacitor bak ad a high displacemet power factor. The figure 6 shows the behavior of the total power factor for differet values of displacemet power factor ad total harmoic curret distortio. Total power factor behavior.00 Total power factor Displacemet power factor THD-I=0% THD-I=0% THD-I=0% THD-I=30% THD-I=40% THD-I=50% THD-I=60% THD-I=70% THD-I=80% THD-I=90% THD-I=00% Figure 6 Power factor with o liear loads ad voltage distortio Whe the loads are o liear ad the voltage is distorted the active, reactive ad apparet power ca ot be calculated usig traditioal methods such as the equatios (3), (4) ad (5). The commo equatio () ca ot be used to calculate the power factor either. The active power is the mea (or average) value of the istataeous power. If the phase agles of the voltage harmoics are eglected, the active power ca be calculated as:
6 P = N = ( ) Now, the power factor ca be calculated usig equatio (): V I cos ϕ (0) pf N ( ) VI cos ϕ = = VI () but, the voltage rms value is a fuctio of the total harmoic voltage distortio ad the rms value of the fudametal compoet of voltage: V = V + THD () V Usig equatios (7), () ad (), the power factor ca be calculated as follows: pf P = (3) S + THD I + THD V There are two terms ivolved i the calculatio of the power factor. The term P S is the relatioship betwee the total active power (icludig harmoics) ad the apparet fudametal power. This term should ot be called displacemet power factor because it ivolves the active power caused by the fudametal compoets ad harmoics. The term ( THD I + THD ) V dist, which depeds o the distortio of voltage ad curret. The power factor calculated as the product of the distortio power factor ad the proportio of the total active power to the fudametal pf : + is the distortio power factor ( pf ) apparet power is the total power factor ( ) T The term P S ca be expressed as: where I cos( ) S P S pf V I cos = S P = (4) T pf dist S ( ϕ ) + N = V I S cos ( ϕ ) V ϕ is the displacemet power factor ( pf ) be calculated as follows: disp (5), so the total power factor ca
7 pf T = pf disp + N = VI cos( ϕ ) pf S dist (6) I a similar way to the case of the o liear loads ad siusoidal voltage, if the reactive power of the loads icreases, the displacemet agle betwee the fudametal compoets of voltage ad curret also icreases ad the total power factor decreases. If the distortio of curret ad voltage icreases the distortio power factor decreases ad the total power factor decreases as well. If we wat to improve the power factor the it is ecessary to:. Reduce the displacemet agle betwee the fudametal compoets of voltage ad curret.. Reduce the total harmoic distortio of both the curret ad the voltage. A vector relatioship, show i figure 7, ca be obtaied from the fudametal active ( P ), reactive ( Q ) ad apparet power ( ) F F improvemet. This vector relatioship allows for the use of a capacitor bak ( ) S before the displacemet power factor Q to correct the displacemet power factor (show i figure 8), but agai, a resoat coditio betwee the capacitor bak ad the mai trasformer must be avoided. I a similar way to the case of power factor with o liear loads ad siusoidal voltage, the total power factor ca be improved by decreasig the harmoic curret distortio, ad this ca be achieved by usig a harmoic filter. The capacitive part of the filter improves the displacemet power factor while the combiatio of the reactor ad the capacitor bak decreases the total harmoic distortio of the curret, thus improvig the distortio power factor, as well as the total power factor. C F Figure 7 Figure 8 The figure 9 shows the behavior of the distortio power factor for differet values of total harmoic curret ad voltage distortio.
8 Distortio power factor behavior.0000 Distortio power factor % 3% 6% 9% % 5% 8% % 4% 7% 30% THD-I THD-V=% THD-V=3% THD-V=5% THD-V=7% THD-V=9% THD-V=% THD-V=3% THD-V=5% Figure 9 Example: Cosider a load of 850 kw, 000 kva with 38% total harmoic curret distortio. The displacemet power factor is calculated as = The rms value of curret is: Therefore the total power factor is: I = I = I pf = cos ϕ so I ( ) = = + THD I Coclusio: While traditioal methods of improvig power factor may help to reduce some utility power factor charges, simple power factor correctio does ot assure adequate improvemet of total power factor or achieve the total beefits of performig total power factor correctio. Oly whe measures are take to correct both the displacemet ad distortio power factors ca eergy efficiecy be achieved. The implemetatio of total power factor improvemet ca achieve sigificat cost savigs such as elimiatio of utility power factor charges, reduced eergy demad, reduced electrical equipmet operatig temperatures ad exteder electrical equipmet life.
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