FUNDAMENTALS OF POWER SYSTEMS

1 FUNDAMENTALS OF POWER SYSTEMS

1 Chapter FUNDAMENTALS OF POWER SYSTEMS INTRODUCTION The three baic element of electrical engineering are reitor, inductor and capacitor. The reitor conume ohmic or diipative energy wherea the inductor and capacitor tore in the poitive half cycle and give away in the negative half cycle of upply the magnetic field and electric field energie repectively. The ohmic form of energy i diipated into heat whenever a current flow in a reitive medium. If I i the current flowing for a period of t econd through a reitance of R ohm, the heat diipated will be I Rt watt ec. In cae of an inductor the energy i tored in the form of magnetic field. For a coil of L henrie and a current of I ampere flowing, the energy tored i given by 1 LI. The energy i tored between the metallic plate of the capacitor in the form of electric field and i given by 1 C where C i the capacitance and i the voltage acro the plate. We hall tart with power tranmiion uing 1-φ circuit and aume in all our analyi that the ource i a perfect inuoidal with fundamental frequency component only. 1.1 SINGLE-PHASE TRANSMISSION Let u conider an inductive circuit and let the intantaneou voltage be v m in ωt (1.1) Then the current will be i I m in (ωt φ) where φ i the angle by which the current lag the voltage (Fig. 1.1). The intantaneou power i given by p vi m in ωt. I m in (ωt φ) m I m in ωt in (ωt φ) (1.) m I m [co φ co (ωt φ)]

FUNDAMENTALS OF POWER SYSTEMS 3 The value of p i poitive when both v and i are either poitive or negative and repreent the rate at which the energy i being conumed by the load. In thi cae the current flow in the direction of voltage drop. On the other hand, power i negative when the current flow in the direction of voltage rie which mean that the energy i being tranferred from the load into the network to which it i connected. If the circuit i purely reactive the voltage and current will be 90 out of phae and hence the power will have equal poitive and negative half cycle and the average value will be zero. From equation (1.) the power pulate around the average power at double the upply frequency. CHAPTER 1 Fig. 1.1 oltage, current and power in ingle phae circuit. Equation (1.) can be rewritten a p I co φ (1 co ωt) I in φ in ωt (1.3) I II We have decompoed the intantaneou power into two component (Fig. 1.). p I II p I co I in Fig. 1. Active, reactive and total power in a ingle phae circuit. (i) The component p marked I pulate around the ame average power I co φ but never goe negative a the factor (1 co ωt) can at the mot become zero but it will never go negative. We define thi average power a the real power P which phyically mean the ueful power being tranmitted.

4 ELECTRICAL POWER SYSTEMS (ii) The component marked II contain the term in φ which i negative for capacitive circuit and i poitive for inductive circuit. Thi component pulate and ha zero a it average value. Thi component i known a reactive power a it travel back and forth on the line without doing any ueful work. Equation (1.3) i rewritten a p P(1 co ωt) Q in ωt (1.4) Both P and Q have the ame dimenion of watt but to emphaie the fact that Q repreent a nonactive power, it i meaured in term of voltampere reactive i.e., Ar. The term Q require more attention becaue of the intereting property of in φ which i ve for capacitive circuit and i +ve for inductive circuit. Thi mean a capacitor i a generator of poitive reactive Ar, a concept which i uually adopted by power ytem engineer. So it i better to conider a capacitor upplying a lagging current rather than taking a leading current (Fig. 1.3). + + C C I lead by 90 I lag by 90 Fig. 1.3 -I relation in a capacitor. Conider a circuit i which an inductive load i hunted by a capacitor. If Q i the total reactive power requirement of the load and Q i the reactive power that the capacitor can generate, the net reactive power to be tranmitted over the line will be (Q Q ). Thi i the baic concept of ynchronou phae modifier for controlling the voltage of the ytem. The phae modifier control the flow of reactive power by uitable excitation and hence the voltage i controlled. The phae modifier i baically a ynchronou machine working a a capacitor when overexcited and a an inductor when underexcited. It i intereting to conider the cae when a capacitor and an inductor of the ame reactive power requirement are connected in parallel (Fig. 1.4). I C I C I L I L Fig. 1.4 Power flow in L-C circuit.

FUNDAMENTALS OF POWER SYSTEMS 5 The current I L and I C are equal in magnitude and, therefore, the power requirement i ame. The line power will, therefore, be zero. Phyically thi mean that the energy travel back and forth between the capacitor and the inductor. In one half cycle at a particular moment the capacitor i fully charged and the coil ha no energy tored. Half a voltage cycle later the coil tore maximum energy and the capacitor i fully dicharged. The following example illutrate the relationhip between the reactive power and the electric field energy tored by the capacitor. Conider an RC circuit (Fig. 1.5). From Fig. 1.5 ωc I R + (/ 1ωC) R ω C + 1 and if voltage i taken a reference i.e., v m in ωt the current i I m in (ωt + φ) m C i ω. in (ωt + φ) (1.6) R ω C + 1 I/ ωc 1 where in φ (1.7) I R + ( I/ ωc) R ω C + 1 Now reactive power Q I in φ (1.8) Subtituting for I and in φ, we have Q. (1.5) ωc 1 ωc. (1.9) R ω C + 1 R ω C + 1 R ω C + 1 ωc Reactive power R ω C + 1 Now thi can be related with the electric energy tored by the capacitor. The energy tored by the capacitor. W 1 Cv (1.10) Now v 1 1 mwc co( w t + f) mco ( w t + f) Ú idt - - C C w + w R C 1 R w C + 1 I R in t m Fig. 1.5 Relationhip between electric field energy and reactive power. C (1.11) CHAPTER 1 W dw dt 1 m co ( w t + f) C co ( w t + f) C. R w C + 1 R w C + 1. co (ωt + φ). in (ωt + φ). ωc R ω C + 1 (1.1) ωc. in (ωt + φ) R ω C + 1 Q in (ωt + φ) (1.13)

6 ELECTRICAL POWER SYSTEMS From thi it i clear that the rate of change of electric field energy i a harmonically varying quantity with a frequency double the upply frequency and ha a peak value equal to Q. In an R-L circuit the magnetic field energy and reactive power in a coil are imilarly related. 1. THE 3-PHASE TRANSMISSION Auming that the ytem i balanced which mean that the three-phae voltage and current are balanced. Thee quantitie can be expreed mathematically a follow: a m in ωt b m in (ωt 10 ) c m in (ωt + 10 ) (1.14) i a I m in (ωt φ) i b I m in (ωt φ 10 ) i c I m in (ωt φ + 10 ) The total power tranmitted equal the um of the individual power in each phae. p a i a + b i b + c i c m in ωt I m in (ωt φ) + m in (ωt 10 ) I m in (ωt 10 φ) + m I m in (ωt + 10 ) in (ωt + 10 φ) I[ in ωt in (ωt φ) + in (ωt 10 ) in (ωt 10 φ) + in (ωt + 10 ) in (ωt + 10 φ)] I[co φ co (ωt φ) + co φ co (ωt 40 φ) + co φ co (ωt + 40 φ)] 3I co φ (1.15) Thi how that the total intantaneou 3-phae power i contant and i equal to three time the real power per phae i.e., p 3P, where P i the power per phae. In cae of ingle phae tranmiion we noted that the intantaneou power expreion contained both the real and reactive power expreion but here in cae of 3-phae we find that the intantaneou power i contant. Thi doe not mean that the reactive power i of no importance in a 3-phae ytem. For a 3-phae ytem the um of three current at any intant i zero, thi doe not mean that the current in each phae i zero. Similarly, even though the um of reactive power intantaneouly in 3-phae ytem i zero but in each phae it doe exit and i equal to I in φ and, therefore, for 3-φ the reactive power i equal to Q 3φ 3I in φ 3Q, where Q i the reactive power in each phae. It i to be noted here that the term Q 3φ make a little phyical ene a would the concept of three phae current I 3φ 3I. Neverthele the reactive power in a 3-phae ytem i expreed a Q 3φ. Thi i done to maintain ymmetry between the active and reactive power.

FUNDAMENTALS OF POWER SYSTEMS 7 1.3 COMPLEX POWER Conider a ingle phae network and let e jα and I I e jβ (1.16) where α and β are the angle that and I ubtend with repect to ome reference axi. We calculate the real and reactive power by finding the product of with the conjugate of I i.e. S I* e jα I e jβ I e j(α β) I co (α β) + j I in (α β) (1.17) Here the angle (α β) i the phae difference between the phaor and I and i normally denoted by φ S I co φ + j I in φ P + jq (1.18) CHAPTER 1 The quantity S i called the complex power. The magnitude of S P + Q i termed a the apparent power and it unit i volt-ampere and the larger unit are ka or MA. The practical ignificance of apparent power i a a rating unit of generator and tranformer, a the apparent power rating i a direct indication of heating of machine which determine the rating of the machine. It i to be noted that Q i poitive when (α β) i poitive i.e. when lead I i.e. the load i inductive and Q i ve when lag I i.e. the load i capacitive. Thi agree with the normal convention adopted in power ytem i.e. taking Q due to an inductive load a +ve and Q due to a capacitive load a negative. Therefore, to obtain proper ign for reactive power it i neceary to find out I* rather than *I which would revere the ign for Q a *I e jα I e jβ I e j(α β) I co (α β) j I in (α β) I co φ j I in φ P jq (1.19) 1.4 LOAD CHARACTERISTICS In an electric power ytem it i difficult to predict the load variation accurately. The load device may vary from a few watt night lamp to multi-megawatt induction motor. The following category of load are preent in a ytem: (i) Motor device 70% (ii) Heating and lighting equipment 5% (iii) Electronic device 5% The heating load maintain contant reitance with voltage change and hence the power varie with (voltage) wherea lighting load i independent of frequency and power conumed varie a 1.6 rather than.

8 ELECTRICAL POWER SYSTEMS For an impedance load i.e. lumped load P and Q R R + ( πfl) + ( πfl). R. (πfl) (1.0) From thi it i clear that both P and Q increae a the quare of voltage magnitude. Alo with increaing frequency the active power P decreae wherea Q increae. The above equation are of the form P P[f, ] (1.1) Q Q[f, ] Compoite load which form a major part of the ytem load are alo function of voltage and frequency and can in general be written a in equation (1.1). For thi type of load, however, no direct relationhip i available a for impedance load. For a particular compoite load an empirical relation between the load, and voltage and frequency can be obtained. Normally we are concerned with incremental change in P and Q a a function of incremental change in and f. From equation (1.1) ΔP ~ P. + P Δ f. Δf and ΔQ ~ Q. + Q Δ f. Δ f (1.) The four partial derivative can be obtained empirically. However, it i to be remembered that wherea an impedance load P decreae with increaing frequency, a compoite load will increae. Thi i becaue a compoite load motly conit of induction motor which alway will experience increaed load, a frequency or peed increae. The need for enuring a high degree of ervice reliability in the operation of modern electric ytem can hardly be over-emphaized. The upply hould not only be reliable but hould be of good quality i.e., the voltage and frequency hould vary within certain limit, otherwie operation of the ytem at ubnormal frequency and lower voltage will reult in eriou problem epecially in cae of fractional hore-power motor. In cae of refrigerator reduced frequency reult into reduced efficiency and high conumption a the motor draw larger current at reduced power factor. The ytem operation at ubnormal frequency and voltage lead to the lo of revenue to the upplier due to accompanying reduction in load demand. The mot eriou effect of ubnormal frequency and voltage i on the operation of the thermal power tation auxiliarie. The output of the auxiliarie goe down a a reult of which the generation i alo decreaed. Thi may reult in complete hut-down of the plant if corrective meaure like load hedding i not reorted to. Load hedding i done with the help of under-frequency relay which automatically diconnect block of load or ectionalie the tranmiion ytem depending upon the ytem requirement.

FUNDAMENTALS OF POWER SYSTEMS 9 1.5 THE PER UNIT SYSTEM In a large interconnected power ytem with variou voltage level and variou capacity equipment it ha been found quite convenient to work with per unit (p.u.) ytem of quantitie for analyi purpoe rather than in abolute value of quantitie. Sometime per cent value are ued intead of p.u. but it i alway convenient to ue p.u. value. The p.u. value of any quantity i defined a the actual value of the quantity (in any unit) the bae or reference value in the ame unit In electrical engineering the three baic quantitie are voltage, current and impedance. If we chooe any two of them a the bae or reference quantity, the third one automatically will have a bae or reference value depending upon the other two e.g., if and I are the bae voltage and current in a ytem, the bae impedance of the ytem i fixed and i given by CHAPTER 1 Z I The rating of the equipment in a power ytem are given in term of operating voltage and the capacity in ka. Therefore, it i found convenient and ueful to elect voltage and ka a the bae quantitie. Let b be the bae voltage and ka b be the bae kilovoltampere, then actual p.u. b The bae current p.u. current Bae impedance p.u. impedance ka b 1000 b Actual current Bae current Actual current ka b 1000 b Bae voltage Bae current b kab 1000 Actual impedance Bae impedance Z. ka b 1000 Z. MA ( k ) b Thi mean that the p.u. impedance i directly proportional to the bae ka and inverely proportional to quare of bae voltage. Normally the p.u. impedance of variou equipment correponding to it own rating voltage and ka are given and ince we chooe one common bae ka and voltage for the whole ytem, therefore, it i deired to find out the p.u. impedance of the variou equipment correponding to the common bae voltage and ka. If the individual quantitie are Z p.u. old, ka old and old and the common bae quantitie are Z p.u. new, ka new and new, then making ue of the relation above, F Z p.u. new Z p.u. old. ka new old. ka H G I K J (1.3) old b b new

10 ELECTRICAL POWER SYSTEMS Thi i a very important relation ued in power ytem analyi. The p.u. impedance of an equipment correponding to it own rating i given by Z p.u. IZ where Z i the abolute value of the impedance of the equipment. It i een that the p.u. repreentation of the impedance of an equipment i more meaningful than it abolute value e.g., aying that the impedance of a machine i 10 ohm doe not give any idea regarding the ize of the machine. For a large ize machine 10 ohm appear to be quite large wherea for mall machine 10 ohm i very mall. Wherea for equipment of the ame general type the p.u. volt drop and loe are in the ame order regardle of ize. With p.u. ytem there i le chance of making mitake in phae and line voltage, ingle phae or three phae quantitie. Alo the p.u. impedance of the tranformer i ame whether referred on to primary or econdary ide of the tranformer which i not the cae when conidering abolute value of thee impedance. Thi i illutrated below: Let the impedance of the tranformer referred to primary ide be Z p and that on the econdary ide be Z, then Z p Z where p and are the primary and econdary voltage of the tranformer. Now Z p p.u. Z p I p F Z pi I p H G F HG p p I KJ Z. Z p I Z p.u. KJ. p Z I ZI. p From thi it i clear that the p.u. impedance of the tranformer referred to primary ide Z p p.u. i equal to the p.u. impedance of the tranformer referred to the econdary ide Z p.u.. Thi i a great advantage of p.u. ytem of calculation. The bae value in a ytem are elected in uch a way that the p.u. voltage and current in ytem are approximately unity. Sometime the bae ka i choen equal to the um of the rating of the variou equipment on the ytem or equal to the capacity of the larget unit. The different voltage level in a power ytem are due to the preence of tranformer. Therefore, the procedure for electing bae voltage i a follow: A voltage correponding to any part of the ytem could be taken a a bae and the bae voltage in other part of the circuit, eparated from the original part by tranformer i related through the turn ratio of the tranformer. Thi i very important. Say if the bae voltage on primary ide i pb then on the econdary ide of the tranformer the bae voltage will be b pb (N /N p ), where N and N p are the turn of the tranformer on econdary and primary ide repectively.