Voltage stability and thermal limit: Constraints on the maximum loading of electrical energy distribution feeders

R.B. Prada L.J.Souza

Indexing terms: Voltuge stat: ility, Voltage collupse, Loading constraints, Distribution systems

Abstract: The loading condition of distribution systems is assessed with respect to voltage stability as well as to thermal limit. Real networks of a diijtribution utility of Brazil are used in the tests performed. Results show that the maximum loading may be limited by voltage stability rather than by thermal limit. The paper shows the necessity of considering voltage stability as a new constraint on the operational loading of electrical energy distribution feeders, including the connection of new consumers. The voltage stability constraint should be considered in studies concerning distribution systems expansion. The new planning constraint may avoid the building, of new lines and transformers which could never be fully used in systems already stressed by voltage stability.

1 Introduction

The voltage instability phenomenon is not new to power system practising engineers and researchers. The phenomenon was well recognised in radial distribution systems and explained in many text books [l]. It seems, however, that the evolution of such systems in both engineering and analytical techniques has put the phe- nomenon aside for years. Where planners and opera- tors are aware of the potential for voltage stability, one or more of several analytical techniques are usually used to assess the risl~ of voltage instability and margin between the postcontingency operating state and the point of collapse [l].

Voltage collapse and abnormally high or low volt- ages have been observed with even greater frequency and severity on distribution systems. A recent (June 1997) major blackout in the S/SE Brazilian system was ascribed to a voltage. instability problem in a distribu- tion network that was widespread to the corresponding transmission system, leading to the failure and trip-off 0 IEE, 1998 IEE Proceedings online no. 19982186 Paper first received 29th October 1997 and in revised form 1st May 1998 R.B. Prada is with the Department of Electrical Engineering, Catholic University of Rio de Janero, Rua Marques de S%o Vicente 225 Gavea, 22450 150 Rio de Janeiro, €U, Brazil L.J. Souza is with the Dcpartment of Electrical Engineering, Federal University of MaranhBo, AV. dos Portugeses s/n, Campus do Bacanga, 65080 040 SBo Luis, MA, B,razil

IEE Proc.-Gener. Tuansm. DiJtrib., Vol. 145, No. 5. September 1558

of a major DC link. Voltage problems are associated with increased loading of the transmission lines, insuffi- cient local reactive supply, and the transmission of power across long geographical distances [2].

This paper attempts to bring back attention to the voltage stability problem in distribution systems through the analysis of real-life networks.

2 Voltage regions of operation and effects of reactive control

The normal operation of electric power systems requires that the voltage magnitude is kept inside a range of about *5% of the nominal value. In certain circumstances an excessive voltage decrease may occur. To restore voltage to the normal range of operation requires use of controls associated with reactive power, such as voltage settings in generators and condensers, taps in on-load tap-changing transformers, shunt capacitor and reactor switching. It may happen, how- ever, that the available controls are not only insuffi- cient to correct the abnormal voltages, but they may even cause the voltage level to deteriorate further.

To demonstrate this situation, a previous paper [3] has made use of a two-bus system consisting of an activeheactive power load fed by a generator with infi- nite capacity through a transmission 7c equivalent cir- cuit with no thermal limit. As the active and reactive load increases, for example by keeping the power factor constant, the two existing voltage solutions get closer to each other until they coincide: only one solution exists for that load. Any other increase in the load would lead to no voltage solution. Since the load is assumed to have no restraint in absorbing power, and the generator has no restraint in generating power, the load corresponding to a unique voltage solution is the maximum flow the network can transmit.

Based on the effectiveness of the reactive control actions in dealing with voltage levels, two voltage regions of operation were defined [3] as follows: (i) Region A, the normal region of operation where voltage corrective actions work as expected. For exam- ple, increasing sending-end voltage, installing a capaci- tor at the receiving end, tapping up an intermediate transformer; all cause an increase in the receiving-end voltage. (ii) Region B, the abnormal region of operation where voltage corrective actions do not work. On the con- trary, they worsen the voltage situation at the receiving end if loads are modelled as constant power.

SI3

A boundary between the regions was detected: it corre- sponds to the unique voltage solution, i.e. to the maxi- mum load the network can supply. The maximum activeheactive power flow the network can transmit is usually associated with voltage stability. That is because instability in the voltage control devices may be experienced when the system is operating near the maximum flow. The maximum transfer of active power, i.e. the usual static stability limit, is a special case of the maximum transfer of activeheactive power

The maximum flow the network can transmit can occur at almost any value of load voltage magnitude. There is a possibility that the magnitude can be inside the normal range of operation. For example, for a shunt capacitive compensated network, the critical volt- age magnitude at the boundary between the two regions may well be inside or above normal voltage rat- ing. That is the most interesting and worrying situa- tion, since a critical voltage magnitude well below the normal range of operation does not normally deserve attention.

Capacitive shunt compensation increases the network transfer capacity, the same being true for increases in generator voltage magnitude and increases in trans- former taps. Therefore it is more difficult to reach the maximum limit. On the other hand, the limit can be reached at a higher value of load voltage magnitude, perhaps inside the normal range of operation.

3 Voltage collapse situations

Based on the previous results it is possible to state three situations leading to voltage collapse: (i) The system is operating in Region A. If the voltage magnitude is too low, reactive controls can be applied to rectify the situation. If reactive sources are depleted, the voltage level can become too low for normal opera- tion. In that case undervoltage relays may come into operation to protect the equipment. The automatic control action may trigger a cascade of events leading to the complete shutdown of the system. This undesira- ble situation is the well-known problem of excessive voltage drop. (ii) The system is operating in Region B where the volt- age level cannot be raised in the usual way for constant power loads. If the system is operating in such a region, actions to raise the voltage magnitude result in even greater voltage deterioration. Available reactive source controls would try unsuccessfully to raise the voltage level, until the voltage is too low and under- voltage relays come into action. The complete shut- down of the system is a possible consequence. The situation is the problem of voltage control stability. The nomenclature is not universally accepted, and a distinction between the two different phenomena can- not be found in the literature. (iii) The system is operating in Region A. There is enough reactive reserve and the voltage magnitude is kept inside the normal range of operation. As the load increases, the operating point approaches the bound- ary. Near the boundary, i.e. the maximum activehex- tive power the network can deliver to the load, sensitivities are of unusual values and small load increase implies very large voltage deviation. A very low voltage leads the system to collapse. In this paper the main interest is on this voltage collapse mechanism.

141.

574

4 Analytical tool

The objective is to know whether a voltage solution for a given load is in Region A, Region B or on or near the boundary. This is mathematically determined for a two-bus system: it is only necessary to check the deter- minant sign and magnitude of jacobian matrix [J] with dimension 2 x 2 of the load flow calculations [3]. This result is extended to a multinode system [5]. The line- arised system of load flow equations is

Rearranging terms by putting the equations related to the bus under analysis to the bottom of the system

where submatrices A, B, C and D originate from a par- tition of full jacobian matrix. Assuming incremental load (or generation) variation A P and AQ only on bus 4, i.e. A P ‘ = AQ’ = 0 the system is reduced to

where

(3)

(4) has dimension 2 x 2. Therefore critical conditions at 1 bus can be verified by the determinant sign and magni- tude of [D’] with dimension 2 x 2 [5]. A positive sign indicates Region A of operation, a negative sign indi- cates Region B, and the magnitude indicates the ’dis- tance’ to the boundary. If only incremental reactive power variation is possible or desirable, e.g. the case of a bus with only reactive compensation, the critical con- dition is to be verified by det[D’] divided by dl1, ele- ment of [D‘] [6].

According to Schur’s formula, if det[D’] = 0 then det[J] = 0. This condition has been extensively used as the characteristic of voltage collapse, e.g. [7] and was first proposed by Venikov [8]. Now a different phrasing is offered: “The voltage stability critical condition det[J] = 0 means that in at least one bus of the system the load cannot increase owing to the risk of large volt- age deviation leading to voltage collapse”. However, det[J] is not an adequate index to be used in large sys- tems, unless voltage stability problems are widespread throughout the network. As voltage problems are usu- ally confined to an area, the global index det[J] is not able to point out the problem numerically. Therefore the local bus index det[D‘] is recommended. Similar indices which are neither global nor local are the small eigenvalues [7] and the minimum singular value [9], both of Jacobian matrix [J].

5 networks

The loading condition of distribution networks was assessed with respect to voltage stability. Several real- life networks of a distribution utility of Brazil were used in the tests performed. In this paper the network shown on Fig. 1 was analysed. All the necessary data for power flow studies are presented in the Appendix (Section 9). These data correspond to the maximum

Voltage stability assessment of distribution

IEE Proc.-Gener. Trnnsm. Distrib.. Vol. 145, No. 5, September 1998

6 V 34.5 kV 13.8 kV 1) 69 kV 13.8 kV 34.5 kV

14km

(5) 0.6 MVAr

4 alim. 2x1 .O MVA

0.3 MVAr

6! 164.80km - 336.4 AA

34.5 kv 34.5 kV 13.8 kV 13.8 kV ,V

336.4 AA 66.17km

516.25 MVA

13.8 kV 2~2.5 MVA -(It 2 alim.

34.5 kV 1.0MVA (17) 2 13.8 kV

1.8 MVAr

Povoado Formosa 70km (18)

s I-+ Celtins (Araguaina)

SE Campestre / 34.5kV 1 IOCAA vi (20)

Tablie 1: Operating point and nodal voltage stability indexes

Branch Power flow MVA I Nodal voltage and stability index

From To Type Actual Limit

1 2 TL 4.13 51.63 2 3 TR 4.13 5.00 4 3 TR 2.70" 2.50 4 5 TL 2.06 12.13 5 6 TL 1.75 12.13 6 7 TR 0.82 1 .oo 6 7 TR 0.82 1.00 1 8 TL 6.59 51.63 8 10 TL 4.88 51.63 10 13 TR 4.70 6.25 11 13 TR 1.43 2.50 11 13 TR 1.43 2.50 11 12 TL 2.96 12.13 12 14 TL 2.50 12.13 14 15 TR 0.63 1.00 14 16 TL 1.77 12.13 16 17 TR 1.63 2.50 8 9 TR 0.70 2.50 19 18 TL 0.00 12.13 4 20 TL 0.73 12.13 18 9 TR 0.28 1.00

Bus Type

1 generation 2 3 load/capac. 4 capacitor 5 load 6 7 load/capac. 8

9 load 10 11 12 load 13 load/capac.

15 load/capac. 16 17 loadlcapac. 18 load 19 20 load

14

Voltage pu

1.029 1.029 1.024 1.020 0.999 0.981 0.996 1.025

1.024 1.015 1.023 0.975 1.015 0.936 0.949 0.874 0.885 1.038 1.043 1.011

det [D'l

0.50E+00

0.47E+00

0.13E-01

0.58E-01

0.1 7E-0 1

0.69E-01

0.36E-02

0.99E-03 0.98E-02

0.4 1 E-0 1

ITR transformer; TL transmission line; * violated thermal limit

load of a typical midweek day. Following the utility these branches. There is a transformer connected policies, active load is modelled as 40% constant power between buses 4 and 3 with an overload of 8%. Table 1 and 60% constant impedance, whereas reactive load is also lists the bus voltage magnitude. The voltage mag- modelled as 100% as constant impedance. nitude on loads varies from 1.038 to 0.885pu, although

Table 1 lists the NWA power flow in all lines and only on bus 17 voltage magnitude is below 0.95pu, the transformers as well as the maximum MVA loading on usual minimum value for normal operation. (The mini-

IEE Proc -Gener Transm Dist,*ib, Val 145, No 5, September 1998 515

mum voltage magnitude limit is often ignored in distri- bution networks attending rural areas.)

Nodal voltage stability indexes varying from 0.5 to 0.001 are also shown in Table 1. Experience with other distribution systems guarantees that these values are small (near zero) therefore indicating poor voltage sta- bility conditions. BLIS 17 is the bus with the worst volt- age stability index as well as the lowest voltage magnitude and is located at the end of a feeder.

The test performed consisted of installing a new transformer unit in parallel with the existing over- loaded one and the simulation of load growth. The new transformer unit alleviates the thermal constraint on branch 4-3 allowing larger power flow. It was verified that critical voltage stability condition, det[D’] = 0 at bus 17, was reached after only 4% increase on total load. Network operation with a load larger than that may be extremely unsatisfactory. For example, addi- tional impedance loads would cause smaller power con- sumption and larger losses. Automatic voltage regulation makes the load’s behaviour as a constant power model, and additional loads would turn unstable the regulation mechanism. Voltage collapse is then a real danger. At the operating point corresponding to the maximum loading, the flow through the two trans- former units was only 12% above the rating value of one of them. In other words, maximum flow thermal capability was increased 100% with the new trans- former unit but only 12% was used owing to the volt- age stability criteria.

The results show that the maximum loading is lim- ited by voltage stability rather than by thermal limit. In the test-case money was spent on a new transformer unit which could not be fully used. Besides buying the new unit it would be necessary to take appropriate measures to improve the voltage stability conditions. One such measure would be the installation of a larger capacitor bank at bus 17 to increase transfer capacity of power flow to that bus. The voltage stability con- straint would be alleviated. With no thermal limit nor voltage stability constraints in the network, load growth is allowed.

The results show that the operational loading of dis- tribution feeders as well as the coiinection of new loads should be done considering voltage stability as a new constraint. Voltage stability should also be considered on the studies concerning distribution systems expan- sion. It would avoid the building of new branches which could never be fully used in systems already stressed by voltage stability.

The minimum voltage magnitude limit is often not taken into account as a constraint on the maximum loading of feeders distributing energy in rural areas. The overload capacity of transformers and the mini- mum value of det[D’] for secure operation are two important points which were not addressed. However, their inclusion in the analysis is straightforward once the utility engineers establish the appropriate values.

6 Conclusions

The loading condition of the distribution systems was assessed with respect to voltage stability. For this pur- pose a local bus voltage stability assessment analytical tool was used. It is based on a straightforward physical characterisation of the voltage stability phenomena. The methodology was able to detect real voltage stabil- ity problems well known to the operators of the Brazil-

516

ian system, which substantiates its applicability and adequacy.

Several real networks of a distribution utility system of Brazil were used in the tests performed. The paper has shown a test case with a network with low voltage magnitude at the feeder end. Nodal voltage stability indexes indicated poor voltage stability conditions.

The test performed consisted of installing a new transformer unit in parallel with the existing over- loaded one and the simulation of load growth. It was verified that critical voltage stability condition was reached after a small increase on total load. Network operation with a load larger than that may be extremely unsatisfactory owing to the real danger of voltage collapse. The maximum flow thermal capability was increased 100% with the new transformer unit but only 12% was used owing to the voltage stability crite- ria.

Results have shown that the maximum loading may be limited by voltage stability rather than by thermal limit. Therefore this paper has shown the necessity of considering voltage stability as a new constraint on the operational loading of electrical energy distribution feeders, including the connection of new consumers. Furthermore, the voltage stability constraint should be considered in studies concerning distribution systems expansion. The new planning constraint may avoid the building of new lines and transformers which could never be fully used in systems already stressed by volt- age stability.

7 Acknowledgments

The authors are grateful to MaranhZo state research agency FAPEMA and the Brazilian research agency CNPq.

8

1

2

3

4

5

6

7

8

9

References

MANSOUR, Y. (Ed.): ‘Voltage stability of power systems: con- cepts, analytical tools, and industry experience’. IEEE workgroup on voltage stability, system dynamic performance subcommittee, 90TH0358-2. 1990 SCHLEUTER, R.A., COSTI, A.G., SEKERKE, J.E. and FORGEY, H.L.: ‘Voltage stability and security assessment’. EPRI EL-5967, Project 1999-8, Final report, August 1988 PRADA, R.B., CORY, B.J., and NAVARRO-PEREZ, R.: ‘Assessment of steady state voltage collapse critical conditions’. Presented at the 10th Power systems computation conference, PSCC 90, Graz, Austria, August 1990 PRADA, R.B., and NAVARRO-PEREZ, R.: ‘Voltage collapse or steady state stability limit’. Presented at the international workshop on Bulk power system voltage phenomena: stability and security, Maryland, USA, August 1991 PRADA, R.B., VIEIRA FILHO, X., GOMES, P., and DOS SANTOS, M.G.: ‘Voltage stability system critical area identifica- tion based on the existence of maximum power flow transmis- sion’. Presented at the 11th conference 011 Power systems computation, PSCC 93, Avignon, France, August 1993 PRADA, R.B., and DOS SANTOS, J.O.R.: ‘Fast evaluation of local bus voltage stability considering contingency analysis’. Pre- sented at the IFAC symposium on Power plants control andpowev systems, Canclin, Mexico, December 1995 GAO, B., MORISON, G.K., and KUNDUR, P.: ‘Voltage stabil- itv evaluation usine modal analvsis’. IEEE Trans.. 1992. PWRS- , , I ,

7,’(4), pp. 1529-15z2 VENIKOV. V.A.. STROEV. V.A.. IDELCHICK. V.I.. and TARASOV; V.I.:’ ‘Estimation ’of electrical power system steady- state stability in load flow calculations’, IEEE Trans., 1975,

LOF. P.-A.. SMED. T.. ANDERSON. G.. and HILL. D.J.: PWRS-94, (3), pp. 1034-1041

‘Fas( calcufation of a voltage stability index’,’IEEE Trans.,’ 1992, PWRS-7, (l), pp. 54-64

IEE Proc.-Gener. Transn?. Distrib., Vol. 145, NO. 5, September 1998

9 Appendix

Table 2: Distribution network bus data

Bus Voltage Load Shunt

Magnitude/ M W MVAr MVAr angle Number Typehame

1 2 Franco-ELN69 1.029 / O 0.00 0.00 0.0 2 0 Franco-CEM69

3 0 Franco13

4 0 Franco34

5 0 E.Rural 34

6 0 Estreito 34

7 0 Estreito 13

8 0 F.Noyuei 69

9 0 F.Noguei 10 0 Balsas69 11 0 Balsas34

12 0 R.Balsas 34

13 0 Balsas13

14 0 Riachao34

15 0 Balsas 13

16 0 Itapecur34

17 0 ltapecur 13

18 0 Formosa 34

19 0 Fort(VZ) 34

20 0 Campestre 34

l.OOO/O l.OOO/O

l . O O O / O

l . O O O / O

1.000 / 0

1.000/0

l . O O O / O

1.000/0

1.000/0 l.OOO/O 1.000 / 0

1.000/0

1.000/0

1.000/0

l . O O O / O

1 .ooo / 0 1 .ooo / 0

l . O O O / O

1.000/0

0.00 0.00 0.0

1.40 0.60 0.6

0.00 0.00 0.6

0.28 0.12 0.0

0.00 0.00 0.0

1.68 0.71 0.3

0.00 0.00 0.0

0.50 0.24 0.0

0.00 0.00 0.0 0.00 0.00 0.0

0.27 0.15 0.0

1.80 0.97 0.6

0.00 0.00 0.0

0.62 0.33 0.6

0.00 0.00 0.0

1.80 0.97 1.5 0.22 0.10 0.0

0.00 0.00 0.0

0.64 0.27 0.0

Type 2: swing bus /Type 0: PQ bus

Table 3: Distribution network branch parameters

Branch parameters

From

1

2

4

4

5

6

6

1

8

10

11

11

11

12

14

14

16

8

19

4

18

- To

2

3

3

5

6

7

7

8

10

13

13

13

12

14

15

16

17

9

18

20

9

-

Tap Max xL('o) (MVAr) (pu) flow

VC R (%) ~

0.04

0

0

87.75

87.75

0

0

65.84

26.43

0

0

0

204.74

204.74

0

420.35

0

0

409.47

116.99

0

0.10

145.40

244.00

65.39

65.39

507.00

507.00

158.33

63.57

129.12

280.40

280.40

152.57

152.57

534.00

313.24

278.80

260.80

305.13

87.18

615.00

0.002

0

0

0.068

0.068

0

0

2.825

1.134

0

0

0

0.158

0.158

0

0.323

0

0

0.315

0.090

0

51.6

1 5.0

1 2.5

12.1

12.1

0.974 1.0

0.974 1.0

51.6

51.6

1 6.25

1 2.5

1 2.5

12.1

12.1

1 1 .o 12.1

1 2.5

1 2.5

12.1

12.1

1 1 .o R: resistance; XL: reactance; YC: total susceptance

IEE Proc.-Gener. Transm. Disrrib., Vol. 145, No. 5, September 1998