Figure 1 Siemens PSSE Web Site

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1 Stability Analyi of Dynamic Sytem. In the lat few lecture we have een how mall ignal Lalace domain model may be contructed of the dynamic erformance of ower ytem. The tability of uch ytem i a matter of critical imortance. The tability of a ytem may be determined by conidering it reone to a mall change in any inut arameter. A table ytem will quickly ettle into a new teady tate with erha a few raidly decaying ring during the tranition. An untable ytem will tart to ocillate and the magnitude of the ocillation will grow rather than decay. Since any real life ytem i alway ubject to ome minor erturbation an untable ytem doe not need any inut to tart ocillating in ractie but will ring away of it own accord.. One method of determining the tability of a ower ytem i through comuter imulation. We have een how a general uroe imulation tool like Simulink can be ued. There are alo a number of dedicated Power Sytem analyi tool that are widely ued by network oerator for modelling their ytem. PSSE from Siemen i one of the mot well known and i ued by Eirgrid in modelling the Irih National Grid.. Figure Siemen PSSE Web Site 3. While comuter are a owerful aid to analying network and ytem there are alo a number of manual technique that may alo be ued. Thee technique often give a better inight into the actual rocee involved and may hel an engineer get a better undertanding of how a ytem work. Of coure it i not ractical to analye very large ytem uch a a comlete electric ower ytem by hand. Neverthele it i often oible to analye maller ection manually or indeed to imlify large ytem by identifying the dominant element and focuing on thoe.

2 4. In the Coure on Power Electronic we looked at the Nyquit tability analyi technique. Nyquit focued on the oen loo reone of a negative feedback control ytem. In thi coure we will look briefly at another technique the Routh Criterion which can be ued to determine whether or not any given ytem i table by looking at the root of the characteritic equation. The Characteritic Equation If we take a ingle inut ingle outut linear ytem whoe tranfer function i: Out() = N()/D() * In() In() N() D() Out() Then it can be hown that the tability of the ytem i determined by the denominator of the tranfer function: D() The Characteritic Equation of the ytem i obtained by etting the denominator to zero: D() = 0 The olution to thi equation are known a the ole of the ytem. (For reference the olution of N()=0 are known a zeroe but thee are not conidered in tability analyi. ) It can be hown that a long a D() ha only real number coefficient (which it will have becaue the coefficient come from real world arameter) then the ole will either be individual real number or air of comlex conjugate. The real number ole correond to firt order low a filter term (the real number rereent the corner frequencie of thee firt order filter in radian er econd). The comlex conjugate air rereent econd order low a filter which may be damed or undamed (ocillatory) and whoe reonant frequency (in radian er econd) i equal to the magnitude of the comlex number.

3 Examle of ole Take a ytem with the following tranfer function: 4 6 Out( ) = In( ) The characteritic equation i D() = =0 We can olve thi by factoriing: (4)(j)(-j) = 0 Setting each factor to zero give u the ole The ole are =-4, =--j and =-j =-4 rereent a firt order low a term at a corner frequency of 4 rad/ the comlex air -/-j rereent a econd order low a filter at a reonant frequency of.3 rad/ (where.3 = magnitude of -j) Routh Stability Criterion The Routh Stability Criterion (often called the Routh Hurwitz Stability Criterion) tate that any ole with negative oitive real coefficient i table while any ole with a oitive real coefficient i untable. Imaginary Axi (j) Left Half Plane Pole Stable Right Half Plane Pole Untable Real If we lot the ole on a comlex number lane we can ay that any ole in the right half lane (oitive real coefficient) i untable. Comlex ole in the left half lane are table damed ocillation but the further they are from the imaginary axi the more table they are. Note that right half lane air of comlex ole rereent untable reonant frequencie while right half lane real ole indicate DC intability (the outut dc level will continually increae or decreae).

4 Alying the Routh Stability Criterion Given a block diagram of a control ytem with one outut the Routh tabililty criterion can be alied a follow:. Ue block diagram maniulation to convert the ytem into a ingle inut/ ingle outut negative feedback ytem.. The tranfer function of the cloed loo ytem may be extracted now: G( ) Out( ) = In( ) G( ) H ( ) 3. Multily out G/(GH) to get the numerator and denominator a traight olynomial 3. Take the denominator of the tranfer function and et D() = 0 a the characteritic equation. 4. Ue the coefficient of the characteritic equation to contruct a Routh Array (exlained later) 5. By examining the Firt column of the Routh array we can tell the following: Each change of ign in column indicate a root with a oitive real art. So for tability every element in the firt column mut have the ame ign (either oitive or negative). 6. It hould be noted that the Routh table imly anwer the quetion: How many untable right half lane ole are there? It can tell u if a ytem i table or not. It doe not give more information about how cloe a table ytem i to intability. You can however ue algebraic exreion in a Routh table to determine the tability limiting value of certain arameter, for examle a gain term.

5 Uing the Routh Method. A number of excellent exlanation of the Routh - Hurwitz method are available online. For convenience I will rerint a fairly uccinct exlanation from Dr. Jame amman from Wet Michigan Univerity available online from htt:// From: Jame ammen: Coure ME 3600 Control Sytem.

6 Samle Problem The Diagram above how a block diagram of the mall ignal dynamic reone of a ower ytem with one generator connected to an infinite bu. R i the generator droo = 5.0Hz / umw G gt () i the combined turbine / generator tranfer function. You may aume that the magnitude of G gt () i.0 and that the time contant of G gt () are mall enough to be neglected for the uroe of thi analyi /(τ ) i a mall ignal linearied model of the mechanical inertia and daming ytem. In thi cae = 00Hz/uMW and τ =5 A mall ignal model of ower tranmitted from the generator to the network (P E ) ha hown that P E ()=. δ(). E ' () where = 0.8 umw/rad and = 0.7 umw/uv at the current oerating oint. No voltage regulator i fitted. Ue the Routh tability criterion to determine whether or not the ytem i dynamically table at thi oerating oint.

7 Solution (a) The firt te i to ue block diagram maniulation to make thi ytem into a ingle inut ingle outut negative feedback ytem hown below. Taking note of the fact that G gt () =.0 and moving the internal umming junction to the left we can eventually arrive at: For the uroe of tability we can conider thi to be a ingle inut ingle outut ytem with inut = P c ()-. E'() and outut = δ() The cloed loo tranfer function of the loo may now be readily extracted = R GH G gainhi feedback gaingand forward function with Cloed loo tranfer ) ( : τ The characteritic equation i got by making the denominator of the loo tranfer function equal to 0: 0 0 ) ( = = R imlifying R τ τ

8 Before contructing a Routh table we will lug in the arameter from the roblem: R = 5.0Hz / umw = 00Hz/uMW τ =5 = 0.8 umw/rad The characteritic Equation become: = = 0 o a=5, a= and a0=50 Now we can contruct a Routh Table: Col Col Col 3 Row a =5 a 0 =50 Row a = Row 0 b =(-/ a)(a.0- a0.a) =50 Now looking at the element of column : 5, and 50 we can ee that there are no change of ign. Therefore thi ytem ha no ole in the right half lane and i table.

9 A comment on Voltage Regulation The ower ytem analyed in the amle roblem above ha no voltage regulator fitted. Fitting a voltage regulator modifie the control loo: We could treat terminal voltage a an internal variable in order to make the ytem ingle outut and aly Routh but it i oible to make ome general obervation without additional analyi. Firt it may be noted that a voltage regulator ue negative feedback to tabilie terminal voltage and thi might be exected to enhance overall ytem tability. However the deendence of terminal voltage on ower angle δ combined with the deendence δ on excitation voltage E' mean that the voltage regulation loo and the ower angle loo are very cloely couled. Notice that without a voltage regulator any mall increae in δ will tend to increae electrical tranmitted ower Pe lowing the machine down which will lead to a tabiliing reduction in δ. However if a voltage regulator i fitted it will ick u the mall increae in voltage caued by a mall increae in δ and reduce E' to comenate. Reducing E' will reduce tranmitted electrical ower Pe leading to a further detabiliing increae in δ. The relative imortance of thee two effect deend on the relative value of and which deend on the co and ine of the oerating ower angle reectively. We can conclude that careful tuning will be required to enure that a voltage regulator doe not detabilie the ytem.

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