State-Space Model for a Multi-Machine System

Transcription

1 State-Space Moel for a Multi-Machine System These notes parallel section.4 in the text. We are ealing with classically moele machines (IEEE Type.), constant impeance loas, an a network reuce to its internal machine terminals. We have foun that the linearize swing equation is given by: H i i n i i j P Sij ij, i=,,n (eq..6) where it is important to observe that we have assume no amping. Let s consier writing the linearize swing equations for a test system (see example. in text), as shown below. The three swing equations are: H H H P S P S P S P S P S P S

2 One important fact: The stability of a power system epens on relative rotor angles ij NOT absolute rotor angles i. This is because synchronism is a relative phenomenon. That is, it makes no sense to say generator is in synchronism. Rather, we must say with what it is in synchronism, i.e., Generator is in synchronism with generators an, or Generator is in synchronism with the rest of the system. So we nee to efine our states in terms of relative rotor angles. In the above equations, the states (erivatives) are in terms of absolute rotor angle. We can eal with this in the following way. First, multiply through each equation by /H i, resulting in: H H H P S P S P S P S P S P S Now subtract the last equation from each of the other two. When we o this, the erivative terms on the left will be affecte as follows (an this is the main motivation for making this subtraction): ( ) which is /. However, this very convenient substitution of variable will not help if amping is moele (D i i ) in the swing equation AND the amping is nonuniform (we will see below what this means) because then it is not possible to combine the corresponing spee variables. Let s look at this issue by re-writing the above equations with amping.

3 D H D H H H P S P S P S P S D P S P S H H Subtracting the last equation from the other two (in the case of the first equation) affects the n erivative terms on the left accoring to: ( ) (as before) an the rst erivative terms on the left accoring to: D D H H Cannot combine angles if D i /H i iffer We can combine angles if ratios D i /H i are the same for all i, which is the conition for uniform amping. In this case, the first erivative terms become (when we subtract last equation from first): D D D D D H H H H H The implication is that we can ALWAYS reuce the number of states by ue to the ability to use relative angles. But an aitional reuction of states by ue to the ability to use relative spees only occurs in the cases of no amping or of uniform amping. In general, D i /H i ratios will be ifferent, an so moeling nonuniform amping is necessary. When this is the case, we are only able to get the state reuction for relative angles, but not for relative spees. The resulting system appears as below.

4 D D PS PS PS H H H H PS D D S S S S H H H H P P P P An replacing the first erivative terms by spee eviations, we get: D D P P P P H H H H S S S S D D P P P P H H H H S S S S In the particular special case of no amping, we obtain: H H H H P P P P S S S S P P P P S S cognizing that =- an that =-, we may change the sign of the secon term in each equation if we also make this change of variables. This results in: H H H H S S P P P P S S S S P P P P S S eq. (#) So we have erivatives on an, an these are our states. But observe that there are two other variables, namely,. S This means we have 4 variables an only equations. Can we express an in terms of an? Clearly, since =-, if we can o it for one, we can o it for the other. This is one by noting first that S 4

5 (eq. *) We can prove this as follows: Therefore, from eq. (*), we can write that versing the subscript orer of the last term on the right-han-sie, an changing signs, we get: (eq. **) Then, since =-, we get (eq. ***) Substituting eq. (**) an (***) into eq (#), we obtain P S PS PS PS PS H H P S PS PS PS PS H H Gathering terms in each variable, we get two ifferential equations: PS PS PS H H H PS PS H H PS PS H H PS PS PS H H H Denote the coefficients of the above ifferential equations as,,, an, where (assuming the last equation, for bus n, is the one that gets subtracte off in the above steps): n ii PSij PSni j Hi H ij PSnj PSij n Hn Hi ji 5

6 Note these expressions iffer from those given in the text s aenum, pg. 65, because my alpha s here are efine on the lefthan-sie of the equation, whereas the text s equations are efine on the right-han-sie of the equation. Using these alphas, I rewrite the ifferential equations as We can now convert these secon orer linear ifferential equations into first orer linear ifferential equations, in orer to evelop a state-space form. We o this by recognizing that, Then, the above two secon orer ifferential equations become four first orer ifferential equations, as follows: So let s efine the state vector as x Note for our machine system, we have only 4 states ue to the state reuction for relative angles an relative spees. 6

7 Then More explicitly, x Ax Your text on page 6 shows the computation of the alphacoefficients for the 9-bus, -generator system of Fig..9. It is shown that =4.96 =59.54 =.84 =5.46 Then, the state-space equation is: Question is, now, what to o with the above in orer to obtain useful information about the small-isturbance behavior of our system. We will investigate this next. 7