Global optimizasyon yöntemi ile asenkron motor hız kontrolu

Abstract

Bu yüksek lisans tezi çalışmasında, doğru geriliş ara devreli frekans çevirici (DGAFÇ) ile skaler kontrol yöntemine göre sürülen, rotoru sargılı (Bilezikli) asenkron motorun hız kontrolü gerçekleştirilmiştir. Sürücü önünde yer alan kontrolör, dijital olarak, PID yapısında oluşturulmuştur. Çalışmada hedeflenen temel amaç, önceden belirlenen bir "Amaç ölçütü"ne göre, PID kontrolör katsayılarını optimum olarak belirleyip, asenkron motorun istenen (referans) hıza ulaşana kadar, geçici rejimini kontrol etmektedir. Optimum PID kontrolör katsayıları, Yeni Global Arama Yöntemi 'ni kullanan bir program yardımı ile, önceden belirlenen sınırlar içinde, aranmaktadır. Kontrolör parametrelerinin optimum değerlerinin arandığı bölgenin sınırları ise, algoritması Nichols-Zeigler yönteminden yararlanılarak oluşturulmuş bir program yardımı ile belirlenmektedir. Kullanılan "Yeni Global Arama Yöntemi", fiziksel sistemin kontrolü için gerekli verileri, sistem çıkış büyüklüklerinden alarak aramayı gerçekleştirdiğinden, sistemin matematiksel modeli ve transfer fonksiyonundan tamamen bağımsız olarak çalışır. Bu nedenle matematiksel modeli kolayca oluşturulabilen doğrusal (Lineer) sistemlere olduğu gibi, matematiksel modeli ve transfer fonksiyonu kolayca elde edilemeyen, doğrusal olmayan (non-lineer) sistemlere de kolayca uygulanabilir. Yapılan işlemi Sistem çıkış büyüklüklerini ( konum, hız, akım ) girdi olarak bilgisayara aktarıp, tanımlanan Amaç ölçütü'ne göre kontrolör katsayılarını belirleyici aramayı yapmak ve bir sonraki adımda yeni katsayılara göre oluşturulmuş kontrol işaretini, fiziksel sisteme tekrar uygulamak şeklinde özetlenebilir. Çalışmanın sonunda bulunan optimum PID kontrolör parametrelerinin, sistemin geçici rejim davranışını istenilen şekilde kontrol edebildiği gözlenmiş olup, sonuçlar, çeşitli hız referansları için, grafiklerde sunulmuştur. Ayrıca değişik hız referansları ve yüklenme durumları için elde edilen optimum PID kontrolör katsayılarından oluşan tablolar da verilmiştir.

The most commanly used machine in electrical drives is induction machine. Its better reliability, low maintenance cost, cheaper price are the main reason why it is preferred in many applications. However its application was limited by the complexity of its control which arises because of the variable-frequency supply, ac signals processing and the complex dynamics of the machine. The recent develop ments in power electronics have solved the variable- frequency supply problem by adequate frequency converters. On the other hand the implementation of microprocessors in the digital control circuits has introduced a wide scope of possibilities to overcome the signal processing. In this work, the system which consist of an induction machine and its variable frequency drive, is controlled by the digital PID controller. The optimum values of digital PID controller parameters were found by the New Global Optimization Method, according to the described Performance Index. The Performance Index for the system is to be a weighted sum of the risetime (tr ), percent overshoot (po), steady state error (e8B ). P.I. - - ( £j the point L is replaced in AA by P. If either P fails to satisfy the constraints of fp > fj or it is not in the search boundaries then the trail is discarded and a fresh point is chosen from the potential trail set. algorithm goes on like that. As the algorithm proceeds the current set of N stored points tends to cluster around the maxima. In 1963, H.J. KUSHNER proposed a search algorithm based upon the assumption that an unknown function could be modelled as a sample function of Gaussian Random Process- specifically a Brownian Motion Process. xvii He showed that the expected value of the unknown function X(t) conditioned on all the measurements taken is a piece wise linear approximation of X(t) itself. By using these two properties he was able to calculate the location of the best point to evaluate next. The location of the next guess is obtained by finding th^e point with the maximum probability that X(t) exceeds Xm ( the maximum X(t) found after m measurements ) by some positive constant Ka or in other words JVAR[£(X)\ $ t Cumulative normal density function. This probability can be maximized by minimizing the quantity belowt (X£(t) + Ka- E[X(t)] )a VAR[X(t)] from this the minimum value of A and its corresponding value of t can be found in closed form over an interval between two sample points. The values of A,^ for each interval then compared and the minimum is taken as the best choise for next guess. If we summarise the overall method, 1. Evaluate the function at its end points. 2. Find the E[f(X)] and the variance for the segments for each succesive points. (At the beginning we have only one segment) 3. Find jij^ for each segmet. 4. Find the function value at t which correspond to the t of AmLa. 5. Form two new segments around the new measurements. B.E.STUCKMAN developed n-dimensional model of the Kushner's method. Search strategy is as followsi xviii 1. Evaluate the function at the 2a vertices of the n- dimensional space. 2. Construct the line segments lst which connects Xs to Xt 3. Suppose that the measurements of the function can be modelled as Brownian Motion Process. 4. Take into account only the points along the line segmets. The expected value can be given as Elf (X) ] = f(XB) + A fiXt\îJ ^ X a point on l8t, X is the X's distance from X8. The conditional varience is VAR[f(X)] = c.X. lJf* ~, k C is the mean square variation of the curve along l8t. Next we will fjLnd the probability that the unknown function will exceed ^m^, the largest value of the function after m guesses by Ka» y/VARl£(X) J We can maximize the above quantity by minimizing (f*(t) «. KB - E[X(t)] )a A VARlX(t)) at the minimum point if we solve for X* » t:iio * Ka - fix.) - x. f (x'\ ; f{x"].LMt\ xix *,? = {f'{J0 +K»- f{x°}) -|Jgtl 2. f*(X) + 2.Ka- f(Xe) - f(Xt) X* is independent of C. The maximum value of A is given as. 4*4x1= tt-t [(*£(*> +**- fi*.)). (£!<*) + *"- f(ze))] \-Let\ He find the line segment with the lowest Amin and the location of the best candidate for the next guess Xm+1 by k*. -K/n+ı has the highest probability of exceeding the largest value found by Ka. 5. The function is evaluated at Xm+1 and we continue in the same iterative manner. The segment containing the new point is divided into two and the segments connecting the new point to the m-2 points are added. The value of 4nin i s found for each line segments including the newest ones. search is terminated after a predetermined number of points. In the new hybrid method which had been developed by Olcay BOZ, proposed we are given a function of N bounded variables and an initial search domain is defined by using the limits of each variable. H trial points are choosen from the grids if the degree of function is more then five than there will be too much initial points and this will make the search very complicated. The function is evaluated at each trial point and the position and the function value corresponding to each point are stored in a table AA. Standart deviation of the table values of the points stored in AA are found and this is used for finding Kn. Line segments which connects each set of two points are constructed. Here we have two ways to go on, 1. To use the whole table for constructing line segments. In this way all the points in the table AA are connected with another point in the table by a line segments. We will have m.(m-l)/2 line segments. ( mi number of points stored in array AA ) 2. To devide the table into sub-tables and construct line segments for each sub-table separately. In this way the table AA is divided into a predetermined number of sub- tables and each point in a subtable is connected with the other points. There will be no line segments between the points in different subtables.We will have q(m/q) (m/q-1 ) /2 line segments. xx For each line segments the point that maximises the probability that the function value will exceed fm(x). ( The largest value of the function value in the table AA at that time.) by Km is found. This is found by maximizing the probability pxob(f(X) * f*(X) +Ka) = 1 - «( f*(X) * Km - E[f(X)) y/VAR[f(X)\ Since $ is a monotonically increasing function, we can maximize the probability by minimizing the sinpler quatitiy. (f£(t) + KB- E[X{t)) )a VARlX(t)] If we solve the location of A^^ yields, jf'(X) * Ka- flXm)).\lst\ 2.f£(X) +2.Km- f(XB) - f(Xt) X is the distance of the newly found point from X8. Function value of this point (P) is compared with the point (L) of the lowest function value in the array AA. If not the new point is discarded and another one is found on another line segment. After changing a point in the table lowest and highest function values in the array AA. are compared if they are identical to a predetermined accuracy we stop and our global maximum is the point with the highest function value in the table. If not we go on processing p. ine segments after finding the new vsalue of Ka. and fa(x), if stopping criteria is not reached we conctract line segments again in the table AA and iteration goes on in this manner. For not restricting the saerch to the points near a local maxima we have to cover the entire search space for global searching. We can fcover the entire search space by extending the line segments to the boundaries. By exten ding the line segments to the boundaries we can find points apart from each other. This decreases the probabilty of getting close to a local maximum.