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# Design of lo order dynamic pr ecompensators using con ex methods Laya Shamg ah Afsoon Nejati Amin Nobakhti Houshang Karimi Electrical Engineering Sharif Uni ersity of echnology Abstract Pr ecompensat PDF document - DocSlides

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Design of lo order dynamic pr e-compensators using con ex methods Laya Shamg ah, Afsoon Nejati, Amin Nobakhti Houshang Karimi Electrical Engineering, Sharif Uni ersity of echnology Abstract Pr e-compensators ar used in multi ariable contr ol systems to educe (or eliminate) open loop system interactions. Classical methods or the design of pr e-compensator ar tra- ditionally based on static designs. Static pr e-compensators ar pr eferr ed or their simplicity ut ar highly unsystematic in the amounts of achie able diagonal dominance. In many ap- plications, only the mor po werful dynamic pr e-compensators ar able to deli er the desir ed amounts of decoupling This paper pr oposes new method or the design of dynamic pr e- compensators which is based on Quadratic Pr ogramming (QP) optimization. Using the pr oposed appr oach the total pr e- compensator is ound thr ough se eral smaller optimization pr oblems, one or each column. The application and effecti e- ness of the QP dynamic design is demonstrated on Distrib uted Generation unit (DG) case study ODU ON Rosenbrock contrib ution to the design of control systems for linear multi ariable plants inspired much acti vity in the de elopment of techniques for achie ving diagonal dominance [1]. The primary objecti of all such techniques is to reduce plant interactions by the introduction of multi ariable pre- compensator so that the control system design can then be completed by using classical techniques to synthesise set of single-loop controllers for the compensated plant [2][3]. raditional techniques de eloped for the achie ement of diagonal dominance by the use of static pre-compensators are the pseudo-diagonalisation [4][5], the function-minimisation method using conjug ate-direction optimisation [6], and the ALIGN algorithm de eloped initially in conjunction with characteristic-locus methods [7]. More recently adv ent of po werful optimization algorithms pa ed the ay for de elop- ment of impro ed techniques based on Ev olution Strate gies [8], -norm [9], and -norm [10]. Dynamic pre-compensation of fers the opportunity to not only aim to routinely achie diagonal dominance for wide range of plants, ut also achie ar higher le els of diagonal dominance. Methods for the design of dynamic pre- compensators should strik the right balance between achie v- able performance and pre-compensator comple xity This is not al ays easily obtained. or xample while Chughtai and Munro [10] xtend their static formulation for dynamic design, the dynamical order of the entual dynamic pre- compensator will be ery high (each element ould be equal to the order of the entire plant plus the weighting functions). Con ersely in [11] method is proposed which can produce lo order pre-compensators, ut the method is not systematic nobakhti@sharif.edu (Corresponding author) and comes at the xpense of considerable design ef fort required for curv ﬁtting problems in case of an system. One of he more po werful recent approaches in solving the dynamic problem has been with the use of Ev olutionary Algorithms [8]. An olutionary optimization of fers lar ge design ﬂe xibility including the ability to set each element of the pre-compensator to ha e a desired order Alas, these user beneﬁts are countered by two important obstacles; huge computational ef fort, and the ‘curse of dimensionality’. The latter is especially inhibiting when it comes to problems of lar ger size or higher dynamical order This paper aims to dra upon the main beneﬁts of the pre vious techniques to propose practical and usable method for the design of dynamic pre-compensators. Using the proposed algorithm, a separate design problem is solv ed for each column of the pre-compensator Each element of the pre-compensator can be either static or ha arbitrary dynamical comple xity ut it may not be set to zero. II A Q UAD OG MM PP OA DYNA ON A. Pr oblem Statement Consider the system to be stable TI transfer function matrix (TFM) ij )] where is the set of rational transfer functions. The design problem is to ﬁnd dynamic precompensator ij )] such that is dominant ov er set of frequencies ,..., [4], where ij =1 il (1) or design considrations it is desirable not to impose an restrictions on the dynamical comple xity of so that the desired order for each element of the pre-compensator may be set independently of others. The pre-compensator is thus deﬁned as, lj =0 lj +1) lj (2) where is matrix of inte gers. Elements of determine the respecti order of the element of the pre-compensator In using this technique, either column or ro dominance can be used; ut in this paper the design problem will be solv ed in the case of column dominance used for direct frequenc designs. The column 2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011 978-1-4577-0079-8/11/$26.00 2011 AACC 3680

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dominance problem solv ed in this paper is to determine dynamic pre-compensator such that the cost function, Γ( Ω) =1 =1 =1 =1 ij jj (3) is minimised. There are two important features of (3) which may be xploited for reduced problem size. Let ,..., )] and ,..., )] It will then follo from (3) that, min Γ( Ω) min =1 =1 min (4) It is therefore possible to solv dif ferent problems, one for each column of If the ro dominance measure as used instead, the same could be sho wn with respect to the ro Secondly note that diagonal scaling does not alter the alue of (3). That is Γ( Ω) Γ( Ω) where diag ,..., It will therefore al ays be possible to scale such that, ii (1) (5) where ij +1) is the coef ﬁcient of th term of the polynomial ij (see (2)). Con ersely min Γ( Ω) =1 min ii (1) (6) Imposing (5) on (2) will mean that minimization of (3) will no longer result in the tri vial zero solution by ensuring ii will be nonzero. Thus, instead of minimizing the ratio deﬁned in (3), the same solution can be obtained by minimizing the modulus of the of f-diagonal terms of subject to (5). This leads to the follo wing optimization problem for the th column of the precompensator min =1 =1 ,j ij (7) subject to, ii (0) (8) where the pre-compensator is deﬁned according to (2). B. The QP optimization pr oblem form the QP problem, we create modiﬁed system whose norm represents the summed interactions of the riginal system as represented by (7). The norm problem can then be easily con erted into QP The ectorization procedure is similar to one laid out for static pre-compensators in Lemma of [12], xtended here for dynamic designs. Application of the ectorization allo ws (7) to be re written as, min || || min (9) where, ,..., mj (10) and, lj lj +1) lj ,..., (1) (11) In (11), is deﬁned as, max ij (12) Similarly we deﬁne as, ,..., (13) The dimensions of are 1) and the are deﬁned as follo ws, il )) ..., (14) and, ,..., (15) In (14), denotes the matrix with its th block deleted. This will correspond to remo ving the diagonal entries of from the minimization as speciﬁed in (7). proceed with the QP formulation, note that, (16) Each ector will contain single (imposed by the constraint 5)) and series of zeros (see (11)). Let be permutation matrix which will bring into this form, =(1 ,..., {z ij +1) ,..., (1) {z ,..., ..., mj mj +1) ,..., mj (1) (17) since an permutation matrix PP then from (16) we ha e, {z (18) 11 21 12 22 (19) since is constant the ﬁrst term does not ef fect the mini- mization. Hence the solution is obtained by minimizing, 12 22 (20) which is in the form of QP Since is Hermitian matrix with comple entries. Therefore it is only necessary 3681

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to consider the real part of Alternati ely note that since the norm is real, then will al ays be necessarily zero. In summary the problem of designing the th column of the dynamic pre-compensator (7) is solv ed by the follo wing QP problem, min lj 22 12 (21) separate QP is solv ed for each column of or ro dominance measure, QP ould be required for each ro of III GO FI ON XA LE demonstrate the essential features of the algorithm, we shall ﬁrst consider single model design problem. The system to consider is gi en by 5) 17 9) 12 45 68 36) 3) 11 33 27) (2 6) 10 26 26 9) 13 57 113 104 36) 12 44 60 27) 8) 10 26 26 9) 7) 13 57 113 104 36) 17 9) (22) The Nyquist Array of is sho wn in Figure 1. The Fig. 1. Nyquist Array for uncompensated ﬁrst column is already dominant, ut the second and third columns are clearly not dominant at all. be gin be setting, 0 0 0 0 0 0 0 0 0 (23) This will correspond to completely static pre-compensator (zero order for all elements). The solution in this case as found to be, 1308 2449 2 1 1 459 1 0 6837 1 (24) The Nyquist Array of is sho wn in Figure 2. Notice that the ﬁrst column has been completely decoupled, ut the second two columns retain some interactions. may therefore wish to increase the order of the elements of the second and third columns and set, 0 1 1 0 1 1 0 1 1 (25) Using the pre-compensator (26) is obtained, Fig. 2. Nyquist Array for 001184 042 01757 1 0 96 385 (26) Ev en though all elements of the third column were allo wed to be up to ﬁrst order only the element is ﬁrst order in and the other two elements in the third column remain static. Clearly from the user point of vie this is important since if the order of an element is set higher than it ought to be, the algorithm will automatically return null coef ﬁcient. The Nyquist Array of is sho wn in Figure 3. The third column has no also been completely decoupled. Since both the ﬁrst and third columns ha been Fig. 3. Nyquist Array for completely decoupled, if we wish to reduce the interactions further the only option left is to increase the order of the 3682

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second column. ﬁnd that for 0 1 0 0 1 1 0 2 0 (27) The solution which is obtained will completely decouple the system. The pre-compensator in this case is, 2 1 (28) The decomposition can be eriﬁed by forming the product as sho wn belo 1) 0 0 0 1 4) 0 0 1 3) (29) TE ON DG UDY A. System Description The use of distrib uted generation (DG) units such as photo- oltaic arrays, wind turbines, and fuel cells pro vides se eral adv antages for the utility distrib ution grid. or instance, the DG systems decrease the cost of ener gy production, increase po wer quality of the distrib ution system, and reduce the en vironmental and economical problems. Figure sho ws the schematic of an electronically-coupled DG unit. The DG unit is represented by DC oltage source, po wer electronics con erter (VSC) which has ast dynamic response and used as an interf ace to connect DG unit to utility grid, series ﬁlter and step-up transformer and represent both the series ﬁlter and the step-up transformer parameters. The local load is represented by balanced three-phase parallel RLC netw ork at the point of common coupling (PCC). arameters of the system sho wn in Figure are summarized in able I. A DG unit normally operates in grid-connected mode when the CB switch in Figure is closed, i.e., the DG unit and its dedicated load are part of the distrib ution grid. In the grid- connected mode, the host grid assumes supervisory role and determines the oltage amplitude and frequenc y v alues of the load at the PCC. Consequently the DG unit is only responsible for control of its real/reacti po wer components. Often this is based on the well-kno wn dq-current control methodology [13]. The DG unit and the local load form an islanded system when switch CB is open. In this mode, grid control of oltage and frequenc is no longer present, leading to possible instabilities [14]. Evidently to maintain oltage and frequenc stability (and desired response characteristics) it is necessary to acti ate replacement control systems. B. DG Model and linearization Application of Kirchhof f oltage and current la ws for the islanded DG system (Figure 4) gi es rise to nonlinear set Fig. 4. Schematic diag am of an islanded DG unit ABLE ETE UDY TE Quantity alue VSC ﬁlter resistance, 1.5 (0.010 pu) VSC ﬁlter inductance, 300 H (0.785 pu) VSC terminal oltage, base oltage 600 (l-l) (1 pu) PWM carrier frequenc 1980 Hz DC us oltage, dc 1500 VSC rated po wer 2.5 MW Load nominal resistance, 76 (1 pu) Load nominal inductance, 111.9 mH (0.554 pu) Load nominal capacitance, 62.86 F (1.805 pu) Frequenc set-point 60 Hz oltage set-point 11267 of equations. The equations are described by dI td dt tq td td dI tq dt td tq tq dV dt td Ld dI Ld dt Lq Ld dI Lq dt Ld Lq tq Lq (30) The state ector control input, and control output, are td tq Ld Lq td tq The state space equations of (30) represent multi-input, multi-output nonlinear autonomous forced system. The chief requirement of the controller ould be to re gulate the output at around the same alue 11267 olts and 60 Hz). The act that the control problem is only re gulatory one permits the use of linear techniques pro vided it can be demonstrated that linear and nonlinear model are matched reasonably well for typical xpected ariations around the speciﬁed output set- points. Therefore, the approach tak en here is to linearize the model at the nominal output and load alues. This gi es rise to the follo wing linear set of equations, (31) 3683

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0.9 1.1 (a) V (pu) 0.3 0.4 0.5 0.6 0.7 0.8 360 370 380 390 (b) Time (s) (rad/s) Fig. 5. Compar ison of the linear appro ximation (dashed) with the nonlinear model (solid) with respect to step changes in control inputs where and are constant matrices as follo ws, (2 0 0 0 0 0 0 0 0 1 0 0 (32) o v erify the accurac of the linearized models, independent set-point changes around the speciﬁed point are. The oltage is subjected to 10% change whilst the change in the frequenc is These are well within permissible ranges of allo wed po wer ﬂuctuations. The results are sho wn in Figure 5. It is clear that the linear model follo ws the non- linear response ery well. Subsequent analysis of the open- loop linear model sho that the system is open-loop stable, ut with signiﬁcant amounts of interaction especially in the frequenc loop. In order to proceed, preliminary processing is carried out in accordance with standard multi ariable design steps. Thus for xample; the inputs and outputs are paired according to the RGA criteria, and the model inputs and outputs are made dimensionless to re eal the true xtend of system interactions [15][3]. C. Pr e-compensator design for the DG system The range of frequenc point for the design of the pre- compensator is chosen to be 50 log arithmically spaced points between 10 and 10 which adequately co er the entire bandwidth of the system. The aim is to eep the order as lo as possible in this case. It as found that purely static design as not suf ﬁcient to achie decent le els of dominance. Ho we er precompensator with comple xity matrix, 1 1 0 0 (33) as found to gi acceptable performance. The TFM of the optimizing QP pre-compensator as found to be, 03806 1 0 002343 1189 9271 1 (34) which is transformed by column scaling into the rational form gi en belo nl QP 0786 0 1189 03806 0197 (35) 0.5 −1 −0.5 0.5 V tq V −0.12 −0.1 −0.08 −0.06 0.01 0.02 0.03 −V td 0.05 0.1 −0.06 −0.04 −0.02 0.1 0.2 −0.1 0.1 Fig. 6. Nyquist Array of uncompensated system 0.96 0.98 1.02 −0.04 −0.02 0.02 V tq V −2 x 10 −6 −2 −1 x 10 −6 −V td 0.005 0.01 −2 x 10 −3 0.1175 0.1175 0.1176 −5 10 x 10 −6 Fig. 7. Nyquist Array of the DG system compensated with (35) Figures and respecti ely sho the Nyquist Array of the DG before and after application of pre-compensator (35). The ﬁgures clearly sho the amount of ained dominance. compare the QP design, we also use Ev olutionary Al- gorithms for parameter optimization [8] of dynamic pre- compensator with similar structure to (35) Each candidate pre-compensator transfer function matrix is encoded in chromosome comprising concatenated sub-chromosomes that each represents an element of the compensator matrix with its respected dynamical comple xity Entire pop- ulations of chromosomes of such candidate pre-compensator matrices are caused to olv subject to the actions of mutation, crosso er and selection: the measure of ﬁtness used in such algorithms in the present conte xt is simply the reciprocal of the cost function deﬁned in equation (3). The 3684

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solution as found to be, nl 9997 0 1189 03968 01971 (36) Comparison of (36) and (35) re eals that the QP solution is xtremely close to the global optimum of the original nonlinear dominance ratio cost function. The second column is almost identical with minor dif ferences in the ﬁrst column. Of course the QP as computed signiﬁcantly aster than the EA solution. see the dif ferences consider Figure which sho ws the plot of the Perron root of the uncompensated plant, together with those compensated by the EA and QP pre- compensators. The column dominance ratios for the three cases are also sho wn in Figures and 10. 10 −2 10 −1 10 10 10 −1 −0.5 0.5 1.5 2.5 Perron Frobenius eigenvalue modulas in dBs log (w), w in rads/sec Fig. 8. erron robenius eigen alue of plant (dashed), and compen- sated plant with QP (solid) and EA algor ithms (dotted) 10 −2 10 −1 10 10 10 −0.02 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Column domninace ratio of voltage loop modulus Fig. 9. Column dominance atio of oltage loop -plant (dashed), QP (solid), EA (dotted) ON ON This paper has proposed ne technique for the design of dynamic pre-compensators. belie the technique repre- sents signiﬁcant and important contrib ution to the xisting amily of methods. In particular it combines the design er satility and ﬂe xibility of direct optimization methodologies (such as EAs or optimization by PSQ, SA, etc..), while at the same time of fering the ef ﬁcienc of computation of ered by the con optimization methods (such as LMIs). The technique has been demonstrated through application to two xamples. In the design eriﬁcation xample it as 10 −2 10 −1 10 10 10 −0.2 0.2 0.4 0.6 0.8 1.2 Column domninace ratio of Frequency loop modulus Fig. 10. Column dominance atio of frequency loop -plant (dashed), QP (solid), EA (dotted) sho wn ho the designer is able to gradually increase the order of each element as required and until suf ﬁcient or desired amounts of diagonal dominance are achie ed. The method as also able to ﬁnd the globally optimal design that completely decoupled the system (b ut there are no guarantees this is al ays possible). In the DG case study design comparison with Ev olutionary Algorithms demonstrated that the QP solution is xtremely close to the global optimum of the original column dominance ratio minimization. The method has also been applied to models of other real- life systems with lar ger size and comple xity with similar xcellent results. [1] H.H Rosenbrock. Design of multi ariable control systems using the in erse yquist array Pr oc. IEE 116:1929–1936, 1969. [2] R.V atel and N. Munro. Multivariable System Theory and Design Per amon Press, Oxford, 1982. [3] J.M. Maciejo wski. Multivariable eedbac Design Addison-W esle 1989. [4] D.J. Ha wkins. Multifrequenc y v ersion of pseudodiagonalisation. Electr onic Letter 8(19):473–474, 1972. [5] D.J. Ha wkins. Pseudodiagonalisation and the in erse yquist array method. Pr oc IEE 119:337–342, 1972. [6] G.G Leininger Diagonal dominance for multi ariable yquist array methods using function minimisation. utomatica 15:339–345, 1979. [7] ouv aritakis. Char acteristic Locus Methods for Multivariable eedbac System Design PhD thesis, Uni ersity of Manchester 1974. [8] A. Nobakhti, N. Munro, and B. Porter Ev olutionary achie ement of diagonal dominance in linear multi ariable plants. Electr onic Letter 39(1):165–166, 2003. [9] A. Nobakhti and H. ang. On ne method for -based decompo- sition. IEEE ansactions on utomatic Contr ol 51(12):1956–1961, 2006. [10] S.S. Chughtai and N. Munro. Diagonal dominance using lmis. IEE Pr oc contr ol Theory and Application 151(2), March 2004. [11] A. Nobakhti and N. Munro. Achie ving diagonal dominance by frequenc interpolation. American Contr ol Confer ence 2004. Boston, USA. [12] A. Nobakhti and N. Munro. ne method for singular alue loop shaping in design of multiple-channel controllers. IEEE ansactions on utomatic Contr ol 49(2):249–253, Feb 2004. [13] Piagi and R.H. Lasseter Autonomous control of microgrids. IEEE ower Engineering Society Gener al Meeting 2006. [14] H. Karimi, H. Nikkhajoei, and M.R. Ira ani. Control of an electronically-coupled distrib uted resource unit subsequent to an is- landing ent. IEEE ansactions on ower Delivery 23(1):493–501, 2008. [15] S. Sk ogestad and I. Postlethw aite. Multivariable eedbac Contr ol: Analysis and Design ile 1996. 3685

Classical methods or the design of pr ecompensator ar tra ditionally based on static designs Static pr ecompensators ar pr eferr ed or their simplicity ut ar highly unsystematic in the amounts of achie able diagonal dominance In many ap plications o ID: 23621

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Design of lo order dynamic pr e-compensators using con ex methods Laya Shamg ah, Afsoon Nejati, Amin Nobakhti Houshang Karimi Electrical Engineering, Sharif Uni ersity of echnology Abstract Pr e-compensators ar used in multi ariable contr ol systems to educe (or eliminate) open loop system interactions. Classical methods or the design of pr e-compensator ar tra- ditionally based on static designs. Static pr e-compensators ar pr eferr ed or their simplicity ut ar highly unsystematic in the amounts of achie able diagonal dominance. In many ap- plications, only the mor po werful dynamic pr e-compensators ar able to deli er the desir ed amounts of decoupling This paper pr oposes new method or the design of dynamic pr e- compensators which is based on Quadratic Pr ogramming (QP) optimization. Using the pr oposed appr oach the total pr e- compensator is ound thr ough se eral smaller optimization pr oblems, one or each column. The application and effecti e- ness of the QP dynamic design is demonstrated on Distrib uted Generation unit (DG) case study ODU ON Rosenbrock contrib ution to the design of control systems for linear multi ariable plants inspired much acti vity in the de elopment of techniques for achie ving diagonal dominance [1]. The primary objecti of all such techniques is to reduce plant interactions by the introduction of multi ariable pre- compensator so that the control system design can then be completed by using classical techniques to synthesise set of single-loop controllers for the compensated plant [2][3]. raditional techniques de eloped for the achie ement of diagonal dominance by the use of static pre-compensators are the pseudo-diagonalisation [4][5], the function-minimisation method using conjug ate-direction optimisation [6], and the ALIGN algorithm de eloped initially in conjunction with characteristic-locus methods [7]. More recently adv ent of po werful optimization algorithms pa ed the ay for de elop- ment of impro ed techniques based on Ev olution Strate gies [8], -norm [9], and -norm [10]. Dynamic pre-compensation of fers the opportunity to not only aim to routinely achie diagonal dominance for wide range of plants, ut also achie ar higher le els of diagonal dominance. Methods for the design of dynamic pre- compensators should strik the right balance between achie v- able performance and pre-compensator comple xity This is not al ays easily obtained. or xample while Chughtai and Munro [10] xtend their static formulation for dynamic design, the dynamical order of the entual dynamic pre- compensator will be ery high (each element ould be equal to the order of the entire plant plus the weighting functions). Con ersely in [11] method is proposed which can produce lo order pre-compensators, ut the method is not systematic nobakhti@sharif.edu (Corresponding author) and comes at the xpense of considerable design ef fort required for curv ﬁtting problems in case of an system. One of he more po werful recent approaches in solving the dynamic problem has been with the use of Ev olutionary Algorithms [8]. An olutionary optimization of fers lar ge design ﬂe xibility including the ability to set each element of the pre-compensator to ha e a desired order Alas, these user beneﬁts are countered by two important obstacles; huge computational ef fort, and the ‘curse of dimensionality’. The latter is especially inhibiting when it comes to problems of lar ger size or higher dynamical order This paper aims to dra upon the main beneﬁts of the pre vious techniques to propose practical and usable method for the design of dynamic pre-compensators. Using the proposed algorithm, a separate design problem is solv ed for each column of the pre-compensator Each element of the pre-compensator can be either static or ha arbitrary dynamical comple xity ut it may not be set to zero. II A Q UAD OG MM PP OA DYNA ON A. Pr oblem Statement Consider the system to be stable TI transfer function matrix (TFM) ij )] where is the set of rational transfer functions. The design problem is to ﬁnd dynamic precompensator ij )] such that is dominant ov er set of frequencies ,..., [4], where ij =1 il (1) or design considrations it is desirable not to impose an restrictions on the dynamical comple xity of so that the desired order for each element of the pre-compensator may be set independently of others. The pre-compensator is thus deﬁned as, lj =0 lj +1) lj (2) where is matrix of inte gers. Elements of determine the respecti order of the element of the pre-compensator In using this technique, either column or ro dominance can be used; ut in this paper the design problem will be solv ed in the case of column dominance used for direct frequenc designs. The column 2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011 978-1-4577-0079-8/11/$26.00 2011 AACC 3680

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dominance problem solv ed in this paper is to determine dynamic pre-compensator such that the cost function, Γ( Ω) =1 =1 =1 =1 ij jj (3) is minimised. There are two important features of (3) which may be xploited for reduced problem size. Let ,..., )] and ,..., )] It will then follo from (3) that, min Γ( Ω) min =1 =1 min (4) It is therefore possible to solv dif ferent problems, one for each column of If the ro dominance measure as used instead, the same could be sho wn with respect to the ro Secondly note that diagonal scaling does not alter the alue of (3). That is Γ( Ω) Γ( Ω) where diag ,..., It will therefore al ays be possible to scale such that, ii (1) (5) where ij +1) is the coef ﬁcient of th term of the polynomial ij (see (2)). Con ersely min Γ( Ω) =1 min ii (1) (6) Imposing (5) on (2) will mean that minimization of (3) will no longer result in the tri vial zero solution by ensuring ii will be nonzero. Thus, instead of minimizing the ratio deﬁned in (3), the same solution can be obtained by minimizing the modulus of the of f-diagonal terms of subject to (5). This leads to the follo wing optimization problem for the th column of the precompensator min =1 =1 ,j ij (7) subject to, ii (0) (8) where the pre-compensator is deﬁned according to (2). B. The QP optimization pr oblem form the QP problem, we create modiﬁed system whose norm represents the summed interactions of the riginal system as represented by (7). The norm problem can then be easily con erted into QP The ectorization procedure is similar to one laid out for static pre-compensators in Lemma of [12], xtended here for dynamic designs. Application of the ectorization allo ws (7) to be re written as, min || || min (9) where, ,..., mj (10) and, lj lj +1) lj ,..., (1) (11) In (11), is deﬁned as, max ij (12) Similarly we deﬁne as, ,..., (13) The dimensions of are 1) and the are deﬁned as follo ws, il )) ..., (14) and, ,..., (15) In (14), denotes the matrix with its th block deleted. This will correspond to remo ving the diagonal entries of from the minimization as speciﬁed in (7). proceed with the QP formulation, note that, (16) Each ector will contain single (imposed by the constraint 5)) and series of zeros (see (11)). Let be permutation matrix which will bring into this form, =(1 ,..., {z ij +1) ,..., (1) {z ,..., ..., mj mj +1) ,..., mj (1) (17) since an permutation matrix PP then from (16) we ha e, {z (18) 11 21 12 22 (19) since is constant the ﬁrst term does not ef fect the mini- mization. Hence the solution is obtained by minimizing, 12 22 (20) which is in the form of QP Since is Hermitian matrix with comple entries. Therefore it is only necessary 3681

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to consider the real part of Alternati ely note that since the norm is real, then will al ays be necessarily zero. In summary the problem of designing the th column of the dynamic pre-compensator (7) is solv ed by the follo wing QP problem, min lj 22 12 (21) separate QP is solv ed for each column of or ro dominance measure, QP ould be required for each ro of III GO FI ON XA LE demonstrate the essential features of the algorithm, we shall ﬁrst consider single model design problem. The system to consider is gi en by 5) 17 9) 12 45 68 36) 3) 11 33 27) (2 6) 10 26 26 9) 13 57 113 104 36) 12 44 60 27) 8) 10 26 26 9) 7) 13 57 113 104 36) 17 9) (22) The Nyquist Array of is sho wn in Figure 1. The Fig. 1. Nyquist Array for uncompensated ﬁrst column is already dominant, ut the second and third columns are clearly not dominant at all. be gin be setting, 0 0 0 0 0 0 0 0 0 (23) This will correspond to completely static pre-compensator (zero order for all elements). The solution in this case as found to be, 1308 2449 2 1 1 459 1 0 6837 1 (24) The Nyquist Array of is sho wn in Figure 2. Notice that the ﬁrst column has been completely decoupled, ut the second two columns retain some interactions. may therefore wish to increase the order of the elements of the second and third columns and set, 0 1 1 0 1 1 0 1 1 (25) Using the pre-compensator (26) is obtained, Fig. 2. Nyquist Array for 001184 042 01757 1 0 96 385 (26) Ev en though all elements of the third column were allo wed to be up to ﬁrst order only the element is ﬁrst order in and the other two elements in the third column remain static. Clearly from the user point of vie this is important since if the order of an element is set higher than it ought to be, the algorithm will automatically return null coef ﬁcient. The Nyquist Array of is sho wn in Figure 3. The third column has no also been completely decoupled. Since both the ﬁrst and third columns ha been Fig. 3. Nyquist Array for completely decoupled, if we wish to reduce the interactions further the only option left is to increase the order of the 3682

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second column. ﬁnd that for 0 1 0 0 1 1 0 2 0 (27) The solution which is obtained will completely decouple the system. The pre-compensator in this case is, 2 1 (28) The decomposition can be eriﬁed by forming the product as sho wn belo 1) 0 0 0 1 4) 0 0 1 3) (29) TE ON DG UDY A. System Description The use of distrib uted generation (DG) units such as photo- oltaic arrays, wind turbines, and fuel cells pro vides se eral adv antages for the utility distrib ution grid. or instance, the DG systems decrease the cost of ener gy production, increase po wer quality of the distrib ution system, and reduce the en vironmental and economical problems. Figure sho ws the schematic of an electronically-coupled DG unit. The DG unit is represented by DC oltage source, po wer electronics con erter (VSC) which has ast dynamic response and used as an interf ace to connect DG unit to utility grid, series ﬁlter and step-up transformer and represent both the series ﬁlter and the step-up transformer parameters. The local load is represented by balanced three-phase parallel RLC netw ork at the point of common coupling (PCC). arameters of the system sho wn in Figure are summarized in able I. A DG unit normally operates in grid-connected mode when the CB switch in Figure is closed, i.e., the DG unit and its dedicated load are part of the distrib ution grid. In the grid- connected mode, the host grid assumes supervisory role and determines the oltage amplitude and frequenc y v alues of the load at the PCC. Consequently the DG unit is only responsible for control of its real/reacti po wer components. Often this is based on the well-kno wn dq-current control methodology [13]. The DG unit and the local load form an islanded system when switch CB is open. In this mode, grid control of oltage and frequenc is no longer present, leading to possible instabilities [14]. Evidently to maintain oltage and frequenc stability (and desired response characteristics) it is necessary to acti ate replacement control systems. B. DG Model and linearization Application of Kirchhof f oltage and current la ws for the islanded DG system (Figure 4) gi es rise to nonlinear set Fig. 4. Schematic diag am of an islanded DG unit ABLE ETE UDY TE Quantity alue VSC ﬁlter resistance, 1.5 (0.010 pu) VSC ﬁlter inductance, 300 H (0.785 pu) VSC terminal oltage, base oltage 600 (l-l) (1 pu) PWM carrier frequenc 1980 Hz DC us oltage, dc 1500 VSC rated po wer 2.5 MW Load nominal resistance, 76 (1 pu) Load nominal inductance, 111.9 mH (0.554 pu) Load nominal capacitance, 62.86 F (1.805 pu) Frequenc set-point 60 Hz oltage set-point 11267 of equations. The equations are described by dI td dt tq td td dI tq dt td tq tq dV dt td Ld dI Ld dt Lq Ld dI Lq dt Ld Lq tq Lq (30) The state ector control input, and control output, are td tq Ld Lq td tq The state space equations of (30) represent multi-input, multi-output nonlinear autonomous forced system. The chief requirement of the controller ould be to re gulate the output at around the same alue 11267 olts and 60 Hz). The act that the control problem is only re gulatory one permits the use of linear techniques pro vided it can be demonstrated that linear and nonlinear model are matched reasonably well for typical xpected ariations around the speciﬁed output set- points. Therefore, the approach tak en here is to linearize the model at the nominal output and load alues. This gi es rise to the follo wing linear set of equations, (31) 3683

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0.9 1.1 (a) V (pu) 0.3 0.4 0.5 0.6 0.7 0.8 360 370 380 390 (b) Time (s) (rad/s) Fig. 5. Compar ison of the linear appro ximation (dashed) with the nonlinear model (solid) with respect to step changes in control inputs where and are constant matrices as follo ws, (2 0 0 0 0 0 0 0 0 1 0 0 (32) o v erify the accurac of the linearized models, independent set-point changes around the speciﬁed point are. The oltage is subjected to 10% change whilst the change in the frequenc is These are well within permissible ranges of allo wed po wer ﬂuctuations. The results are sho wn in Figure 5. It is clear that the linear model follo ws the non- linear response ery well. Subsequent analysis of the open- loop linear model sho that the system is open-loop stable, ut with signiﬁcant amounts of interaction especially in the frequenc loop. In order to proceed, preliminary processing is carried out in accordance with standard multi ariable design steps. Thus for xample; the inputs and outputs are paired according to the RGA criteria, and the model inputs and outputs are made dimensionless to re eal the true xtend of system interactions [15][3]. C. Pr e-compensator design for the DG system The range of frequenc point for the design of the pre- compensator is chosen to be 50 log arithmically spaced points between 10 and 10 which adequately co er the entire bandwidth of the system. The aim is to eep the order as lo as possible in this case. It as found that purely static design as not suf ﬁcient to achie decent le els of dominance. Ho we er precompensator with comple xity matrix, 1 1 0 0 (33) as found to gi acceptable performance. The TFM of the optimizing QP pre-compensator as found to be, 03806 1 0 002343 1189 9271 1 (34) which is transformed by column scaling into the rational form gi en belo nl QP 0786 0 1189 03806 0197 (35) 0.5 −1 −0.5 0.5 V tq V −0.12 −0.1 −0.08 −0.06 0.01 0.02 0.03 −V td 0.05 0.1 −0.06 −0.04 −0.02 0.1 0.2 −0.1 0.1 Fig. 6. Nyquist Array of uncompensated system 0.96 0.98 1.02 −0.04 −0.02 0.02 V tq V −2 x 10 −6 −2 −1 x 10 −6 −V td 0.005 0.01 −2 x 10 −3 0.1175 0.1175 0.1176 −5 10 x 10 −6 Fig. 7. Nyquist Array of the DG system compensated with (35) Figures and respecti ely sho the Nyquist Array of the DG before and after application of pre-compensator (35). The ﬁgures clearly sho the amount of ained dominance. compare the QP design, we also use Ev olutionary Al- gorithms for parameter optimization [8] of dynamic pre- compensator with similar structure to (35) Each candidate pre-compensator transfer function matrix is encoded in chromosome comprising concatenated sub-chromosomes that each represents an element of the compensator matrix with its respected dynamical comple xity Entire pop- ulations of chromosomes of such candidate pre-compensator matrices are caused to olv subject to the actions of mutation, crosso er and selection: the measure of ﬁtness used in such algorithms in the present conte xt is simply the reciprocal of the cost function deﬁned in equation (3). The 3684

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solution as found to be, nl 9997 0 1189 03968 01971 (36) Comparison of (36) and (35) re eals that the QP solution is xtremely close to the global optimum of the original nonlinear dominance ratio cost function. The second column is almost identical with minor dif ferences in the ﬁrst column. Of course the QP as computed signiﬁcantly aster than the EA solution. see the dif ferences consider Figure which sho ws the plot of the Perron root of the uncompensated plant, together with those compensated by the EA and QP pre- compensators. The column dominance ratios for the three cases are also sho wn in Figures and 10. 10 −2 10 −1 10 10 10 −1 −0.5 0.5 1.5 2.5 Perron Frobenius eigenvalue modulas in dBs log (w), w in rads/sec Fig. 8. erron robenius eigen alue of plant (dashed), and compen- sated plant with QP (solid) and EA algor ithms (dotted) 10 −2 10 −1 10 10 10 −0.02 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Column domninace ratio of voltage loop modulus Fig. 9. Column dominance atio of oltage loop -plant (dashed), QP (solid), EA (dotted) ON ON This paper has proposed ne technique for the design of dynamic pre-compensators. belie the technique repre- sents signiﬁcant and important contrib ution to the xisting amily of methods. In particular it combines the design er satility and ﬂe xibility of direct optimization methodologies (such as EAs or optimization by PSQ, SA, etc..), while at the same time of fering the ef ﬁcienc of computation of ered by the con optimization methods (such as LMIs). The technique has been demonstrated through application to two xamples. In the design eriﬁcation xample it as 10 −2 10 −1 10 10 10 −0.2 0.2 0.4 0.6 0.8 1.2 Column domninace ratio of Frequency loop modulus Fig. 10. Column dominance atio of frequency loop -plant (dashed), QP (solid), EA (dotted) sho wn ho the designer is able to gradually increase the order of each element as required and until suf ﬁcient or desired amounts of diagonal dominance are achie ed. The method as also able to ﬁnd the globally optimal design that completely decoupled the system (b ut there are no guarantees this is al ays possible). In the DG case study design comparison with Ev olutionary Algorithms demonstrated that the QP solution is xtremely close to the global optimum of the original column dominance ratio minimization. The method has also been applied to models of other real- life systems with lar ger size and comple xity with similar xcellent results. [1] H.H Rosenbrock. Design of multi ariable control systems using the in erse yquist array Pr oc. IEE 116:1929–1936, 1969. [2] R.V atel and N. Munro. Multivariable System Theory and Design Per amon Press, Oxford, 1982. [3] J.M. Maciejo wski. Multivariable eedbac Design Addison-W esle 1989. [4] D.J. Ha wkins. Multifrequenc y v ersion of pseudodiagonalisation. Electr onic Letter 8(19):473–474, 1972. [5] D.J. Ha wkins. Pseudodiagonalisation and the in erse yquist array method. Pr oc IEE 119:337–342, 1972. [6] G.G Leininger Diagonal dominance for multi ariable yquist array methods using function minimisation. utomatica 15:339–345, 1979. [7] ouv aritakis. Char acteristic Locus Methods for Multivariable eedbac System Design PhD thesis, Uni ersity of Manchester 1974. [8] A. Nobakhti, N. Munro, and B. Porter Ev olutionary achie ement of diagonal dominance in linear multi ariable plants. Electr onic Letter 39(1):165–166, 2003. [9] A. Nobakhti and H. ang. On ne method for -based decompo- sition. IEEE ansactions on utomatic Contr ol 51(12):1956–1961, 2006. [10] S.S. Chughtai and N. Munro. Diagonal dominance using lmis. IEE Pr oc contr ol Theory and Application 151(2), March 2004. [11] A. Nobakhti and N. Munro. Achie ving diagonal dominance by frequenc interpolation. American Contr ol Confer ence 2004. Boston, USA. [12] A. Nobakhti and N. Munro. ne method for singular alue loop shaping in design of multiple-channel controllers. IEEE ansactions on utomatic Contr ol 49(2):249–253, Feb 2004. [13] Piagi and R.H. Lasseter Autonomous control of microgrids. IEEE ower Engineering Society Gener al Meeting 2006. [14] H. Karimi, H. Nikkhajoei, and M.R. Ira ani. Control of an electronically-coupled distrib uted resource unit subsequent to an is- landing ent. IEEE ansactions on ower Delivery 23(1):493–501, 2008. [15] S. Sk ogestad and I. Postlethw aite. Multivariable eedbac Contr ol: Analysis and Design ile 1996. 3685

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