Dynamic modeling and stability analysis of power networks using dq0 transformations with a unified reference frame

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1 Proceeings of the Estonian Acaemy of Sciences, 08, 67, 4, Available online at Dynamic moeling an stability analysis of power networks using q0 transformations with a unifie reference frame Juri Belikov a an Yoash Levron b a Department of Computer Systems, Tallinn University of Technology, Akaeemia tee 5a, 66 Tallinn, Estonia b The Anrew an Erna Viterbi Faculty of Electrical Engineering, Technion Israel Institute of Technology, Haifa 0000, Israel Receive February 08, accepte 5 May 08, available online October 08 c 08 Authors. This is an Open Access article istribute uner the terms an conitions of the Creative Commons Attribution- NonCommercial 4.0 International License ( Abstract. The q0 reference frame has become popular for moeling an control of traitional electric machines an small power sources. However, its wiesprea use for moeling an analysis of large-scale, general power systems is still a pening issue. One problem that arises when consiering q0 moels is that they are typically base on local reference frames, an therefore linking ifferent moels is not straightforwar. In this paper we propose to approach this problem by moeling the network an its components using a q0 transformation that is base on a unifie reference frame. We emonstrate this iea on the basis of synchronous machines an photovoltaic generators, an we also establish a q0-base ynamic moel of a transmission network. The resulting moels all use a unifie reference frame, an therefore can be irectly linke to each other in simulation an analytically. The paper is accompanie by a free software package (Levron, Y. an Belikov, J. Toolbox for Moeling an Analysis of Power Networks in the DQ0 Reference Frame that constructs the propose ynamic moels an provies tools for ynamic simulations an stability stuies base on q0 quantities. Key wors: power sytems, q0 transformation, renewable energy, stability.. INTRODUCTION Corresponing author, juri.belikov@ttu.ee In recent years, the increasing penetration of small istribute generators an fast power electronics base evices has given rise to new challenges in moeling the ynamics of power systems. This becomes evient when consiering large-scale systems for which it is especially important to maintain the balance between accuracy an complexity. This challenge has le to new ynamic moels of power systems that are base on irect-quaraturezero (q0) quantities []. Such moels are not as general as three-phase (abc)-base (EMTP-like) moels an are avantageous mainly when the transmission network an units are balance. However, q0 moels combine two properties of interest: similar to transient moels, q0 moels are erive from physical representations, an are therefore accurate at high frequencies, an as a result the assumption of slowly varying signals is not require. In aition, q0 moels are time-invariant, which allows efining an operating point, an enables smallsignal analysis. A etaile comparison of simulation techniques base on the abc an q0 reference frames can be foun in [,]. The q0 transformation has been traitionally use in transient analysis of electric machines [4] an is increasingly use toay for moeling istribute sources, complex loas, renewable generators, an on evices base on power electronics [5 7]. However, wiesprea use of this transformation for moeling large-scale power systems is still a pening issue. Towar this en, one problem that arises when consiering q0 moels is that units are typically base on local reference frames, an therefore linking ifferent moels is not straightforwar. The iea of moeling a transmission network an loas in a common q reference frame was presente in [,8], assuming that the network consists only of resistors an

2 J. Belikov an Y. Levron: Moeling an analysis of power systems in a unifie reference frame 69 inuctors. These works consier a common reference frame, meaning that all the units an the network share the same reference, see [9]. Work [0] presents q0- base moels of three-phase networks with RL elements. Work [] evelops a small-signal q0 moel of a microgri that inclues synchronous machines an electronically interface istribute generators. In aition, an attempt to erive a moel of a general network is presente in [], where it is shown how to construct a state-space moel of the network that is nonminimal. A further extension of this paper is presente in []. In this paper we continue these ieas an propose a metho for moeling the network an its components using a q0 transformation that is base on a unifie reference frame, where the main objective is to create q0- base moels of large-scale power systems that are both accurate an time-invariant. The resulting moels all use a unifie reference frame, an therefore can be irectly linke to each other in simulation an analytically. We open the paper by recalling the basic transformation from one reference frame to another an emonstrate this process using the classic example of a synchronous machine connecte to an infinite bus. The paper then extens this example an presents a q0-base ynamic moel of a general transmission network, using ieas presente in [0,4] an base on the network topology efine in MATPOWER [5]. We also evelop moels of several typical units. This approach is emonstrate numerically in a eicate software tool available in [6]. Several numeric examples are escribe in etail: a 4-bus system showing comparison between quasi-static, abc-, an q0-moels; a 4-bus system illustrating the joint ynamics of synchronous machines, photovoltaic generators, an the transmission network; an a large 00-bus system emonstrating small-signal analysis.. PRELIMINARIES AND A MOTIVATING EXAMPLE Consier a reference frame rotating with an angle of θ(t). For instance, in a synchronous machine, θ(t) is typically selecte to be the rotor electrical angle. Let ζ represent the quantity to be transforme (current, voltage, or flux), an use the compact notation ζ abc = [ζ a,ζ b,ζ c ] T, ζ q0 = [ζ,ζ q,ζ 0 ] T. Then, the q0 transformation can be written as [4, Appenix C] where ω s is the steay-state system frequency or the frequency of an infinite bus, see [7] for more etails. We will show that such selection provies several avantages in the connectivity of a large variety of system components an also can be use to erive compact an easyto-use small-signal moels escribing the power system ynamics. The q0 transformation can be reefine with respect to a new reference frame rotating with an angle ω s t by irect substitution of θ = ω s t in () as ζ q0 = T ωs ζ abc. () A formula that allows conversion of signals from the stanar reference frame (efine with respect to θ) to the new frame (efine with respect to ω s t) is given as [ ζ ζ q ζ 0 ] = [ sin(δ) cos(δ) 0 cos(δ) sin(δ) ] ζ ζ q, () ζ 0 where δ(t) = θ(t) ω s t + π/, the variables ζ, ζq are efine with respect to θ, an ζ, ζ q are efine with respect to ω s t. To explain the iea of a unifie reference frame, we start by recalling a classical moel of a synchronous machine connecte to an infinite bus, an present both moels using a unifie reference frame. Assume a simplifie synchronous machine represente as an ieal voltage source behin a synchronous inuctance L. The machine is connecte to an infinite bus, as shown in Fig.. Both the infinite bus an the internal synchronous machine are represente as voltage sources. The synchronous machine voltage is ṽ = 0, ṽ q = V e, v 0 = 0, with a reference angle of θ. In this example θ is the electric angle of the rotor in respect to the stator. The infinite bus has a constant frequency of ω s, so its voltage is given by v = V g, v q = 0, v 0 = 0, with a reference angle of ω s t. Now the goal is to construct a ynamic moel of the complete system base on q0 signals. However, a potential problem is that the two voltage sources are efine with respect to two ifferent reference frames (θ an ω s t). To solve this, we choose ω s t as a unifie reference frame for both the infinite bus an synchronous machine, an construct a moel of the synchronous machine synchronous inuctance infinite bus with T θ = cos(θ) cos ( θ π ) sin(θ) sin ( θ π ) ζ q0 = T θ ζ abc () cos ( θ + π ) sin ( θ + π ). ṽ = 0 ṽ q = V e v 0 = 0 SG synchronous machine L v = V g v q = 0 v 0 = 0 Further in this paper we aim to explore an alternative reference frame that rotates with the unifie angle ω s t, Fig.. Single-phase iagram of a simple synchronous machine (SG) connecte to an infinite bus ( ).

3 70 Proceeings of the Estonian Acaemy of Sciences, 08, 67, 4, using signals in this reference frame. The synchronous machine voltage is now obtaine by substituting ṽ = 0, ṽ q = V e, v 0 = 0 in (), which yiels [ v v q v 0 ] = [ ] Ve cos(δ) V e sin(δ), 0 δ = θ ω s t + π/. (4) In aition, the ynamic behavior of the angle δ is escribe by t δ = poles ( ) P φ + P m K Jω s t δ, (5) which is the classic swing equation. The term J is the rotor constant of inertia, poles is the number of machine poles (must be even), P m is the mechanical power, an K is the amping constant. The three-phase power can be compute as P φ = (v i + v q i q + v 0 i 0 ). Let δ = φ, then the combination of (4) an (5) results in a state space moel given as t φ =φ, t φ = poles ( Jω s V e (cos(φ )i + sin(φ )i q ) + P m K φ ), v =V e cos(φ ), v q = V e sin(φ ), v 0 = 0. (6) Note that the moel represents only the machine s voltage source an oes not inclue the synchronous inuctance, which is moele separately in the next section.. MODELING POWER SYSTEM COMPONENTS USING A UNIFIED REFERENCE FRAME In this section we show that this iea can be extene an that complex networks an various power system units can be connecte to each other by using signals that are efine with respect to the unifie reference frame. This allows us to obtain a complete moel of the system that is base entirely on q0 quantities, an thus provies a constructive metho to analyze the system s ynamic behavior... State-space moels of linear passive transmission networks We open this section by eveloping moels of single passive components [8], using the unifie reference frame. The iscussion that follows shows how to combine these moels to escribe the ynamics of general transmission networks. The ynamic moel of an inuctor is given in the abc reference frame as L t I abc, = V abc, V abc,, (7) an can be converte to the q0 frame as follows. Observe that the ifferentiation of T ωs results in t I q0 = T ω s t I abc + T ωs t I abc, (8) an the erivative of T ωs can be expresse as t T ω s = [ 0 ωs ] 0 ω s 0 0 T ωs = W T ωs. (9) Substitute (7) an (9) into (8), an use relations () to get t I q0, = W I q0, + ( ) Vq0, V q0,. (0) L This equation escribes a state-space moel of the threephase inuctor. Analogously, the moel of a capacitor C is given as ( ) ( ) Vq0, V q0, = W Vq0, V q0, + t C I q0,. An for a resistor R the moel is given by simple static relations V q0 = I RI q0, () where I enotes the ientity matrix. Similarly, by combining elementary passive components, any balance transmission network can be moele base on the unifie reference frame. Define the q0 signals v,n, v q,n, v 0,n to be the voltages on bus n; i,n, i q,n, i 0,n to be the injecte currents to bus n; an V (t) = [v, (t),...,v,n (t)] T, I (t) = [i, (t),...,i,n (t)] T, etc., where N is the number of buses. Assume a network with the stanar branch [5] as in Fig.. This network can be represente in the minimal state-space form using q0 signals as [] t ξ = A ξ ξ + B ξ u, y = C ξ ξ + D ξ u, where u = [V,V q,v 0 ] T an y = [I,I q,i 0 ] T. shunt element bus i (from) y i ieal transformer τ ik : L ik R ik shunt element Fig.. Stanar branch connecting buses i an k. y k () bus k (to)

4 J. Belikov an Y. Levron: Moeling an analysis of power systems in a unifie reference frame 7 In aition, often isconnecte buses shoul be eliminate. The nee for this arises in several situations: Sometimes certain buses are not connecte to either a generator or a loa. In this case the current injecte into the bus is zero. Frequently loas are moele as shunt elements an are integrate into the network moel. In this case loa buses appear as isconnecte buses with zero current. In many scenarios there is a nee to analyze the ynamics or stability of a certain subset of units in the network, typically only the generators. In such cases elimination of the isconnecte buses results in a simpler ynamic moel in which the inputs an outputs relate only to the require subset of buses. This is usually one after integrating the loas into the network moel as shunt elements. The ynamic moel after isconnecte buses have been eliminate is t ξ = Ã ξ ξ + B ξ ũ, ỹ = C ξ ξ + D ξ ũ, () where ũ an ỹ represent the input/output following the reuction proceure. Bus elimination is achieve by controlling the inputs of the buses being eliminate, so that the corresponing outputs are zeroe. Assume for instance that the ith input an ith output are eliminate. This is one using the ynamic moel output equation y = C ξ ξ + D ξ u. In case the matrix D ξ of the original q0 moel is iagonal, the output y i may be zeroe by controlling the ith input so that u i = D i,i C iξ, where C i is the ith row in matrix C ξ. Thus, the input u i an the output y i are eliminate an will not appear in the reuce moel. In more complex systems, there is a nee to transform the state vector an to compute a new ynamic moel such that the corresponing rows in C ξ are also zero. This may be achieve by a LU ecomposition of the relevant rows in C ξ. More etails regaring this proceure may be foun in [6]. t φ = R al f f Lβ φ + φ φ 5 + R al a f Lβ φ 4 + sin(φ 6 )v cos(φ 6 )v q, t φ = R a φ φ φ 5 + cos(φ 6 )v + sin(φ 6 )v q, L q t φ = R a φ + v 0, (4) L 0 t φ 4 = R f L a f φ R f L φ 4 + v f, L β t φ 5 = poles J ( t φ 6 =φ 5 ω s with outputs efine as i = L f f L β i q = L f f L β i 0 = L 0 φ, ω = φ 5, δ = φ 6, L β T m + L β 6L f f L q Lβ L φ φ + L a f q Lβ φ φ 4 ), sin(φ 6 )φ L q cos(φ 6 )φ + L a f L β cos(φ 6 )φ L q sin(φ 6 )φ L a f L β i f = L a f L β φ + L Lβ φ 4, sin(φ 6 )φ 4, cos(φ 6 )φ 4, where Lβ = L L f f La f. In this moel, the state variables are φ = λ, φ = λ q, φ = λ 0, φ 4 = λ f, φ 5 = ω, δ = φ 6 ; the inputs are v, v q, v 0, v f, T m ; an the outputs are i, i q, i 0, i f,ω. Unlike the simplifie moel (6), this moel inclues the inuctance terms L, L q, L 0. A convenient property of the moel presente above is that its inputs an outputs are efine with respect to the unifie reference frame rotating with ω s t, an therefore it can be irectly connecte to the network. For instance, connecting the synchronous machine moel to Table. Nomenclature: synchronous machine.. Physical synchronous machine For completeness, consier a more sophisticate (physical) moel of a synchronous machine [4]. The moel presente herein captures the interaction of the irect-axis magnetic fiel with the quarature-axis mmf an of the quarature-axis magnetic fiel with the irect-axis mmf, as well as the effects of resistances, transformer voltages, fiel wining ynamics, an salient poles. The parameters are explaine in Table. By following [4] an omitting laborious algebraic manipulations, the resulting state-space moel of a synchronous machine in the q0 reference frame (with respect to ω s t) is given by Variable λ, λ q, λ 0 λ f ṽ, ṽ q, v 0 ĩ, ĩ q, i 0 v f, i f L, L q, L 0 L a f L f f R a,r f J T m Description Flux linkages Fiel wining flux linkage Stator voltages Stator currents Fiel winings voltage an current Synchronous inuctances Mutual inuctance between the fiel wining an phase a Self-inuctance of the fiel wining Armature an fiel wining resistance Rotor moment of inertia Mechanical torque

5 7 Proceeings of the Estonian Acaemy of Sciences, 08, 67, 4, an infinite bus is immeiate. An infinite bus with a frequency of ω s is efine in this reference frame by v = V g, v q = v 0 = 0, where V g is the infinite bus RMS voltage. Therefore, a synchronous machine connecte to an infinite bus can be moele by irect substitution of these voltages into (.)... Photovoltaic generator Similarly to the synchronous machine, photovoltaic inverters can also be represente in the unifie reference frame that rotates with ω s t. Several moels of photovoltaic generators are available in the recent literature, as reviewe in [0]. Regarless of the moel use, q0 signals efine in respect to a local reference frame can be converte to a unifie reference frame. Here we emonstrate this process using a photovoltaic inverter moel base on [9]. The inverter is moele by t v b = v b C b [ P pv ( ) ] v î + v q î q, k = K p (v b v re f ) + K i (v b v re f ), î = kv, î q = kv q, î 0 = 0, where P pv enotes the DC power at the output of the photovoltaic array, C b is the bus capacitor (connecte at the input of the switching evices), v b is the bus capacitor voltage, an v re f is the reference voltage for the bus capacitor control circuitry. The term (v î +v q î q ) is the inverter output power in terms of q0 signals. Base on [9], a simple PI controller is use here to control the bus voltage v b. Hence, K p an K i enote the coefficients for the proportional an integral terms, respectively. The secon part escribes the inverter output capacitor as t φ = ) (î i + ωs φ, C sh t φ = ) (îq i q ωs φ, C sh t φ = ) (î0 i 0 C sh with v = φ, v q = φ, v 0 = φ, an where C sh is the inverter output capacitance. 4. NUMERIC TEST CASES This section shows several test cases that illustrate the main iea of moeling an analysis in the unifie reference frame. Three networks with 4, 4, an 00 buses, whose parameters are taken from [5], are consiere. The first example presents a brief comparison between the quasi-static, abc, an q0 moels. The secon an thir examples are evote to the small-signal analysis of the 4- an 00-bus networks. 4.. Comparison of quasi-static, abc, an q0 moels Figure illustrates comparison between conventional abc an q0 moels presente in terms of sparsity an total number of nonzero elements for various test-case networks taken from the MATPOWER atabase [5]. The sparsity inices are compute as the ratio between zero elements in all system matrices an the total number of elements. Observe that both moels have approximately the same number of nonzero elements; however, the abc moel is slightly sparser. Consier now a simple 4-bus network. Its single-line iagram is shown in Fig. 4. The quasi-static an q0 moels were constructe using the software package available in [6]. The abc moel was implemente using MATLAB toolbox Simscape Power Systems. The transient behavior is analyze uner changing operating conitions as epicte in Fig. 5. The input voltage is subsequently change from the initial value as follows. The ramp signal with the slope of 00 an upper limit of 0 kv is use to change the component of the input voltage v, at time t = 0.0 s. Then the input is steppe from 0 to 0 kv at t = 0. s. The output current q components are measure on bus. Observe that all moels coincie in the steay state as expecte. However, the abc an q0 moels are able to more accurately escribe the transient behavior as they capture high-frequency phenomena. Sparsity (%) Nonzero elements abc q Number of buses abc q Fig.. Comparison between abc an q0 moels. The top plot presents the sparsity inex, an the bottom plot illustrates the total number of nonzero elements. 4 Fig. 4. Single-line iagram of a 4-bus system.

6 J. Belikov an Y. Levron: Moeling an analysis of power systems in a unifie reference frame 7 V q0, Î q0, SG + P m, Î q0, SG + SC SC V q0, I q0, V q0, 4-bus Transmission Network t ξ = A ξ ξ + B ξ V q0 I q0 = C ξ ξ + D ξ V q0 V q0,6 I q0,6 V q0,8 SC SC Î q0,6 + PV P pv,6 Î q0,8 + PV P m, I q0, I q0,8 P pv,8 Fig. 5. Transient response of a 4-bus network. The lines correspon to quasi-static (- -), abc ( ), an q0 ( ) moels. 4.. A 4-bus network Consier now the IEEE 4-bus system. This network was moifie by replacing synchronous machines connecte to buses 6 an 8 by photovoltaic generators. Figure 6 shows a single-line iagram of the system, which is compose of two 00 an 50 MW synchronous generators an two 0 MW photovoltaic energy sources. In this figure, the values of P in MW an Q in MVar represent the operating point, which is obtaine by solving the power flow equations. Bus represents an infinite bus moele by a voltage source. The complete system is constructe in the unifie q0 reference frame. A sketch of the signal flow iagram of the complete system is shown in Fig. 7. Simulation parameters of synchronous machines are given in Table. Note that mechanical input powers (P m T m ω s ) are use as new external inputs. P =.7 Q =. P = 6.0 Q = 4. 4 Fig. 7. Sketch of the signal flow iagram for the complete system. The following blocks are use: synchronous generator (SG), shunt capacitor (SC), photovoltaic generator (PV), an infinite bus ( ). Signals entering the network (ashe lines) are subject to rerouting. Parameters of the infinite bus are v, = V an v q, = v 0, = 0 V. Moreover, ω s = π50 ra/s is use to efine the frequency of the unifie reference frame. Simulation parameters for photovoltaic generators are given in Table. Further we perform a small-signal stability analysis of an interconnecte system, which is one in several steps: first the system s operating point is compute by solving the system power flow equations an converting the resulting complex voltages an currents to q0 quantities. Next, unit moels are linearize in the neighborhoo of this operating point, an the overall system is escribe using state equations. We start with the nominal case an use the initial values provie in Tables an. An array of Boe plots representing the frequency responses of the linear moel is illustrate in Fig. 8 for P m, P an P m, P input output pairs. These figures emonstrate that synchronous machines are weakly couple at low frequencies but nonetheless affect each other at a certain resonance frequency of about 0 ra/s. P = 65. Q = 76. P = 0.6 Q = 50. P = 9. Q =. P = 7.,Q =.7 P = 54.6 Q = 6. P =. Q = 8. P = 0.0 Q = 0.0 PV P = 6.5,Q = 8.6 P =.6 Q = 8. 5 P = 4.9,Q = 9.7 P = 5.6,Q = 7.9 P = 7.8 Q = 7. 6 P = 4.5 Q = P = 45.4,Q = 4.5 P =.6 Q = 0. P =. Q = 5.6 P = 9.6 Q = 0. 7 P = 7.5 Q = PV P = 6.4 Q =.5 8 P = 0.0 Q = 0.0 P = 0.0,Q = 9.8 P = 6.4 Q =. P = 5.0 Q =.5 Table. Simulation parameters: synchronous machine Parameter SG SG Units R a Ω R f Ω L f f H L a f H J kg.m L = L q = 0.L H P m 0 60 MW SG SG Fig. 6. Single-line iagram of a 4-bus system with two synchronous machines (SG) an two renewable energy sources (PV). Values of P an Q enote the operating point (power flow solution). Table. Simulation parameters: photovoltaic generator Parameter C b v re f C sh P pv K p K i PV, PV Units µf V µf MW

7 74 Proceeings of the Estonian Acaemy of Sciences, 08, 67, 4, Fig. 8. Frequency responses in a 4-bus network. Imag Real (a) Root locus of the eigenvalues SG mech. power [MW] PV DC power [MW] Pm, Pm, Ppv,6 Ppv, Time [s] (b) Change in P m,,p m,,p pv,6 In aition, as expecte, high frequencies in the mechanical powers are filtere by the network inuctances an synchronous machines inertia. The transient process is shown in Fig. 9 for active powers. As expecte, the infinite bus reacts to the changes in power generation. Table 4 provies numeric information for the largest eigenvalues. These results are then visualize in Fig. 0(a), which shows the root locus of ominant poles for intervals of changing operating conitions. In aition, steps are epicte in Fig. 0(b). Next, the ynamic behavior is analyze uner the changing operating conitions as presente in Fig. 0(b). The nonlinear moels of units are then re-linearize in the neighborhoo of new operating points when the transient process is complete an the system is in the steay state. Time instances are selecte as t {.9,.9,5.9,8}. Signals are subsequently change from their initial values as follows. The mechanical powers P m, an P m, of SG an SG at buses an are increase from 0 to 80 MW at t = s an from 60 to 90 MW at t = 6 s, respectively. The DC power P pv,6 at the output of the photovoltaic array PV connecte to bus 6 is ecrease from 0 to 5 MW at t = 4 s, while P pv,8 is kept constant (0 MW). Fig. 9. Transient process of single-phase active powers. Table 4. Change of the ominant eigenvalues Eig. # Initial P m, P pv,6 P m, Fig. 0. Eigenanalysis: root locus of the ominant eigenvalues when external input signals are change. Dotte arrows inicate the irection of poles change. From Fig. 9 it can be seen that the provie changes in input powers irectly affect the steay-state conitions an cause a change of the operating point. The presente analysis may assist in unerstaning the ynamic behavior of the system. For example, from Fig. 0(a) it follows that poles move towar the origin, which means that the response of the system becomes slower with the increase of external powers. In aition, observe that a change in the DC power P pv,6 of the photovoltaic array connecte to bus 6 has a minor effect (zoome area in Fig. 0(a)) on the overall ynamics of the system, because poles are slightly move from their positions. This is ue to the fact that the photovoltaic inverter connecte to the gri is less powerful than the respective synchronous machines. Clearly, if the amount or the overall capacity of photovoltaic energy sources is increase, the effect will be more significant. This, in particular, means that the propose approach may help to estimate the amount of renewable energy prouce by PVs within the stability limits. 4.. A 00-bus network In this section, we show that the propose approach can be applie for the analysis of larger systems. In particular, consier a 00-bus system. A etaile escription an numeric values can be foun in [5] base on [0]. Moels of the transmission network, generators, an loas are constructe using []. Generators are escribe using the swing equation (6), an loas are represente base on the balance series RL impeances. The small-signal moel is obtaine similarly to the case of the 4-bus network. The complete (with attache units an loas) q0 moel has 470 states an 48 external input signals (mechanical powers). All the components are moele in the unifie reference frame rotating with ω s = π50 ra/s. The sparsity pattern of the matrix A (sparsity inex is 0.49%) an the istribution of the ominant eigenvalues are shown in Fig.. The ynamic

8 J. Belikov an Y. Levron: Moeling an analysis of power systems in a unifie reference frame 75 (a) Matrix A Imag Real (b) Distribution of eigenvalues Fig.. Sparsity pattern of the matrix A an the istribution of the ominant eigenvalues for the q0 moel of the feebackconnecte 00-bus system. behavior is further analyze uner changing operating conitions to emonstrate similarities an ifferences between quasi-static an q0 moels. Scenario (similarities): In the first test case the system is simulate uner the nominal conitions, an then the overall active power eman of all loas is increase by 5% an 5% from 9 to 56 an 009 MW, respectively. Table 5 provies numeric information, in which the first three eigenvalues with the largest real parts are shown for both quasi-static (qs) an q0 moels. Observe that both quasi-static an q moels provie almost ientical ominant poles. Scenario (ifferences): Consier first the case when the moment of inertia (J) of the synchronous machine connecte to bus 96 is change. The moment of inertia is first increase an then ecrease by 4 times from the nominal value. The ominant poles of the quasi-static an q moels are similar for higher values but iffer for a smaller moment of inertia. Table 6 provies numeric information, in which the first several eigenvalues with the largest real part are shown. Observe that the q moel preicts that the system is unstable, but the quasistatic moel fails to preict this instability. Consier now the case when several line resistances are change. In particular, we change the branch resistances for the lines connecting buses 5, 6, an First, branch resistances are increase by.5 times an then ecrease by 5 times from their nominal values. Table 7 provies the numeric information, in which the first three eigenvalues with the Table 5. Dominant poles: increase in active power consumption Moel Eig. # Nominal Step (5%) Step (5%) q qs q qs q qs Table 6. Dominant poles: change in the moment of inertia Moel Eig. # Nominal Increase Decrease q ± 67.8 j qs q qs q qs Table 7. Dominant poles: change in branch resistances Moel Eig. # Nominal Increase Decrease q ± 0.98 j qs q qs q qs largest real part are shown. Observe that although the quasi-static moel provies quite similar results for the increase case, it fails to preict instability when resistances are ecrease. This example shows that the traitional small-signal stability analysis base on quasi-static approximation of the network has to be use carefully as in some cases it may give misleaing results. 5. DISCUSSION AND CONCLUSIONS A eveloping solution to moel the ynamics of largescale power systems is to use q0 quantities, assuming that the transmission network is symmetrically configure. This approach combines two properties of interest: similar to transient moels, q0-base moels are erive from physical moels an are therefore accurate at high frequencies. In aition, similarly to timevarying phasor moels, q0 moels are time invariant. This property allows one to efine an operating point an enables small-signal analysis. As a result, q0-base analysis is expecte to be beneficial when consiering the stability of large-scale power networks that inclue a variety of istribute an renewable power sources. A current challenge is to merge various q0-base moels appearing in the literature to obtain a complete moel of a large power system. In this paper we approach this problem by representing various q0 moels in a reference frame rotating with a unifie angle. This enables irect connections between units of ifferent types an the network an provies a means to perform a smallsignal stability analysis of large-scale systems. This approach is emonstrate on the basis of two stanar units (synchronous machine an photovoltaic generator), an in aition, a moel of the transmission network base on the unifie reference frame is provie. The paper is

9 76 Proceeings of the Estonian Acaemy of Sciences, 08, 67, 4, accompanie by a free software package available in [6] that constructs the propose ynamic moels an provies tools for ynamic simulations an stability stuies base on q0 quantities. Three particular examples are presente in this paper. ACKNOWLEDGMENTS J. Belikov was supporte by the Estonian Research Council grant MOBTP6. Y. Levron was partly supporte by the Gran Technion Energy Program (GTEP). The publication costs of this article were covere by the Estonian Acaemy of Sciences. REFERENCES. Ilić, M. an Zaborszky, J. Dynamics an Control of Large Electric Power Systems. Wiley, New York, Demiray, T. an Anersson, G. Comparison of the efficiency of ynamic phasor moels erive from ABC an DQ0 reference frame in power system ynamic simulations. In The 7th IET International Conference on Avances in Power System Control, Operation an Management. Hong Kong, China, 006, 8.. Yang, T., Bozhko, S. V., an Asher, G. M. Moeling of uncontrolle rectifiers using ynamic phasors. In Electrical System for Aircraft, Railway an Ship Propulsion (ESA RS). IEEE, 0, Fitzgeral, A. E., Kingsley, C., an Umans, S. D. Electric Machinery. McGraw-Hill, New York, Ei, A. Utility integration of PV-win-fuel cell hybri istribute generation systems uner variable loa emans. Int. J. Elec. Power, 04, 6, Huang, B. an Hanschin, E. Characteristics of the ynamics of istribution electrical networks. Int. J. Elec. Power, 008, 0(9), Teoorescu, R., Liserre, M., an Roriguez, P. Gri Converters for Photovoltaic an Win Power Systems. John Wiley & Sons, Sauer, P. W., Lesieutre, B. C., an Pai, M. A. Transient algebraic circuits for power system ynamic moelling. Int. J. Elec. Power, 99, 5(5), Krause, P. C., Wasynczuk, O., Suhoff, S. D., an Pekarek, S. Analysis of Electric Machinery an Drive Systems. Wiley-IEEE Press, Schiffer, J., Zonetti, D., Ortega, R., Stanković, A. M., Sezi, T., an Raisch, J. A survey on moeling of microgris From funamental physics to phasors an voltage sources. Automatica, 06, 74, Katiraei, F., Iravani, M. R., an Lehn, P. W. Small-signal ynamic moel of a micro-gri incluing conventional an electronically interface istribute resources. IET Gener. Transm. Dis., 007, (), Levron, Y. an Belikov, J. Observable canonical forms of multi-machine power systems using q0 signals. In IEEE International Conference on the Science of Electrical Engineering. Eilat, Israel, 06, 6.. Belikov, J. an Levron, Y. A sparse minimal-orer ynamic moel of power networks base on q0 signals. IEEE Trans. Power Syst., 08, (), Levron, Y. an Belikov, J. Moeling power networks using ynamic phasors in the q0 reference frame. Electr. Pow. Syst. Res., 07, 44, Zimmerman, R. D., Murillo-Sánchez, C. E., an Thomas, R. J. MATPOWER: steay-state operations, planning, an analysis tools for power systems research an eucation. IEEE Trans. Power Syst., 0, 6(), Levron, Y. an Belikov, J. DQ0 ynamics Software manual. Technion Israel Institute of Technology, Haifa, Israel, pf. 7. Sauer, P. W. an Pai, M. A. Power System Dynamics an Stability. Prentice Hall, Upper Sale River, New Jersey, Szcześniak, P., Feyczak, Z., an Klytta, M. Moelling an analysis of a matrix-reactance frequency converter base on buck-boost topology by DQ0 transformation. In The th International Power Electronics an Motion Control Conference. Poznan, Polan, 008, Levron, Y., Canaay, S., an Erickson, R. W. Bus voltage control with zero istortion an high banwith for single-phase solar inverters. IEEE Trans. Power Electron., 06, (), Birchfiel, A. B., Xu, T., Gegner, K. M., Shetye, K. S., an Overbye, T. J. Gri structural characteristics as valiation criteria for synthetic networks. IEEE Trans. Power Syst., 07,, Levron, Y. an Belikov, J. Open-source software for moeling an analysis of power networks in the q0 reference frame. In IEEE PES PowerTech Conference. Manchester, UK, 07, 6.

10 J. Belikov an Y. Levron: Moeling an analysis of power systems in a unifie reference frame 77 Energiavõrkue ünaamiline moelleerimine ja stabiilsuse analüüs, kasutaes q0 teisenust ning ühtlustatu koorinaaisüsteemi Juri Belikov ja Yoash Levron q0 koorinaaisüsteem on muutunu väga populaarseks traitsiooniliste masinate ja väikeste energiaallikate moelleerimiseks ning juhtimiseks. Kui selle lai kasutus suuremõõtmeliste energiasüsteemie moelleerimiseks ja analüüsiks on ikka veel lahtine küsimus. Üheks tüüpiliseks q0 mueli probleemiks on lokaalse koorinaaisüsteemi, mistõttu erinevate muelite ühenamine pole alati otsene. Artiklis on välja pakutu lähenemine, kus võrk ja selle komponente on moelleeritu, kasutaes q0 teisenust, mis põhineb unifitseeritu koorinaaisüsteemil. See iee on illustreeritu, kasutaes sünkroonsete masinate ja päikesegeneraatorite muelei. Kõik mueli kasutava unifitseeritu koorinaaisüsteemi, mistõttu nee kõik saava otseselt üksteisega ühenua nii simulatsioonis kui ka analüütiliselt. Artikliga on kaasas tasuta tarkvarapakett, mis konstrueerib pakutu ünaamilise mueli.