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1 TEMPUS OF ENERGY, Course: Distribution of Power Energy Power flow analysis for distribution networks with DG Jordan Radosavljević Faculty of Technical Sciences, University of Priština in Kosovska Mitrovica, Serbia. [email protected] 1. Introduction Given all generators and loads in a system, a power flow calculation provides the voltages at all the nodes in the system. Once these voltages are known, calculating the flows in all the branches is straightforward. Power flow studies are simply the application of power flow calculations to a variety of load and generation conditions and network configurations. The efficiency of the applied power flow computation method in a radial distribution network is of an extreme importance. A fast and efficient backward/forward sweep method for the computation of power flows is applied. It is assumed that the distribution network is three-phase and balanced. 2. Branch numbering schemes The solution method used for radial distribution networks is based on the direct application of the KVL and KCL. To enhance the numerical performance of the backward/forward sweep solution method, an efficient branch numbering scheme should be used. Figure 1 shows two typical branch numbering schemes: (a) by branches and (b) by layers. These numbering schemes are very simple and straightforward. The branch numbering scheme (a) has been implemented in the power flow algorithm. The selection scheme does not affect the accuracy and speed of calculation. Fig. 1 Branch numbering schemes: (a) by branches, (b) by layers. 3. Line modeling The π equivalent circuit is most extensively used in practical applications. Figure 2 is the nominal π equivalent circuit of the line or cable, between nodes m and i, where the serial impedance 0 Z Vi and shunt admittance Y Vi . 2 Fig. 2. The π equivalent circuit of lines and cables. 0 The lines in distribution networks are short and the shunt admittance Y Vi are usually neglected in the power flow calculation. 4. Transformer modeling To model the transformer the Г equivalent is used, as shown in Figure 3. Transformers are modelled through the impedance Z Ti and the non-rated transmission ratio t 1 nt , where n is an integer which represents the tap changer position. For manually controlled transformers n is usually n 2 , and for transformers with automatic control n is usually equal to n 10 12 . t could range 0.025, 0.015 or 0,0125. In the power flow calculations, the magnetizing admittance o of the transformer Y Ti are usually neglected. Fig. 3. Equivalent model of transformer. 5. Load modeling Behavior of loads can be modeled by studying the changes in their active and reactive power requirements due to changes in system voltages, either analytically or experimentally. There are three well established models for loads representation in power system studies: Constant power: for which load active power and reactive power are assumed to be constant irrespective of bus load voltage; which means they are usually voltage independent. P P0 , Q Q0 (1) Constant current: in which the active and reactive powers are assumed to be proportional to the load bus voltage. Usually it represents the combination of constant impedance and constant power loads. 3 V P P0 , V0 V Q Q0 V0 (2) Constant impedance: this is the traditional representation, in which, at system nominal frequency, the active and reactive consumed powers are assumed to be proportional to the square of supply voltage. The load is represented by a constant admittance added to the system admittance matrix at the concerned load node. 2 V P P0 , V0 V Q Q0 V0 2 (3) where V0 represents the initial operating voltage and P0, Q0 are active and reactive power corresponding to the initial operating voltage. 5.1 Uniformly distributed load lumped model Fig. 4 shows the general configuration of the exact model of uniformly distributed loads. The node at one-fourth of the way from the source end has been presented as a dummy bus. Fig. 4. Exact lumped load model of uniformly distributed loads 6. Distributed generation modeling The ways of connection of DG units to the grid are summarized as: Wind turbines: The grid connected wind turbines are divided into fixed and variable speed groups. In the first group, a propeller through a gear box rotates the rotor of a squirrel cage induction generator, which is directly connected to the grid. In the second group, a doubly fed induction generator or a synchronous generator, either permanent magnet or conventional one, is used. The AC output power of these units is converted via a power electronic based rectifier and inverter to grid compatible AC power. Fuel cells: Fuel cells by electro-chemical process convert directly the stored chemical energy in the fuel to electrical and thermal energy without any electric machine. The output DC power of the fuel cell is converted via an inverter to grid compatible AC power. 4 Photovoltaic systems: Photovoltaic systems convert solar energy into electricity and like fuel cells; their output DC power is converted via an inverter to grid compatible AC power. Internal combustion engines: These units convert chemical energy derived from liquid or gas fuels into mechanical one. Then it rotates either a synchronous or an induction generator which is directly connected to the grid. Gas turbines: These units convert the potential energy saved in fossil fuels from chemical to heat and then heat to mechanical. Afterwards, it rotates a synchronous generator that is directly connected to the grid. Micro-turbines: These units work like gas turbines. The only difference is to rotate a highspeed permanent magnet synchronous generator. Hence, the generator is connected to the grid via power electronic interface devices. In accordance with the units mentioned above, the primary energy of DG units may be injected to the grid via either a synchronous or an asynchronous electric machine which is directly connected to the grid, a combination of an electric machine and a power electronic interface, or only via a power electronic interface. If the electric machine is directly connected to the grid, its operation determines the model of DG for power flow studies. In other cases, the characteristics of the interface control circuit determine the DG model. These models are extracted in following. Induction generator model: Generally, in an induction generator both active and reactive powers are functions of slip: P P V , s Q Q V , s (4) where P and Q are produced active and reactive power, respectively, s is the slip of induction generator speed, and V is the bus voltage. Assuming P is constant and neglecting the very low dependency of reactive power to the slip, the expression (4) can be reduced as follows: P Ps const. Q f V (5) The expressed model by (5), which is so-called Static Voltage Characteristic Model (SVCM), is an appropriate model of squirrel cage induction generator for PFSs. Since the bus voltages are near 1.0 p.u. in steady state cases, squirrel cage induction generator can be modeled as a PQ bus for simplicity. P Ps const . Q Qs const (6) Synchronous generator model: Depending on the excitation system, synchronous generators are divided into two categories: (A) with regulating excitation voltage and (B) with fixed excitation voltage. The first one can also be subdivided into: (A1) voltage control mode (constant terminal voltage) and (A2) power factor control mode (fixed power factor). The DGs in sub-categories A1 and A2 are modeled by PV and PQ nodes, respectively. Consider an example of a wound rotor synchronous generator to model the fixed excitation voltage synchronous generators (category B). The following equation describes it: 5 Eq Q Xd 2 V2 P 2 Xd (7) where, P and Q are the active and reactive power of the distributed generator respectively, Eq is the no-load voltage and maintains constant, Xd is the synchronous reactance, V is the generator terminal voltage. Assuming P is constant: P const. Q f V (8) This expression is similar to (5), but Q is positive in expression (8), which means that synchronous generators without excitation voltage regulation may inject reactive power to the grid. Thus, the SVCM may be used as model of synchronous generators without excitation voltage regulation too. Power electronic interfaces: Generated power of photovoltaic systems, fuel cells, microturbines and some wind units are injected to the grid via power electronic interfaces. In such cases, the DG model in PFSs depends on the control method which is used in the converter control circuit. As a general rule, in case the control circuit of the converter is designed to control P and V independently, the DG model shall be as a PV node and when it is designed to control P and Q independently, the DG model shall be as a PQ node. 7. Power flow algorithm The iterative algorithm for solving the radial system consists of four steps. These steps at iteration k are: 1. Initialization: Let the root node be the slack node with known voltage magnitude and angle, and let initial voltage for all other nodes be equal to the root node voltage. Vi 0 1; θi0 0; i 0,1, ... , N (9) 2. Backward sweep to sum up line section current: starting from the end line section and moving towards the root node, the current in line section i, in according with Fig. 5 is: k k k k 1 J i I Pi I Ci Y i V i o J k ; i N, N-1, ... , 0; k 1,2, ... i i Fig. 5. A part of the distribution network. (10) 6 k k where: I Pi are current injections at node i corresponding to power load; I Ci are current k are currents in line section connected injections at node i corresponding to capacitor; J to node i; Y i are sum of shunt admittances at node i; i set of line sections connected to o k 1 node i; V i are voltage at node i. 3. Forward sweep to update nodal voltage: starting from the first node and moving towards the last node, the voltage at node i, in according with Fig. 5 is: k k k Vi V m Zi Ji (11) k k where: Z i is the impedance of branch i; V m , V i k are voltages at nodes m and i; J i is the current in line i. 4. The voltage mismatches calculations: after above three steps are executed, the voltage mismatch at each node (say node i) are calculated as below: Vi k Vi ( k ) Vi ( k 1) , i 0,1, ... , N (12) If any of these voltage mismatches is greater than a convergence criterion, the three steps are repeated until convergence is achieved. 7.1 Incorporating DG units in power flow algorithm Appreciation of DG in the power flow algorithm is achieved by modifying the expression (10), according to Figure 6, k k k k k 1 J i I Pi I Ci I DGi Y i V i o k J ; i N, N-1, ... ,0; k 1,2,... (13) i i k where I DGi are current injections at node i corresponding to DG. Fig. 6. A part of the distribution network with DG connected at node i. The DG units, which are modeled as PQ nodes can be treated as negative PQ loads in power flow algorithm without any problem: 7 k I DGi sp sp PDGi jQDGi V * k 1 i sp sp PDGi const; QDGi const; ; (14) sp sp , Q DGi where PDGi are produced (specified) active and reactive power of the DG connected at node i. However, handling PV nodes in the power flow algorithm requires some additional processes. It is to be noted that the generator terminal voltage is typically controlled by the specification of the scheduled voltage magnitude. So, for a PV node, the active power output and voltage magnitude of the generator are specified. In order to handle PV nodes in a power flow algorithm, the backward/forward sweeps are performed considering them as negative PQ loads. When the power flow is converged the following three steps are done: 1. Calculate voltage magnitude mismatch for all PV nodes k Vi k V i V i sp sp where V i (15) is the specified voltage magnitude for node i. If any of these mismatches is greater than a threshold, then perform the next step: 2. Calculate current injections at node i corresponding to DG, k I DGi sp k 1 PDGi jQDGi * k 1 Vi (16) k The reactive power injection QDGi required to maintain the voltage at the generator bus i on the specified magnitude, can be calculated using: 1 k (k) (k 1 ) QDGi QDGi Im V i Z PVi V i V i sp sp * (17) where: QDGi is the reactive power injection of the DG connected at node i; V i is the calculated voltage at node i, V i Vi i V sp i Vi sp i ; ; sp Vi is the specified voltage at node i, Z PVi is equal to the sum of the complex impedances of all line sections between the PV node i and the root node (substation bus). In case there are n PV nodes in the distributin network, the reactive power injections at these PV nodes are determined by vector equation: 1 (k 1 ) Q(k) DG Q DG Im V Z PV V V sp sp k * (18) where: Q DG QDG1 , QDG 2 ..., QDGn is the vector of the reactive power injections at PV nodes; V is the vector of the calculated voltages at PV nodes; V sp is the vector of the specified voltages at PV nodes; Z PV is the PV node sensitivity matrix. The dimension of Z PV is equal to the number of PV nodes. The diagonal entry, Z PVii , in Z PV is equal to the sum of the complex impedances of all line sections between PV node i and the root node (substation bus). If two PV nodes, i 8 and j, have completely different paths to the root node, then the off-diagonal entry Z PVij is zero. If i and j share a piece of common path to the root node, then Z PVij is equal to the sum of the complex impedances of all line sections on this common path. 3. QDGi then is compared with the reactive power generation limits. If QDGi is within the limits, ie., min k max QDGi QDGi QDGi then the corresponding DG currents, are injected to PV node i according to (16). In subsequent iterations, these currents will be combined with other nodal current injections. Otherwise, if QDGi violates any reactive power generation limit, it will be set to that limit, and combined with the reactive load at this node. Subsequently, the rows and columns in the PV node sensitivity matrix, Z PV , corresponding to this node are removed and the LU factors of Z PV are updated. The iteration described in steps 1-3 will continue until the voltage magnitude mismatches for all PV nodes as calculated in (15) become less than a threshold. 9 8. Example The described power flow algorithm is applied on a radial distribution network shown in Fig. 7. Data for branch and loads of the distribution network are reported in Table 1 and Table 2, respectively. Fig. 7. Distribution network. Table 1. Branch data. Branch 0–1 1-2 2-3 3-4 4-5 4–6 3–7 3–8 2–9 1 – 10 R (p.u.) 0.000963 0.001073 0.006409 0.005106 0.007506 0.002467 0.003005 0.001143 0.002760 0.002106 X (p.u.) 0.003219 0.000673 0.004609 0.002876 0.004229 0.001390 0.002612 0.000994 0.002399 0.001187 B t 0 0 0 0 0 0 0 0 0 0 0.9875 Table 2. Load data. Node Type 0 1 2 3 4 5 6 7 8 9 10 BAL PQ PQ PQ PQ PV PQ PQ PQ PQ PQ V (p.u.) 1 / / / / 1.01 / / / / / PG (p.u.) / 0 0 0 0 2 0 0 0 0 0 QG (p.u.) / 0 0 0 0 (-1 ÷1) 0 0 0 0 0 Pp (p.u.) / 0 0.522 0.936 0.336 0 0.477 0.672 0.207 1.116 0.882 QP (p.u.) / 0 0.174 0.312 0.112 0 0.159 0.224 0.069 0.372 0.294 Sbase = 1 МVA, Vbase = 23 kV. Table 3 shows the results for two cases: (а) With DG at node 5, (б) Without DG at node 5. 10 Table 3. Results of the power flow calculations. (а) With DG at node 5 (б) Without DG at node 5 Node Voltages: -------------------------Node V theta [r.j.] [deg] 1.0000 1.0035 -0.4811 2.0000 1.0000 -0.4738 3.0000 0.9909 -0.2749 4.0000 0.9957 0.0387 5.0000 1.0100 0.5711 6.0000 0.9943 0.0230 7.0000 0.9883 -0.3382 8.0000 0.9906 -0.2823 9.0000 0.9960 -0.5688 10.0000 1.0013 -0.5055 11.0000 1.0000 0 Node Voltages: -------------------------Node V theta [p.u.] [deg] 1.0000 1.0018 -0.8551 2.0000 0.9962 -0.9320 3.0000 0.9746 -1.3156 4.0000 0.9696 -1.3735 5.0000 0.9696 -1.3735 6.0000 0.9681 -1.3900 7.0000 0.9720 -1.3811 8.0000 0.9743 -1.3233 9.0000 0.9921 -1.0277 10.0000 0.9996 -0.8796 11.0000 1.0000 0 Currents in Branch: -------------------------Branch !I! fi [p.u.] [deg] 1.0000 3.7047 -30.7409 2.0000 2.8023 -34.6261 3.0000 1.2275 -56.9930 4.0000 1.2314 -160.6165 5.0000 1.9836 -176.0484 6.0000 0.5057 -18.4119 7.0000 0.7167 -18.7732 8.0000 0.2203 -18.7173 9.0000 1.1811 -19.0037 10.0000 0.9285 -18.9404 11.0000 3.7516 -30.7409 Currents in Branch: -------------------------Branch !I! fi [p.u.] [deg] 1.0000 5.5178 -19.5966 2.0000 4.5877 -19.6538 3.0000 2.8497 -19.7889 4.0000 0.8847 -19.8181 5.0000 0 0 6.0000 0.5194 -19.8249 7.0000 0.7288 -19.8160 8.0000 0.2239 -19.7582 9.0000 1.1857 -19.4626 10.0000 0.9301 -19.3145 11.0000 5.5876 -19.5966 Power Flow: -------------------------Branch P Q [r.j.] [r.j.] 1.0000 3.2113 1.8735 2.0000 2.3190 1.5732 3.0000 0.6675 1.0169 4.0000 -1.1568 0.4061 5.0000 -2.0000 0.1181 6.0000 0.4770 0.1590 7.0000 0.6720 0.2240 8.0000 0.2070 0.0690 9.0000 1.1160 0.3720 10.0000 0.8820 0.2940 11.0000 3.2245 1.9177 Power Flow: -------------------------Branch P Q [r.j.] [r.j.] 1.0000 5.2346 1.7761 2.0000 4.3282 1.4669 3.0000 2.6343 0.8801 4.0000 0.8137 0.2714 5.0000 0 0 6.0000 0.4770 0.1590 7.0000 0.6720 0.2240 8.0000 0.2070 0.0690 9.0000 1.1160 0.3720 10.0000 0.8820 0.2940 11.0000 5.2640 1.8741 Power Loss in Branch: -------------------------Branch ploss qloss [p.u.] [p.u.] 1.0000 0.0407 0.1359 2.0000 0.0253 0.0159 3.0000 0.0290 0.0208 4.0000 0.0232 0.0131 5.0000 0.0886 0.0499 6.0000 0.0019 0.0011 7.0000 0.0046 0.0040 8.0000 0.0002 0.0001 9.0000 0.0116 0.0100 10.0000 0.0054 0.0031 11.0000 0 0 Total active power loss: 0.230 [p.u.] Total reactive power loss: 0.254 [p.u.] Power Loss in Branch: -------------------------Branch ploss qloss [p.u.] [p.u.] 1.0000 0.0902 0.3015 2.0000 0.0678 0.0425 3.0000 0.1561 0.1123 4.0000 0.0120 0.0068 5.0000 0 0 6.0000 0.0020 0.0011 7.0000 0.0048 0.0042 8.0000 0.0002 0.0001 9.0000 0.0116 0.0101 10.0000 0.0055 0.0031 11.0000 0 0 Total active power loss: 0.350 [p.u.] Total reactive power loss: 0.482 [p.u.] 11 References [1] N. Jenkins, R. Allan, P. Crossley, D. Kirscher and G. Strbac, Embedded Generation, IET, London, United Kingdom, 2008. [2] Shirmohammadi, H.W. Hong, A. Semlyen and G. X. Luo, A Compensation-Based Power Flow Method for Weakly Meshed Distribution and Transmission Networks, IEEE Transactions on Power Systems, Vol. 3, No. 2, 1988, pp. 753-762. [3] C.S. Cheng and D. Shirmohammadi, A Three-Phase Power Flow Method for Real-Time Distribution System Analysis, IEEE Transactions on Power Systems, Vol 10, No. 2, 1995, pp. 671-679. [4] S.M. Moghaddas and E. Mashhour, Distributed Generation Modeling for Power Flow Studies and a Three-Phase Unbalanced Power Flow Solution for Radial Distribution Systems Considering Distribution Generation, Electric Power Systems Research, Vol. 79, 2009, pp. 680– 686. [5] H.E. Farag, E.F. El-Saadany, R. El Shatshat and A. Zidan, A Generalized Power Flow Analysis for Distribution Systems with High Penetration of Distributed Generation, Electric Power Systems Research, Vol. 81, 2011, pp. 1499–1506.