Journal of Power and Energy Engineering, 2014, 2, 361367 Published Online April 2014 in SciRes. http://www.scirp.org/journal/jpee http://dx.doi.org/10.4236/jpee.2014.24048 How to cite this paper: Gong, X., et al. (2014) Research on the Method of Calculating Node Injected Reactive Power Based on L Indicator. Journal of Power and Energy Engineering, 2, 361367. http://dx.doi.org/10.4236/jpee.2014.24048 Research on the Method of Calculating Node Injected Reactive Power Based on L Indicator Xue Gong1, Bingcheng Zhang2, Bing Kong1, Anan Zhang1, Hongwei Li1, Wei Fang1 1School of Electrical Engineering and Information, Southwest Petroleum University, Chengdu, China 2Baikouquan Oil Production Plant, Petro China Xinjiang Oilfield Company, Karamay, China Email: gxsnowswpu@163.com, 1013215627@qq.com, kongbsc@163.com, ananzhang@swpu.edu.cn, lhwmail@126.com, weifang@163.com Received Sep te mber 2013 Abstract With the power grid load increasing, the problem of grid voltage stability is increasingly promi nent, and the possibility of voltage instability is also growing. In order to improve the voltage sta bility, this paper an al yz ed how the voltage stability was influenced by different reactive power in jection based on the simplified Lindicator of online voltage stability monitoring. According to the basic differential property of the simplified Lind icator, a general and normative analytical algo rithm about reactive power optimization was deduced. The analytical algorithm can calculate the load node injected reactive power, and then the network can run in the optimal steady state on the basis of the calculat i on results. According to the simulation results of IEEE14, IEEE30, IEEE57 and IEEE118, the feasibility and effectiveness of the proposed algorithm to i m prov e voltage sta bility and reduce the risk of grid collapse were verified. Keywords Voltage St ability; LIndic ator; Injected Reactive Power; Reactive Power Optimization; Fast Calculation 1. Introduction With the extending of power grids and the increasing of electricity demands, the challenge for voltage stability is increasing. One of the challenges, the system voltage declining, can reduce the power grid transmission capacity and increase the power losses as well, and this is not conducive to the electrical equipment operation. What’s worse, it may also lead to system voltage collapse and even cause more serious accidents of the whole power grids [1] [2]. In order to ensure the safety of power grid operation and avoid voltage instability incidents, it is necessary to do some research on how to monitor voltage stability online and improve voltage stability quickly [3][5]. Voltage stability evaluation indicator achieved by the voltage stability study is a significant progress, it is an effective method to measure the system voltage stability as well as the basis for the implementation of the voltage stability control [6][9]. The proposed Lindicator in article [10] [11], which can be used in online monitoring, has attracted much attention for its accuracy, linearity, quickness, etc., and it has been applied in the actual pow
X. Gong et al. er grids. Considering that the value of reactance is far greater than that of resistance in actual power grids and the bus voltage phase is small, the Lindicator has been simplified and an LQ sensitivity analysis method has been proposed in [12]. The LQ sensitivity analysis method facilitates the quantitative analysis of voltage influ ence between different nodes. However, this method requires a large amount of computation, and its reaction speed to solve voltage instability still needs to be improved. Considering the above problems, this paper analyzed how the voltage stability was influenced by different reactive power injection based on the simplified Lindicator of online voltage stability monitoring. According to the basic differentia l property of the simplified Lindicator, a general and normative analytical algorithm about reactive power optimization was deduced. The analytical algorithm can calculate the load node reactive power injection volume, and then the network can run in the optimal steady state on the basis of the calculation results. It can ensure the safety of power grid operation and avoid voltage instability incidents. 2. Simplified LIndicator Voltage stability local Lindicator was derived by kessel, etc. according to the simplest two nodes system. Ap plying the Lindicator into common multinode systems needs dividing the nodes into two categories. One is characterized by the behavior of the PQnode which stands for a type consumer node and is defined as αL. The other comprises the generator node which may be given by PVnode or by a slack node and is defined as αG. For any load node j (j ∈ αL), the equation for local Lindicator (Lj) can be described as follows[10] [11]: ** *** 222 1 LL ij iij i corr ii ii ij ij j jjj jjj jj jjj jjj j ZS ZS VV S SSS Lvv Yv Yv Yv αα +∈∈ ≠≠ +++ + == =+=+ ⋅⋅⋅ ∑∑ (1) where Sj is nodal power of load node j, Sjcorr is equivalent power which stems from other loads in the system. Sjcorr is given by (2) In Equation (1), is the node selfadmittance and it equals to 1/Zjj. is conjugate value of selfad mittance. is the collection of load nodes in the system. Vi, Vj are the load node voltage vectors. vj is the voltage amplitude value of load node, and vj2 is equal to Vj* Vj*. is conjugate value of the mutual imped ance between load node i and j. Si is the equivalent load of the system for consumer node i. In order to ensure the voltage stability, each node j in the system must satisfy the condition Lj ≤ 1. The global Lindicator of the system is defined as follows. (3) Thereby, L ranges from 0 to 1.0. A smaller L means a more steady system. When L is close to 1.0, the system tends to critical stable state. In order to guarantee system stability, the value of L must be less than 1.0, and the difference between L value and 1.0 can be used as the voltage stability margin of the system. Equation (1) is a complex expression with complex operation, and the computation will increase sharply with the expansion of power grids. Because the value of reactance is far greater than that of resistance in actual power grids, and bus voltage phase is small, etc., the literature [12] has proposed a simplified Lindicator without considering the reactance and voltage phase. Based on Equation (1), Equation (4) can be obtained. (4) According to Equation (4), Equation (5) can be introduced.
X. Gong et al. * * 22 1 () 1 1 =1 L L ij i ii ij j j ij i ii ij jj ZS Vrejecti o n v L ZS Vfg vv α α ∈ ≠ ∈ ≠ + = + ∑ ∑ (5) According to the simplified method, Equation (6) can be obtained. L L i ii ij ii ji i ii ij ii ji f fv f QX g gv g PX α α ∈ ≠ ∈ ≠ = = = = − ∑ ∑ (6) where Pi and Qi represent the active power injection and reactive power injection respectively. Xji is the reac tance between load node i and j. Therefore, the Equation (7) is obtained. 222 2 11 11() () LL i jiiji jia ia jji i ij ij QX PX L fg vvv v ∈∈ ≠≠ − = +=−+ ∑∑ (7) 3. Optimal Reactive Power Compensation The initial value of Lindicator is set in Equation (8). 22 1 1 ()() LL i jii ji jori ia ia ji i ij ij QX PX Lvv v ∈∈ ≠≠ − =−+ ∑∑ (8) When reactive power compensation is needed, the injected reactive power is set as △Qi. Then the reactive power is updated to Qi + △Qi, and the relationship between reactive power injection and Lj can be described as Equation (9). 22 () 1 1 ()() LL iijii ji jia ia ji i ij ij QQ XPX Lvv v ∈∈ ≠≠ +∆ − =−+ ∑∑ (9) When the first order partial derivative function value of local indicator Lj is set to zero, the extreme value of Lj is obtained. As Lj is a convex function, the extreme value is the minimum. The power grid can run in optimum steady state with the minimum Lj. The first order partial derivative function is described as follows. 1 j j j m L Q L QL Q ∂ ∂∆ ∂= = ∂∆ ∂ ∂∆ 0 (10) where m is the number of load nodes.
X. Gong et al. Because there are usually more than one node in the power grid, solving the extreme value needs to be carried out repeatedly. To simplify the computation and facilitate the understanding, the value of Lj corresponding to each node is added and the sum is set to Lsum, then multiple minima (Lj, j = 1, 2,…, m) can be converted into one minimum (Lsum) [13]. Its expression is as follows. (11) 22 1 () 1 [1()() ] LL miijii ji sum j iaia ji i ij ij QQ XPX Lvv v =∈∈ ≠≠ +∆ − =−+ ∑∑ ∑ (12) with () , LL iijii ji ia ia ii ij ij QQ XPX v vv µ ∈∈ ≠≠ +∆ − = = ∑∑ (13) Therefore, there is 1 22 1 1 1 22 1 1 L L j sum ja j j sum sum jm mja jm jm X Lvv Q L QLX Qvv µ µν µ µν ∈ ≠ ∈ ≠ ∂−⋅ ⋅ + ∂∆ ∂ == ∂∆ ∂ −⋅ ⋅ ∂∆ + ∑ ∑ (14) When the value of Equation (14) is equal to zero vector, it can be converted into matrix form as Equation (15). (15) (16) 1 1 1 0 0 m m m X v X v = K (17) Finally, get (18) 4. Simulation Results According to the above derivation, the IEEE14, IEEE30, IEEE57, and IEEE118 systems have been tested on matpower platform, and the simulation results are shown as below. In Figure 1, with the original load of IEEE14 standard model as a benchmark and one fifth of the original load as the step length, the load changes from 0.2 times to 4.2 times of the original load. The curve L in Figure 1 is the global Lindicator value without reactive power compensation, while Lc is global Lindicator value with reactive power compensation. Ls and Lsc in F igure 1 are the summation of all local Lindicator Lj (j = 1, 2,…, m) without and with reactive power compensation respectively. Contrasting curves L and Lc, as depicted in Figure 1, it can be seen that the effect of improving voltage stability is obvious, and the heavier the load, the more sig nificant the effect is, which illustrate that the proposed method is conducive to ensure the safety of powe r
X. Gong et al. Figure 1. L/Lsum of IEEE14 with load changes. grid operation and reduce the risk of grid collapse, especially when the grid voltage stability is poorer. For ex ample, corresponding to 3.8 of the horizontal axis, Lindicator value of L curve is close to 0.5, which means that the system is in so sensitive a state that a slight increase of load will augment the possibility of voltage instabili ty, while Lindicator value of L curve is about 0.3 enough to avoid voltage instability. Comparing curves L and Lc to curves Ls and Lsc, what can be concluded is that the two groups of curves are the samethe shape and the trend. Therefore, L and Lc can be enlarged as Ls and Lsc, which is more convenient for observation. In Figure 2, the IEEE30 standard model has been tested. The load changes from 0.2 times to 4.0 times of the original load. The effect of improving voltage stability is more apparent, and load power margin is also im proved. According to Figures 3(a) and (b), the conclusions obtained are similar to the above. The simulation results of the IEEE14, IEEE30, IEEE57 and IEEE118 have illustrated the universal applicability of the proposed method. However, the difference between the above models and the actual power grids can not be ignored, which makes the voltage drop proportion of model greater than that of the actual power grids. In actual power grids, the resistance is far less than the reactance and the voltage drop caused by the active load is almost negligible. If the proposed method were applied to one model more similar to the actual power grid, the results would be more ideal. 5. Conclusion This paper analyzed how the voltage stability was influenced by different reactive power injection based on the simplified Lindicator of online voltage stability monitoring. According to the basic d ifferential property of the simplified Lindicator, a general and normative analytical algorithm about reactive power optimization was de duced. The simulation results that the proposed method applied to all systems of IEEE14, IEEE30, IEEE57 and IEEE118 can improve voltage stability and reduce the risk of power grid collapse significantly and effec tively, and the heavier load, the more significant effect, have verified that the proposed method could ensure the safety of power grids operation and avoid voltage instability incidents with its universal applicability.
X. Gong et al. Figure 2. IEEE30 L/Lsum with load change. (a) (b) Figure 3. L(a)/Lsum(b) curves of 4 systems corresponding to each load changes. Funding The National Natural Science Fund Project (51107107). Acknowledgements We thank the support received from the National Natural Science Foundation of China (51107107) and the Chi nese National Key Basic Research and Development Program (973) (2013CB228203). References [1] IEEE Committee Report (19 90) Voltage Stability of Power Systems. Concepts, Analytical Tools, and Industry Expe rience, IEEE Publication, New York. [2] IEEE/CIGRE Joint Task Force on Stability Terms and Definition (2004) Definition and Classification of Power Sys tem Stability. IEEE Transactions on Power System, 19, 13871401.
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