This paper presents a newly developed proximity indicator for voltage stability assessment which can be used to predict critical real system load and voltages at various load buses at critical loading point. The proximity indicator varies almost parabolic with total real load demand and reaches orthogonally to real load axis. This relation has been utilized to predict critical loading point. It has been shown that two operating points are needed for estimating critical point and proper selection of operating points and variation of proximity indicator near collapse point highly affect the accuracy of estimation. Simulation is based on load flow equations and system real and reactive loadings have been increased in proportion with base case scenario for IEEE 14 and IEEE 25 bus test systems to demonstrate the behaviour of proposed proximity indicator. CPF has been used as benchmark to check the accuracy of estimation.
Transmission of bulk power over long distances using existing network has forced power systems to operate closed to their extreme capabilities. This causes heavy stress even in mature systems and leads to voltage instability. It is always required to have control over voltage in system operation especially when the system is stressed. Voltage stability problem is load driven and considered as a major threat to power system security. Static voltage stability analysis is used to assess the margin to voltage instability and also to identify weak buses and lines of any system. It helps system planners and operators to identify voltage instability prone areas of the system where attention is needed and also to exercise corrective actions to maintain suitable real and reactive margins in case of heavy loading conditions. Real and reactive power transfer from generators to load buses is the main focus of voltage stability analysis. Continuous monitoring of the system status and fast and accurate determination of proximity to voltage collapse point is a necessary requirement to prevent voltage instability.
In static analysis, power flow simulations have been carried out with various loading conditions. The analysis checks the real and reactive power flows and voltages at various load buses when the system loading has been increased up to critical loading. Static voltage stability may be direct or indirect. P-V and V-Q curves are widely used as a fundamental tool to assess the voltage stability margin (VSM) at any operating point [
Load flow Jacobian which is available at the end of Newton Rapson power flow convergence, contains many useful information about voltage stability and hence it has been area of interest for many researchers. It is well established that Jacobian becomes singular at collapse point [
In order to have fast and accurate estimation of voltage collapse point, many researchers have proposed various voltage stability indices. A voltage stability index with inverse Jacobian matrix has been proposed in [
Continuous monitoring of the system status and fast and accurate determination of proximity to voltage collapse point is a necessary requirement to prevent voltage instability. Various researchers have proposed proximity indicators based on load flow Jacobian [
The objective of the presented research is to develop a methodology for estimation of critical loading point based on a new proximity index and to investigate accuracy of proposed algorithm.
In static analysis, solution of these of linearised load flow equation is obtained by iterative techniques. It provides a solution in stable operating region. Ill-conditioning of Jacobian based linearised load flowequation results in falling to obtain solution in the stable region. This fact has been utilised for the development of a load flow equations based proximity indicator and the proposed algorithm to investigate voltage stability.
Consider a set of n linear homogenous equations for
Such that
If
In
Excluding
Now,
If
And
The substitution error function
It can be seen that if
The concept is applied to load flow Jacobian
Jacobian is computed in N-R load flow solution when the load flow is converged to final solution.
Let
At solution point, we get linearised power flow equation as
When the system is stressed up to critical loading, one of the eigen value of load flow jacobian becomes zero and the Jacobian becomes singular.
The substitution error function
Hence this function has been taken as proximity indicator for system wise assessment of voltage stability.
As
As discussed earlier, the magnitude of function
Let us assume that two system loading conditions are available in stable equilibrium region
Further, the
The developed algorithm has been used to estimate the critical loading point for IEEE 14 bus and IEEE 25 bus test systems [
The first operating condition is taken as half of base case loading to demonstrate the importance of selection of two loading conditions for enhancing accuracy of estimation of SNBP.
Load bus no. | PLS | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.045 | 4.0900 | 4.4990 | 4.7035 | 4.9080 | 5.1125 | 5.3170 | 5.5215 | 5.7260 | 5.9305 | 6.1350 | 6.3395 | 6.5440 | |
PI | |||||||||||||
3.2800 | 2.7982 | 2.6719 | 2.6032 | 2.5302 | 2.4518 | 2.3678 | 2.2769 | 2.1779 | 2.0687 | 1.9466 | 1.8077 | 1.6453 | |
4 | 0.9491 | 0.8826 | 0.8658 | 0.8568 | 0.8473 | 0.8372 | 0.8266 | 0.8152 | 0.8030 | 0.7898 | 0.7754 | 0.7595 | 0.7414 |
5 | 0.9541 | 0.8941 | 0.8791 | 0.8710 | 0.8626 | 0.8536 | 0.8442 | 0.8342 | 0.8234 | 0.8118 | 0.7992 | 0.7852 | 0.7695 |
6 | 0.9742 | 0.9415 | 0.9334 | 0.9290 | 0.9245 | 0.9197 | 0.9146 | 0.9093 | 0.9036 | 0.8975 | 0.8909 | 0.8837 | 0.8756 |
7 | 0.9541 | 0.8941 | 0.8791 | 0.8710 | 0.8626 | 0.8536 | 0.8442 | 0.8342 | 0.8234 | 0.8118 | 0.7992 | 0.7852 | 0.7695 |
8 | 0.9762 | 0.9453 | 0.9375 | 0.9333 | 0.9289 | 0.9243 | 0.9194 | 0.9142 | 0.9087 | 0.9028 | 0.8963 | 0.8893 | 0.8815 |
9 | 0.9463 | 0.8765 | 0.8592 | 0.8499 | 0.8402 | 0.8299 | 0.8190 | 0.8074 | 0.7951 | 0.7817 | 0.7672 | 0.7511 | 0.7330 |
10 | 0.9427 | 0.8687 | 0.8504 | 0.8405 | 0.8303 | 0.8194 | 0.8079 | 0.7957 | 0.7826 | 0.7686 | 0.7532 | 0.7363 | 0.7172 |
11 | 0.9440 | 0.8715 | 0.8534 | 0.8437 | 0.8336 | 0.8228 | 0.8114 | 0.7993 | 0.7864 | 0.7724 | 0.7571 | 0.7403 | 0.7213 |
12 | 0.9411 | 0.8649 | 0.8459 | 0.8357 | 0.8250 | 0.8136 | 0.8016 | 0.7889 | 0.7752 | 0.7604 | 0.7442 | 0.7264 | 0.7062 |
13 | 0.9388 | 0.8599 | 0.8402 | 0.8296 | 0.8186 | 0.8068 | 0.7944 | 0.7812 | 0.7671 | 0.7519 | 0.7352 | 0.7168 | 0.6959 |
14 | 0.9325 | 0.8457 | 0.8243 | 0.8128 | 0.8008 | 0.7881 | 0.7747 | 0.7604 | 0.7452 | 0.7287 | 0.7107 | 0.6908 | 0.6684 |
Load bus no. | PLS | ||||||||||||
6.6258 | 6.7076 | 6.7894 | 6.8712 | 6.9530 | 6.9939 | 7.0348 | 7.0757 | 7.1166 | 7.1575 | 7.1984 | 7.2393 | 7.2678 | |
PI | |||||||||||||
1.5714 | 1.4909 | 1.4006 | 1.3000 | 1.1844 | 1.1189 | 1.0467 | 0.9657 | 0.8728 | 0.7627 | 0.6201 | 0.4119 | 0.0011 | |
4 | 0.7334 | 0.7249 | 0.7155 | 0.7053 | 0.6940 | 0.6877 | 0.6809 | 0.6735 | 0.6652 | 0.6557 | 0.6439 | 0.6277 | 0.5995 |
5 | 0.7625 | 0.7551 | 0.7470 | 0.7382 | 0.7284 | 0.7230 | 0.7172 | 0.7108 | 0.7037 | 0.6956 | 0.6856 | 0.6719 | 0.6482 |
6 | 0.8721 | 0.8684 | 0.8644 | 0.8600 | 0.8553 | 0.8527 | 0.8499 | 0.8468 | 0.8435 | 0.8397 | 0.8351 | 0.8289 | 0.8185 |
7 | 0.7625 | 0.7551 | 0.7470 | 0.7382 | 0.7284 | 0.7230 | 0.7172 | 0.7108 | 0.7037 | 0.6956 | 0.6856 | 0.6719 | 0.6482 |
8 | 0.8781 | 0.8744 | 0.8705 | 0.8663 | 0.8616 | 0.8591 | 0.8564 | 0.8534 | 0.8501 | 0.8465 | 0.8420 | 0.8360 | 0.8259 |
9 | 0.7250 | 0.7165 | 0.7071 | 0.6970 | 0.6857 | 0.6795 | 0.6728 | 0.6654 | 0.6572 | 0.6479 | 0.6363 | 0.6205 | 0.5929 |
10 | 0.7088 | 0.6998 | 0.6900 | 0.6793 | 0.6674 | 0.6609 | 0.6538 | 0.6461 | 0.6375 | 0.6277 | 0.6155 | 0.5988 | 0.5698 |
11 | 0.7129 | 0.7039 | 0.6941 | 0.6834 | 0.6715 | 0.6649 | 0.6579 | 0.6501 | 0.6415 | 0.6316 | 0.6194 | 0.6026 | 0.5734 |
12 | 0.6972 | 0.6876 | 0.6772 | 0.6658 | 0.6531 | 0.6460 | 0.6385 | 0.6302 | 0.6209 | 0.6103 | 0.5971 | 0.5790 | 0.5472 |
13 | 0.6867 | 0.6769 | 0.6661 | 0.6543 | 0.6412 | 0.6340 | 0.6262 | 0.6176 | 0.6080 | 0.5971 | 0.5834 | 0.5648 | 0.5319 |
14 | 0.6585 | 0.6478 | 0.6362 | 0.6235 | 0.6094 | 0.6016 | 0.5932 | 0.5839 | 0.5736 | 0.5617 | 0.5470 | 0.5267 | 0.4910 |
As discussed, the proposed proximity indicator has been used to estimate the critical loading point of the system.
As shown in
Load bus no. | PLS | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.6500 | 7.3000 | 8.7600 | 10.9500 | 12.7750 | 13.5050 | 14.6000 | 14.9650 | 15.3300 | 15.6950 | 16.0600 | 16.4250 | 16.7900 | 17.1550 | 17.5200 | |
PI | |||||||||||||||
4.0485 | 3.7592 | 3.6241 | 3.3931 | 3.1658 | 3.0631 | 2.8921 | 2.8298 | 2.7643 | 2.6954 | 2.6224 | 2.5448 | 2.4621 | 2.3731 | 2.2769 | |
6 | 0.9877 | 0.9745 | 0.9690 | 0.9603 | 0.9528 | 0.9498 | 0.9451 | 0.9435 | 0.9419 | 0.9403 | 0.9386 | 0.9370 | 0.9354 | 0.9337 | 0.9320 |
7 | 0.9871 | 0.9716 | 0.9645 | 0.9531 | 0.9425 | 0.9381 | 0.9311 | 0.9287 | 0.9262 | 0.9237 | 0.9211 | 0.9185 | 0.9158 | 0.9131 | 0.9104 |
8 | 0.9861 | 0.9693 | 0.9618 | 0.9496 | 0.9384 | 0.9336 | 0.9262 | 0.9236 | 0.9210 | 0.9183 | 0.9156 | 0.9128 | 0.9100 | 0.9071 | 0.9042 |
9 | 0.9811 | 0.9593 | 0.9497 | 0.9342 | 0.9202 | 0.9143 | 0.9052 | 0.9020 | 0.8988 | 0.8956 | 0.8923 | 0.8889 | 0.8855 | 0.8820 | 0.8785 |
10 | 0.9868 | 0.9711 | 0.9641 | 0.9527 | 0.9424 | 0.9381 | 0.9313 | 0.9290 | 0.9266 | 0.9242 | 0.9217 | 0.9192 | 0.9167 | 0.9141 | 0.9115 |
11 | 0.9858 | 0.9690 | 0.9614 | 0.9492 | 0.9381 | 0.9334 | 0.9261 | 0.9236 | 0.9210 | 0.9184 | 0.9158 | 0.9131 | 0.9104 | 0.9076 | 0.9047 |
12 | 0.9862 | 0.9696 | 0.9621 | 0.9499 | 0.9388 | 0.9340 | 0.9266 | 0.9241 | 0.9215 | 0.9188 | 0.9161 | 0.9134 | 0.9106 | 0.9077 | 0.9048 |
13 | 0.9874 | 0.9740 | 0.9684 | 0.9598 | 0.9523 | 0.9492 | 0.9446 | 0.9430 | 0.9414 | 0.9398 | 0.9382 | 0.9365 | 0.9349 | 0.9333 | 0.9316 |
14 | 0.9697 | 0.9343 | 0.9184 | 0.8923 | 0.8682 | 0.8579 | 0.8414 | 0.8356 | 0.8297 | 0.8237 | 0.8175 | 0.8111 | 0.8045 | 0.7978 | 0.7908 |
15 | 0.9690 | 0.9333 | 0.9175 | 0.8918 | 0.8681 | 0.8580 | 0.8419 | 0.8363 | 0.8306 | 0.8247 | 0.8187 | 0.8125 | 0.8061 | 0.7996 | 0.7928 |
16 | 0.9757 | 0.9479 | 0.9356 | 0.9156 | 0.8973 | 0.8894 | 0.8770 | 0.8727 | 0.8682 | 0.8637 | 0.8590 | 0.8542 | 0.8493 | 0.8442 | 0.8389 |
17 | 0.9874 | 0.9722 | 0.9653 | 0.9540 | 0.9437 | 0.9393 | 0.9324 | 0.9301 | 0.9277 | 0.9252 | 0.9227 | 0.9202 | 0.9176 | 0.9149 | 0.9122 |
18 | 0.9852 | 0.9678 | 0.9600 | 0.9473 | 0.9357 | 0.9308 | 0.9230 | 0.9204 | 0.9176 | 0.9148 | 0.9120 | 0.9091 | 0.9061 | 0.9031 | 0.9000 |
19 | 0.9956 | 0.9897 | 0.9868 | 0.9819 | 0.9773 | 0.9752 | 0.9720 | 0.9708 | 0.9696 | 0.9684 | 0.9672 | 0.9659 | 0.9645 | 0.9631 | 0.9617 |
20 | 0.9849 | 0.9683 | 0.9611 | 0.9497 | 0.9394 | 0.9350 | 0.9281 | 0.9257 | 0.9233 | 0.9208 | 0.9182 | 0.9155 | 0.9128 | 0.9100 | 0.9070 |
21 | 0.9752 | 0.9475 | 0.9354 | 0.9157 | 0.8977 | 0.8899 | 0.8776 | 0.8733 | 0.8688 | 0.8642 | 0.8594 | 0.8545 | 0.8494 | 0.8440 | 0.8385 |
22 | 0.9665 | 0.9285 | 0.9117 | 0.8843 | 0.8588 | 0.8478 | 0.8301 | 0.8238 | 0.8174 | 0.8107 | 0.8037 | 0.7965 | 0.7889 | 0.7810 | 0.7727 |
23 | 0.9794 | 0.9558 | 0.9454 | 0.9282 | 0.9122 | 0.9052 | 0.8940 | 0.8901 | 0.8860 | 0.8818 | 0.8774 | 0.8729 | 0.8682 | 0.8632 | 0.8580 |
24 | 0.9604 | 0.9147 | 0.8942 | 0.8603 | 0.8283 | 0.8142 | 0.7914 | 0.7832 | 0.7748 | 0.7660 | 0.7568 | 0.7471 | 0.7370 | 0.7263 | 0.7150 |
25 | 0.9647 | 0.9233 | 0.9045 | 0.8731 | 0.8431 | 0.8298 | 0.8080 | 0.8002 | 0.7921 | 0.7836 | 0.7748 | 0.7654 | 0.7556 | 0.7452 | 0.7341 |
Load bus no. | PLS | ||||||||||||||
17.8850 | 18.2500 | 18.6150 | 18.9800 | 19.3450 | 19.7100 | 20.0750 | 20.4400 | 20.5130 | 20.5276 | 20.5349 | 20.5386 | 20.5393 | 20.5407 | 20.5411 | |
PI | |||||||||||||||
2.1719 | 2.0560 | 1.9263 | 1.7772 | 1.6020 | 1.3853 | 1.0911 | 0.5555 | 0.3110 | 0.2077 | 0.4103 | 0.0952 | 0.0846 | 0.0614 | 0.0551 | |
6 | 0.9303 | 0.9286 | 0.9269 | 0.9252 | 0.9217 | 0.9199 | 0.9181 | 0.9177 | 0.9176 | 0.9176 | 0.9175 | 0.9175 | 0.9175 | 0.9175 | 0.9175 |
7 | 0.9075 | 0.9046 | 0.9017 | 0.8986 | 0.8923 | 0.8890 | 0.8854 | 0.8846 | 0.8844 | 0.8843 | 0.8843 | 0.8843 | 0.8843 | 0.8842 | 0.8842 |
8 | 0.9012 | 0.8981 | 0.8950 | 0.8917 | 0.8851 | 0.8815 | 0.8778 | 0.8769 | 0.8767 | 0.8766 | 0.8766 | 0.8766 | 0.8765 | 0.8765 | 0.8765 |
9 | 0.8749 | 0.8712 | 0.8675 | 0.8637 | 0.8558 | 0.8517 | 0.8474 | 0.8465 | 0.8463 | 0.8462 | 0.8461 | 0.8461 | 0.8461 | 0.8461 | 0.8461 |
10 | 0.9088 | 0.9060 | 0.9033 | 0.9004 | 0.8946 | 0.8916 | 0.8884 | 0.8878 | 0.8876 | 0.8876 | 0.8875 | 0.8875 | 0.8875 | 0.8875 | 0.8875 |
11 | 0.9019 | 0.8989 | 0.8959 | 0.8928 | 0.8865 | 0.8832 | 0.8797 | 0.8789 | 0.8788 | 0.8787 | 0.8786 | 0.8786 | 0.8786 | 0.8786 | 0.8786 |
12 | 0.9018 | 0.8988 | 0.8957 | 0.8925 | 0.8858 | 0.8823 | 0.8786 | 0.8777 | 0.8775 | 0.8774 | 0.8774 | 0.8774 | 0.8774 | 0.8773 | 0.8773 |
13 | 0.9299 | 0.9282 | 0.9265 | 0.9248 | 0.9213 | 0.9195 | 0.9177 | 0.9173 | 0.9172 | 0.9172 | 0.9172 | 0.9172 | 0.9172 | 0.9172 | 0.9172 |
14 | 0.7836 | 0.7761 | 0.7684 | 0.7604 | 0.7434 | 0.7343 | 0.7249 | 0.7230 | 0.7226 | 0.7225 | 0.7224 | 0.7224 | 0.7224 | 0.7223 | 0.7223 |
15 | 0.7859 | 0.7786 | 0.7712 | 0.7634 | 0.7470 | 0.7382 | 0.7290 | 0.7272 | 0.7268 | 0.7267 | 0.7266 | 0.7266 | 0.7266 | 0.7265 | 0.7265 |
16 | 0.8335 | 0.8279 | 0.8221 | 0.8161 | 0.8033 | 0.7965 | 0.7893 | 0.7879 | 0.7876 | 0.7875 | 0.7874 | 0.7874 | 0.7874 | 0.7874 | 0.7874 |
17 | 0.9095 | 0.9067 | 0.9038 | 0.9009 | 0.8947 | 0.8915 | 0.8880 | 0.8872 | 0.8870 | 0.8869 | 0.8868 | 0.8868 | 0.8868 | 0.8868 | 0.8868 |
18 | 0.8968 | 0.8935 | 0.8902 | 0.8867 | 0.8792 | 0.8751 | 0.8704 | 0.8692 | 0.8688 | 0.8686 | 0.8685 | 0.8685 | 0.8685 | 0.8684 | 0.8684 |
19 | 0.9602 | 0.9586 | 0.9570 | 0.9552 | 0.9514 | 0.9492 | 0.9464 | 0.9455 | 0.9453 | 0.9451 | 0.9450 | 0.9450 | 0.9450 | 0.9449 | 0.9449 |
20 | 0.9040 | 0.9007 | 0.8973 | 0.8937 | 0.8853 | 0.8800 | 0.8725 | 0.8697 | 0.8687 | 0.8681 | 0.8677 | 0.8677 | 0.8676 | 0.8674 | 0.8673 |
21 | 0.8326 | 0.8263 | 0.8196 | 0.8124 | 0.7952 | 0.7841 | 0.7674 | 0.7611 | 0.7587 | 0.7572 | 0.7562 | 0.7562 | 0.7560 | 0.7555 | 0.7554 |
22 | 0.7639 | 0.7545 | 0.7444 | 0.7332 | 0.7066 | 0.6890 | 0.6618 | 0.6514 | 0.6474 | 0.6448 | 0.6431 | 0.6431 | 0.6427 | 0.6419 | 0.6417 |
23 | 0.8526 | 0.8467 | 0.8405 | 0.8338 | 0.8178 | 0.8076 | 0.7926 | 0.7871 | 0.7851 | 0.7838 | 0.7830 | 0.7830 | 0.7828 | 0.7823 | 0.7822 |
24 | 0.7028 | 0.6897 | 0.6755 | 0.6596 | 0.6205 | 0.5937 | 0.5504 | 0.5329 | 0.5260 | 0.5216 | 0.5187 | 0.5187 | 0.5180 | 0.5165 | 0.5161 |
25 | 0.7222 | 0.7093 | 0.6952 | 0.6795 | 0.6403 | 0.6132 | 0.5689 | 0.5508 | 0.5436 | 0.5390 | 0.5360 | 0.5360 | 0.5353 | 0.5338 | 0.5334 |
Loading point no. | PLS | V14 | Sε | PLC | % error PLC | V14C | % error V14C |
---|---|---|---|---|---|---|---|
1 | 2.0450 | 0.9325 | 3.2800 | 9.5578 | −30.8762 | 0.6136 | −19.2655 |
2 | 4.0900 | 0.8457 | 2.7982 | ||||
2 | 4.0900 | 0.8457 | 2.7982 | 8.7253 | −19.4778 | 0.6032 | −17.2334 |
3 | 4.4990 | 0.8243 | 2.6719 | ||||
3 | 4.4990 | 0.8243 | 2.6719 | 8.5275 | −16.7691 | 0.5978 | −16.1821 |
4 | 4.7035 | 0.8128 | 2.6032 | ||||
4 | 4.7035 | 0.8128 | 2.6032 | 8.4016 | −15.0449 | 0.5958 | −15.8009 |
5 | 4.9080 | 0.8008 | 2.5302 | ||||
5 | 4.9080 | 0.8008 | 2.5302 | 8.2598 | −13.1036 | 0.5926 | −15.1879 |
6 | 5.1125 | 0.7881 | 2.4518 | ||||
6 | 5.1125 | 0.7881 | 2.4518 | 8.1490 | −11.5858 | 0.5891 | −14.5056 |
7 | 5.3170 | 0.7747 | 2.3678 | ||||
7 | 5.3170 | 0.7747 | 2.3678 | 8.0326 | −9.9916 | 0.5848 | −13.6655 |
8 | 5.5215 | 0.7604 | 2.2769 | ||||
8 | 5.5215 | 0.7604 | 2.2769 | 7.9254 | −8.5241 | 0.5817 | −13.0657 |
9 | 5.7260 | 0.7452 | 2.1779 | ||||
9 | 5.7260 | 0.7452 | 2.1779 | 7.8177 | −7.0496 | 0.5764 | −12.0369 |
10 | 5.9305 | 0.7287 | 2.0687 | ||||
10 | 5.9305 | 0.7287 | 2.0687 | 7.7156 | −5.6507 | 0.5716 | −11.0941 |
11 | 6.1350 | 0.7107 | 1.9466 | ||||
11 | 6.1350 | 0.7107 | 1.9466 | 7.6210 | −4.3557 | 0.5661 | −10.0287 |
12 | 6.3395 | 0.6908 | 1.8077 | ||||
12 | 6.3395 | 0.6908 | 1.8077 | 7.5312 | −3.1260 | 0.5603 | −8.8956 |
13 | 6.5440 | 0.6684 | 1.6453 | ||||
13 | 6.5440 | 0.6684 | 1.6453 | 7.4755 | −2.3636 | 0.5557 | −8.0004 |
14 | 6.6258 | 0.6585 | 1.5714 | ||||
14 | 6.6258 | 0.6585 | 1.5714 | 7.4452 | −1.9482 | 0.5513 | −7.1565 |
15 | 6.7076 | 0.6478 | 1.4909 | ||||
15 | 6.7076 | 0.6478 | 1.4909 | 7.4040 | −1.3840 | 0.5490 | −6.7149 |
16 | 6.7894 | 0.6362 | 1.4006 | ||||
16 | 6.7894 | 0.6362 | 1.4006 | 7.3800 | −1.0563 | 0.5445 | −5.8307 |
17 | 6.8712 | 0.6235 | 1.3000 | ||||
17 | 6.8712 | 0.6235 | 1.3000 | 7.3525 | −0.6799 | 0.5405 | −5.0591 |
18 | 6.9530 | 0.6094 | 1.1844 | ||||
18 | 6.9530 | 0.6094 | 1.1844 | 7.3333 | −0.4163 | 0.5369 | −4.3485 |
19 | 6.9939 | 0.6016 | 1.1189 | ||||
19 | 6.9939 | 0.6016 | 1.1189 | 7.3214 | −0.2531 | 0.5343 | −3.8565 |
20 | 7.0348 | 0.5932 | 1.0467 | ||||
20 | 7.0348 | 0.5932 | 1.0467 | 7.3097 | −0.0931 | 0.5307 | −3.1473 |
21 | 7.0757 | 0.5839 | 0.9657 | ||||
21 | 7.0757 | 0.5839 | 0.9657 | 7.2990 | 0.0531 | 0.5277 | −2.5579 |
22 | 7.1166 | 0.5736 | 0.8728 | ||||
22 | 7.1166 | 0.5736 | 0.8728 | 7.2896 | 0.1817 | 0.5233 | −1.7021 |
23 | 7.1575 | 0.5617 | 0.7627 | ||||
23 | 7.1575 | 0.5617 | 0.7627 | 7.2782 | 0.3388 | 0.5183 | −0.7453 |
24 | 7.1984 | 0.5470 | 0.6201 | ||||
24 | 7.1984 | 0.5470 | 0.6201 | 7.2716 | 0.4287 | 0.5107 | 0.7443 |
25 | 7.2393 | 0.5267 | 0.4119 | ||||
25 | 7.2393 | 0.5267 | 0.4119 | 7.2676 | 0.4834 | 0.4918 | 4.4121 |
26 | 7.2676 | 0.4918 | 0.0012 |
Estimation of critical loading point has been done with the help of proposed Proximity indicator. As shown in
Loading point no. | PLS | V14 | Sε | PLC | V14C | % error PLC | % error V14C |
---|---|---|---|---|---|---|---|
1 | 3.6500 | 0.9604 | 4.0485 | 30.1356 | 0.6288 | 47.2694 | 19.2916 |
2 | 7.3000 | 0.9147 | 3.7592 | ||||
2 | 7.3000 | 0.9147 | 3.7592 | 27.9842 | 0.6243 | 36.7556 | 18.4352 |
3 | 8.7600 | 0.8942 | 3.6241 | ||||
3 | 8.7600 | 0.8942 | 3.6241 | 26.5047 | 0.6195 | 29.5256 | 17.5340 |
4 | 10.9500 | 0.8603 | 3.3931 | ||||
4 | 10.9500 | 0.8603 | 3.3931 | 25.0437 | 0.6132 | 22.3860 | 16.3303 |
5 | 12.7750 | 0.8283 | 3.1658 | ||||
5 | 12.7750 | 0.8283 | 3.1658 | 24.2119 | 0.6074 | 18.3209 | 15.2335 |
6 | 13.5050 | 0.8142 | 3.0631 | ||||
6 | 13.5050 | 0.8142 | 3.0631 | 23.5939 | 0.6041 | 15.3009 | 14.6139 |
7 | 14.6000 | 0.7914 | 2.8921 | ||||
7 | 14.6000 | 0.7914 | 2.8921 | 23.1643 | 0.5990 | 13.2014 | 13.6401 |
8 | 14.9650 | 0.7832 | 2.8298 | ||||
8 | 14.9650 | 0.7832 | 2.8298 | 22.9419 | 0.5996 | 12.1145 | 13.7588 |
9 | 15.3300 | 0.7748 | 2.7643 | ||||
9 | 15.3300 | 0.7748 | 2.7643 | 22.7444 | 0.5960 | 11.1494 | 13.0795 |
10 | 15.6950 | 0.7660 | 2.6954 | ||||
10 | 15.6950 | 0.7660 | 2.6954 | 22.5260 | 0.5938 | 10.0822 | 12.6582 |
11 | 16.0600 | 0.7568 | 2.6224 | ||||
11 | 16.0600 | 0.7568 | 2.6224 | 22.3200 | 0.5904 | 9.0754 | 12.0164 |
12 | 16.4250 | 0.7471 | 2.5448 | ||||
12 | 16.4250 | 0.7471 | 2.5448 | 22.1335 | 0.5891 | 8.1643 | 11.7696 |
13 | 16.7900 | 0.7370 | 2.4621 | ||||
13 | 16.7900 | 0.7370 | 2.4621 | 21.9316 | 0.5863 | 7.1775 | 11.2262 |
14 | 17.1550 | 0.7263 | 2.3731 | ||||
14 | 17.1550 | 0.7263 | 2.3731 | 21.7501 | 0.5840 | 6.2905 | 10.8025 |
15 | 17.5200 | 0.7150 | 2.2769 | ||||
15 | 17.5200 | 0.7150 | 2.2769 | 21.5709 | 0.5796 | 5.4145 | 9.9603 |
16 | 17.8850 | 0.7028 | 2.1719 | ||||
16 | 17.8850 | 0.7028 | 2.1719 | 21.3987 | 0.5767 | 4.5731 | 9.4085 |
17 | 18.2500 | 0.6897 | 2.0560 | ||||
17 | 18.2500 | 0.6897 | 2.0560 | 21.2372 | 0.5735 | 3.7839 | 8.8001 |
18 | 18.6150 | 0.6755 | 1.9263 | ||||
18 | 18.6150 | 0.6755 | 1.9263 | 21.0677 | 0.5687 | 2.9558 | 7.8837 |
19 | 18.9800 | 0.6596 | 1.7772 | ||||
19 | 18.9800 | 0.6596 | 1.7772 | 20.9272 | 0.5641 | 2.2691 | 7.0206 |
20 | 19.3450 | 0.6417 | 1.6020 | ||||
20 | 19.3450 | 0.6417 | 1.6020 | 20.7920 | 0.5577 | 1.6085 | 5.7964 |
21 | 19.7100 | 0.6205 | 1.3853 | ||||
21 | 19.7100 | 0.6205 | 1.3853 | 20.6714 | 0.5499 | 1.0191 | 4.3270 |
22 | 20.0750 | 0.5937 | 1.0911 | ||||
22 | 20.0750 | 0.5937 | 1.0911 | 20.5677 | 0.5352 | 0.5122 | 1.5461 |
23 | 20.4400 | 0.5504 | 0.5555 | ||||
23 | 20.4400 | 0.5504 | 0.5555 | 20.5463 | 0.5249 | 0.4077 | −0.4154 |
24 | 20.5130 | 0.5329 | 0.3110 | ||||
24 | 20.5130 | 0.5329 | 0.3110 | 20.5394 | 0.5204 | 0.3736 | −1.2626 |
25 | 20.5276 | 0.5260 | 0.2077 | ||||
25 | 20.5276 | 0.5260 | 0.2077 | 20.5410 | 0.5179 | 0.3818 | −1.7440 |
26 | 20.5349 | 0.5216 | 0.1403 | ||||
26 | 20.5349 | 0.5216 | 0.1403 | 20.5418 | 0.5162 | 0.3854 | −2.0631 |
27 | 20.5386 | 0.5187 | 0.0952 | ||||
27 | 20.5386 | 0.5187 | 0.0952 | 20.5419 | 0.5154 | 0.3862 | −2.2251 |
28 | 20.5393 | 0.5180 | 0.0846 | ||||
28 | 20.5393 | 0.5180 | 0.0846 | 20.5423 | 0.5148 | 0.3878 | −2.3277 |
29 | 20.5407 | 0.5165 | 0.0614 | ||||
29 | 20.5407 | 0.5165 | 0.0614 | 20.5428 | 0.5144 | 0.3902 | −2.4008 |
30 | 20.5411 | 0.5161 | 0.0551 |
In view of above results, it can be concluded that proposed proximity indicator is useful in estimating the maximum loadability of the system and voltages at various load buses at critical point. The approximation of Sε-PLS plots as parabolic works well if two operating points taken are very near to collapse point and preferably near to convergence limit of NR power flow program. The estimation is even fair if half of base case loading point (light loading) and operating point of last convergence are taken.
This paper has proposed a new load flow equation based proximity indicator and simulation study has been carried out with the help of developed algorithm (
For estimation of critical power and voltage two loading points are taken and computed values of proximity index at these points are used. It has been concluded from
With the work in this paper, it is concluded that:
The proximity index variation with system loading can be approximated as regular curve. Linear and parabolic plots are suitable for simple and quick estimation of critical loading point. The variation of proximity indicator near critical point and its close matching with the approximated curve is the key factor for accurate prediction of critical loading point on PV curve. Accuracy of estimation could be increased by taking two loading points near the point of extreme convergence of NR load flow program.
As the voltage stability phenomenon is load driven and depends upon how system reactive power varies with real system power, the proportionate load increase scenario as taken only give only an insight. Load P-Q characteristics can vary at each load bus and must be considered in simulation for accurate prediction with the proposed algorithm.
To avoid voltage instability, continuous system monitoring and accurate prediction of critical loading point is necessary. In this study, two operating points preferably near the point of last convergence are taken to reduce the error in estimating critical point with the help of proposed proximity indicator. Newton's divided difference formula can also be used by selected more number of points. Artificial neural networks can also be used to predict the collapse point. Load P-Q characteristics must be considered in simulation to improve the accuracy of estimation.
Voltage stability margin
Substitution error function
Proximity Indicator
Total real system demand
Total real demand at critical loading point
Critical value of ith bus
Continuation Power Flow
We are very thankful to Administration of Ujjain Engineering College Ujjain, India, Faculty and Staff of Department of Electrical Engineering, Ujjain Engineering College, Ujjain, India and RGPV Bhopal India providing us all support to complete this research work. Valuable guidance from Dr. L. D. Arya, Retd. Professor, S.G.I.T.S Indore, India is also acknowledged.