Archive for December, 2012

The Electrical Power Grid

December 25, 2012

The  Electrical Power Grid

L V Nagarajan

Mr. Contractor and Mr. Engineer are great friends. They are running a reasonably successful construction business for the last twenty years. They decided to construct houses for their own use, but in a small village far away from maddening crowd. The village they chose, did not even have electricity supply as yet. They decided to start construction of their houses adjacent to each other. They had a small diesel generator for construction purposes. After completing the construction they promptly applied for electricity connection. Due to lack of infrastructure, the same was delayed for more than two years. However they wanted to live in their new houses, hence they commissioned two  diesel generators, one for each house and connected the same to their individual main switch boards. They both moved to their new houses

Mr. Contractor started the generator and switched on his lights, fans and heaters etc and as he switched them on, one by one, he observed the whine of the rotating generator getting louder and shriller. But there were other changes happening in the machine which he could not observe. As he put on one appliance after the other, the extra load slowed down the rotating speed of the machine. The speed governor sensed this speed drop and picked up the speed to normal by automatically increasing the fuel input. Just as the speed of the machine is an indicator of generation-load balance, speed of the machine also decides the frequency of AC-power output. The frequency should be maintained close to 50 Hz, as in all AC systems. As the load on the generator kept varying with the use of different appliances, the speed/frequency also followed the changes to maintain the generation-load balance.

There was a time once, when Mr. Contractor’s gen-set developed a problem and got shut down. He suffered a loss of power. He immediately called his neighbor Mr. Engineer, to look into the problem. Mr. Engineer’s gen-set was up and running. He suggested to connect his neighbor’s load also on to his machine, at least till the faulty machine is repaired. He installed a pair of cables to jumper the outputs of both the machines, but took care to put it through a circuit breaker. He requested Mr. Contractor to switch on only essential loads to keep the load on the single machine within its limits. Mr. Engineer checked the faulty machine and found it was a minor fault. He repaired the same and started the machine. He took the machine in service. But he forgot to remove the jumper cable. Then he decided to leave the same on, to increase the reliability of power supply to both of them. Thus was born a GRID, a Power Grid.

When both gen-sets operated as a GRID, they enjoyed the increased reliability and stability of power supply, but they also experienced a few problems:

i) Any load changes in one system affected both the gen-sets.

ii) The response of the two speed-governors being slightly different, there were oscillations in actual power sharing between the two gen-sets.

iii) The power flow through the jumper cable (normally called Tie-line in GRID terminology) was varying widely, some time very close to its full capacity.

Mr. Engineer being very smart, introduced some changes in the governor systems as below:

a) He made the speed governors less sensitive by introducing a speed-load droop response in their systems.

b) He introduced a secondary frequency-control equipment in both the gen-sets, which responds based on ‘Speed/Frequency plus Tie-line flow’, as seen from respective systems. This secondary control helped individual systems in keeping generation/load balance on their own systems. (This secondary control is called Tie-line Bias Control in GRID terminology).

Under normal conditions, the GRID was operating quite well. they derived following advantages:

– Any minor restrictions on the individual gen-sets could be easily managed

– Any gen-set could be released for routine maintenance easily, with only minor load restrictions

– Voltage and frequency were better regulated, even during load surges and drops

There were also a few new problems:

– lack of proper accounting of Tie-line power flows either way, which could enable proper sharing of energy/fuel expenses.

– Mr. Contactor’s gen-set was inherently more prone to frequent failures. This created problems for Mr. Engineer also. Can he be compensated in any way?

– Once, Mr. Contractor’s gen-set-1 started dropping load gradually due to some mechanical problem. But Mr. Contractor ignored to take action and did not switch of his non-essential loads. This lack of discipline resulted in tripping of both the gen-sets giving a total shut down to the GRID.

– Of course there were other times when Gen-set-1 one failed instantly with the same result. Mr. Contractor could not do anything to prevent this and Mr. Engineer had to accept it.

They together discussed these problems and decided on the following solutions:

1. Tie line power flow will be metered either way and the net flow of energy from either system to the other will be billed to the receiver at an agreed rate.

2. The above is fine for inadvertent exchanges of power, happening due to system dynamics. For scheduled assistance from one system to the other, the charges could be agreed at a higher rate.

3. For unscheduled short-term emergency assistance of power, the charges could be agreed at a still higher rate.

4. Proper discipline should be followed during abnormal situations, to keep the loads to be within the limits of  ‘own generation plus agreed assistance’.

5. They also agreed to commission automatic load shedding (on under frequency and reverse power) and system separation (i.e. tripping the tie-line) in cases where such load-generation balance could not be achieved by self-discipline.

Both Mr. Engineer and Mr. Contractor (somewhat reluctantly) agreed and implemented these ideas. Surely, there were frequent bickering between the two, but the GRID basically worked well. Before the GRID could be affected by this bickering, fortunately they received the electrical connection from the power utility. Yes, now they are connected to a common grid managed by the utility. Mr. Contractor’s gen-set was disposed off. Mr. Engineer’s gen-set was retained as a stand-by supply till their connections were fully stabilized.

Though the above GRID was a small one, the principles of grid operations are the same even when two bigger systems are connected as a super GRID. To ensure that these principles are always upheld, usually a separate GRID operation department, (or a Load Dispatch Centre) is established, common and neutral to all the systems interconnected. It monitors the grid operation on a 24×7 basis and instructs the component systems to adhere to the principles of grid operation.  Failure by one or many of these connected systems to adhere to these principles will lead to major system black-outs, as occurred in the Northern Grid of India on 31 July 2012, which affected 500 million people. But out of these large population, even 0.01% would not have understood what exactly led to this shut down. With this note, I sincerely hope, at least 0.02% may now understand how any lack of discipline as described above could lead to failure of GRID operation and eventually lead to a black-out. (which means at least 50,000 people should read this blog !). I have kept this write-up intentionally simple to help even non engineers to understand.

(Refer: http://en.wikipedia.org/wiki/2012_northern_India_power_grid_failure).

25 Dec 2012

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Sine of an angle – by Hindu Maths

December 3, 2012

Sine of an angle – by Hindu Maths

L V Nagarajan

Aryabhata (AD 476-550) was the first in the line of great mathematician-astronomers known to us from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Āryabhaṭīya and the Arya-siddhanta. The following stanza in Āryabhaṭiya gives a series of 24 numbers and calls them as Ardha-Jya differences.

मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||

Makhi Bhakhi Phakhi Dhakhi Nnakhi Nyakhi

Ngakhi Hasjha Skaki Kishga Sghaki Kighva

Ghlaki Kigra Hakya Dhaki Kicha Sga

Jhasa Ngava Kla Ptha Pha Cha kala-ardha-jyas

When decoded into numbers it reads thus:

225, 224, 222, 219, 215, 210; 

                               205, 199, 191, 183, 174, 164

154, 143, 131, 119, 106, 93;  

                               79, 65, 51, 37, 22, 7 : Ardha Jya Differences.

You may imagine a bow. The string tying the two ends of the bow is called Jya (or rope) in Sanskrit. (This sanskrit word Jya, for rope, is the root for Geometry, Geology, Geography, as we know them now). Jya-ardha is half of this length.

Please see the figures below. In Figure-1, ABC is an arc of a circle. AC is its Jya. AM is ardha-Jya or half-Jya. It is seen clearly that half-Jya, AM  is nothing but Sine of angle AOB, multiplied by radius OA.

Considering the Figure-2 above, all the vertical steps from bottom to top add up to respective Jya-ardhas or half-jyas of the increasing angles. In the above diagram, the angles are in steps of 15 degrees. Hence, Step 1 = R Sin 15; step1+ Step2 = R Sin 30; Step 1 +2 +3 = R Sin 45 and so on up to 90 degrees. These steps are called half-Jya differences. The above sanskrit verse gives the step-sizes or half-Jya differences for 24 steps of 3.75 degrees each to add up to 90 degrees. Thus it gives a table RSines for 0 to 90 degrees in steps of 3.75 degrees, with R= R sin90 = Sum of all steps 1 to 24 = 3438. These values are found to be highly accurate with the present day values of Sines, as shown in the table and chart given at the end.

The genius of Aryabhata defined length-MB (refer to Figure-1) as Utkrama-Jya, reverse-Sine or Versine. Aryabhata proposed accumulation of the above Jya-differences in the reverse order to get the successive Utkrama-Jyas, as it is obvious from the above step-diagram where steps are symmetrical about 45 degrees. Hence Koti-Jya or (Cos x) was defined by Aryabhata as (1 – Utkrama-Jya).

Aryabhata has actually devised an algorithm to develop this Sine table. The second section of Āryabhaṭiya, titled Ganitapāda, contains the following stanza indicating a method for the computation of the sine table.

rasi lipthashtamo bhaga: prathamam jya-arda muchyathe

thath dwibhakta labdhon mishritham thath dwitheeyakam

aadyenaivam kramaath pindaan bhaktwa labdhon samyutha:

khandaka: syu: chaturvimsa jya-ardha pinda: kramadami.

There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse (by Katz) wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.

“When the second half-chord partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.”

With my highly limited knowledge of Sanskrit, I can guess a few parts of the above Sutra.

Rasi – 12th part of a full circle (30 degrees).

Ashtomo Bhaga – 8th part (of Rasi), 3.75 degrees

prathamam – First; Jya-ardha – Sine; Muchyate – obtained

Dwibhakta – Double or  Add to itself;

Labdhon – Profit, Dividend, Quotient;

Mishritam – together;  Dwiteeyakam -The second;

aadyenaivam – In a similar way

kramath – Successive; Pindan – difference

chaturvimsa – twenty four; Jya-ardha Pinda – Sine differences;

Kramadami; recursively

Now let me try to give a context-based English translation of the above Sutra:

Arc of an eighth of a rasi gives you the first Jya;

That doubled and divided – together gives the second;

Same way successive Jya-differences, together with quotient,

recurrently give all the twenty four Jya-differences.

There are 12 Rasis in Earth’s trajectory around the Sun. Hence each Rasi is 30 degrees. A quadrant arc of a circle subtends an angle of 90 degrees at the centre, or 3 rasis of 30 degrees each. 8th part of rasi will be 3.75 degrees. There will be twenty four such 3.75 degree arc-sectors in a quadrant, totaling to 90 degrees. An arc (or a bow) of 2 x 3.75 degrees will have a Jya (or a rope). Half of this, is Jya-ardha and corresponds to Sine of 3.75 degrees; which will be same as this arc of 3.75 degrees, as per the Sutra. Yes, in modern mathematics, it only means Sin(x) = x, when the angle is as small as 3.75 degrees. Evidently, a basic circle of certain radius must have been considered for this purpose and hence Jya really means Rsin(x). For some reason (to be explained later) the first Jya is taken as 225 :– 4 x 24 x 225 gives, the total circumference of the circle considered as 21,600, with corresponding radius of 3438.  The value of Jya of the first angle having been initiated as 225, the Jyas of all the twenty four angles can be found following the above Sutra. Interestingly the Sutra above gives different rules for the first two Jyas and prescribes the recursive rule only from Jya3. (Jya is used synonymously with Jya-ardha, which actually represents Sine).

Jya1 = Jya-Diff 1 = 225; Jya2 = (225+225) – (225/225) = 449; Jya-Diff 2 = 224

Jya-Diff 3 (as per the Sutra) =  Jya-Diff 2 – Jya2/225 = 224 – (449/225) = 222.0044; Jya3 = 671.0044

Jya-Diff 4 =  Jya-Diff 3 – Jya3/225 = 222.0044 – (671.0044/225) = 219.0222; Jya4 = 819.0266

The whole table of 24 Jya-diff’s, as developed by using the above sutra is given in a table below. Even this table gives good values for RSines, though not as accurate as the earlier table as could be seen from the plot.

Now let us check this recursive calculation

Difference, D(n) = Sin (n+1)x – Sin nx

= Sin nx Cos x + Cos nx Sin x – Sin nx

Similarly, D(n -1) = Sin nx – Sin (n -1)x

= Sin nx – Sin nx Cos x + Cos nx Sin x

D(n) – D(n-1) = 2 Sin nx Cos x – 2 Sin nx = -2 Sin nx (1 – cos x)

We know,  Jya(n) = J(n) = R Sin nx

Hence Jya Difference, DJ(n) = R * D(n)

Similarly, DJ(n -1) = R * D(n-1)

Hence, DJ(n) – DJ(n-1) = R * [D(n) – D(n-1)] = – 2 J(n) (1- cos x)

i.e., DJ(n) – DJ(n-1) = -J(n)/K,

K being a constant and equals [1/2(1-cosx)]

With x = 3.75 degrees, It works out that K = 233.5374.

If we start the recursive process with J(1) = K = 233.5374, we will get a very accurate table of Jyas, but with R = K x 48/Pi = 3568. However with values of Pi and (Cos x) as available in ancient Hindu period, values of 225 and 3438 for K and R were good enough, as can be seen from the plot below. Or was there a reason to choose these values? The whole circle is 360 degrees. Each degree can be divided into 60 minutes. Now the whole circle is 360×60 = 21600 minutes. Hence perhaps, the radius of the circle was taken as 21600/2Pi = 3438. An arc of 3.75 degrees will be 225 units long.

The ancient Hindus however knew the exact values of Jya for angles of 30, 45, 60 and 90 degrees and Aryabhata could have very well used them to apply corrections to the above table as required. Hence the value of R may not be very critical. Aryabhatta’s table of Sines, given earlier, is the corrected and improved version of the table as developed by his own formula, and hence is much closer to actual values, especially at 30, 45, 60, 90 degrees.

The whole table looks as below:

As per the recursive fromula

Aryabhata’s  Final Table

Actual

Angle   x

Quotient   Jx/225

 Ardha Jya Diff RSine(n)  Jya-n  Ardha Jya Diff RSine(n)  Jya-n

R=3438   RSine(n)

0

0.00     0   0 0

1

3.75 1.0000 225 225 225 225 224.86
2 7.50 1.9956 224 449 224 449

448.75

3

11.25 2.9822 222.004 671.004 222 671 670.72

4

15.00 3.9557 219.022 890.027 219 890 889.82
5 18.75 4.9115 215.067 1105.093 215 1105

1105.11

6

22.50 5.8455 210.155 1315.248 210 1315 1315.67
7 26.25 6.7536 204.309 1519.558 205 1520

1520.59

8

30.00 7.6316 197.556 1717.114 199 1719 1719.00
9 33.75 8.4757 189.924 1907.038 191 1910

1910.05

10 37.50 9.2822 181.449 2088.486 183 2093

2092.92

11

41.25 10.0473 172.166 2260.653 174 2267 2266.83

12

45.00 10.7679 162.119 2422.772 164 2431 2431.03
13 48.75 11.4405 151.351 2574.123 154 2585

2584.83

14

52.50 12.0624 139.911 2714.033 143 2728 2727.55
15 56.25 12.6306 127.848 2841.882 131 2859

2858.59

16 60.00 13.1427 115.218 2957.099 119 2978

2977.40

17

63.75 13.5963 102.075 3059.174 106 3084 3083.45

18

67.50 13.9896 88.479 3147.653 93 3177

3176.30

19 71.25 14.3206 74.489 3222.142 79 3256

3255.55

20

75.00 14.5880 60.168 3282.310 65 3321

3320.85

21 78.75 14.7906 45.580 3327.891 51 3372

3371.94

22

82.50 14.9275 30.790 3358.681 37 3409

3408.59

23 86.25

14.9980

15.862 3374.543 22 3431

3430.64

24 90.00

15.0018

0.864 3375.407 7 3438

3438.00

The above is a complete table of Sines as per ancient Hindu mathematicians. The last column gives the value of Rsine, (i.e. Jya) as calculated using the current accurate values. The closeness of the values can be observed in the following chart.

Dedication:

When I was researching for this blog, I came across the works of late Sri T S Kuppanna Sastri, an expert in Sanskrit and Ancient astronomy. I was naturally feeling proud, since I have met him about 30 years back. He is an uncle of my wife and he is the father of my friend Dr. T K Balasubramanian, a retired scientist of BARC. Sri Kuppanna Sastri was a professor of Sanskrit and Astronomy in several colleges. His renowned major works are two books namely Pañcasiddhāntikā of Varāhamihira and Vedāṅga jyotiṣa of Lagadha.   I dedicate this blog to the memory of late Sri T S Kuppanna Sastri.

References:

1. http://en.wikipedia.org/

2. History of Ancient Indian Mathematics, C N Srinivasiengar, The World Press Private Ltd. Calcutta (1967)

LVN/26 Nov 2012