I'm sorry, I did not mean to break your foot.  I care.  Yes, literally. Are you guys ready for this? Oh my gosh, mint condition. So how this works is, if you guys wanna be entered in the giveaway, the only way we are gonna be able to tell you if you win and contact you is you have to go and follow our Instagrams. So right now, before you watch this video, go follow me and Mariah's Instagrams. That is the only way we're gonna be able to message you guys. We are gonna be doing this giveaway at the end of November. And it could be yours. You don't even have to do anything except follow our Instagrams, comment and like our posts and that is it. Today, I am pulling the best prank that anyone ever has on their girlfriend, right? I see all these videos that are like, oh, I destroyed my boyfriend's girlfriend's phone and I surprised them with a new one. No. No! I am going to be destroying her phone. Oh, no, no, no, no. No! Ha, ha, ha, ha! And literally not replacing it. So a little background, guys. I have an identical phone. Look at that. Can any of you even tell which one is the real phone? Look at that! This one's real, this one's fake. This is just a fake phone. I can literally do whatever I want with it. But it's a replica to the iPhone X Max. This is the exact phone, just fake. So I hope you guys enjoyed this video. Make sure to give it a thumbs up. Once again, if you are new, go down and hit that subscribe button.  So Mariah's gonna be here in a little bit. She is currently driving to my house. It is snowing outside. Thank you, Chicago, for that. I had to film this intro really quick and make it short and simple, just how we like it. And I will see you all in a little bit. Peace! All right, guys, so a little change of plans. I changed into a nicer outfit because Mariah texted me about five minutes ago saying that she wants to go take pictures today. So while we're out taking pictures, I'm gonna accidentally drop her phone and then I'm just gonna take it and whip it at the ground. Be like, it's already broke, baby. I'm sorry, and whip it at the ground. I got the fake phone right here. Again, I'm gonna go hide it in my pocket until I make the swap. Hashtag team building the freaking comments, Mama Squad, every single person in here. Thank you for watching. I will see you guys in a little bit wherever we go to take photos. See you soon. What is good, Mama Squad? We are here with the one, the only Mariah Klobberrubius. Weighing in at 110 pounds. She's fierce, she is angry, but she's the kindest girlfriend in the world. And that's why I love you. No, you're not angry. You're only angry when you don't get food. Yeah, I'm actually not angry. She's literally a cookie monster. Did you tell me fall?  Oh, we better film fast then, guys. I'm in a non-safe zone right now. The morning lights are going on, babe. We are just arriving to the spot. But why are you filming? Huh? Why are you filming? Why am I filming? Because I want to make memories. We are going to be having a good time, taking pictures, nothing's getting broken. What? Huh? No, nothing. Hold on, Mariah, I have a question. What's your question? Does the back of my head look puffy?  Like this looks puffy. Like a puffer fish? Oh my God, forget it.  It's like a stick shift. Okay, instead of this, it can be like, reeeeeeeeeeeeeeee, reeeeere, reeeeere, reeeeere. Yum. Oh my gosh. You just tried pushing out my buns, stuf. Oh my God, I love touching your buns. Baby, let me touch them buns. Stop! Let me touch them buns. Knock it off. They're like magnets. Yes. Anyways, we're gonna zoom on over to the picture spot. It's you there. Woo, you're looking pretty. I think some two pictures of me are the way. We look good. I think some pictures of me are the way.  Did you just ask me to take pictures of you? Okay, I'll do that. Oh my god. Oh my god. You are you are adorable.  Yeah, now she has to see what they look like Can you take up what?  I'm actually not trying to be mean to you Work it. Oh my god bust out that hip. You got to peep it more break that back break that back Yes, that's what I'm saying. That was actually cute. Yes. That was can you take off your case for me? Your case is kind of annoying when I take pictures What? Why? Because All right. All right, babe ready Ready Mariah. All right, babe. Look pretty look pretty say cheese. Hey cheese Are you kidding me? Babe? I am so sorry. Oh Oh my bill. Are you kidding me right now? Babe? I'm sorry. I'm sorry Give me it. Let me fix it. We what broke off of it some peaceful It's literally not even turning on I can't even right. I'm sorry I did not mean to drop it. Are you kidding me right now? Babe? I'm sorry. Oh, there's a piece to it. Oh my god There's a piece Look, this isn't funny. Please tell me this is a joke, right? What I'm sorry, why? Stupid and it's not going on the video stop. Okay, right. I'm sorry Just let me see it. Let me see it. Let me see it. Why would you write this is destroyed? There's no it's not turning on the screen separated from the actual phone. I'm sorry. I did not mean to break your phone Are you serious? I'm not buying you. I'm sorry.  Why would you do that? I'm sorry. Oh my god. It shattered Oh my god Babe I'm sorry Right, I got off. I'm sorry, babe. You made me mad. I just threw come here Filming me. I'm sorry Babe, okay guys. We're gonna go take pictures anywhere and take pictures I'm waiting to book in myself a new phone as I'm sorry. I did not mean to Why would you do that? Don't cry. Why would you do that? Why would you do stop pushing me? Raya this camera's exciting Right, I'm just kidding. It's not your phone babe. That's not your phone Oh wait Mariah wait Mariah wait Mariah. This is the fake phone.       I don't think I'm being mean to Mariah I don't think I'm being harsh, but you already know this is probably one of my best pranks I have ever done to Mariah mama squad put a team bill down below I literally want to see everyone spamming the comments with how crazy that foam crack was because I don't even know How that shattered I thought it was freaking plastic. I guess not I guess I was wrong guys. She's just standing here She's been silent. What are you doing? Not talking? You're looking at the pictures of yourself. Yeah, babe Talk to your model squad talk to the people that love you My phone's my baby, too I was literally thinking cuz all the youtubers do like oh, I broke my boyfriend girlfriend's phone And I got them a new one, but no, what about I just straight up broke your freaking phone And that was it and the story Mariah's phone Yeah, we got to go clean up Mariah clean up your mess. I'm gonna go wait in the car.  We're gonna go take thumbnail be back in a sec Here we go. We're back and we did clean up the mess. I threw it in the neighboring Trash can so none of you got to worry about the class He got me good. I literally was can you one more time for the model squad repeat that phrase I will not repeat that phrase. Can you one more time one more time one more time? No, say it Say it or I only said it once and I'll only say it once because mobile squad you already know, okay? Well to repeat what she said he got me good You were crying are you kidding me?  How do you feel about this prank? I think it turned out 110,000 percent Squared great Thank you all for watching We are heading to the store right now to get the supplies for our tiktok once again Like I said in the beginning our Apple Watch giveaway only people that are gonna be following our instagrams That is the only way that we are gonna be able to contact you and that's where we're doing the poll from so every Single person that wants to be entered in this Apple Watch giveaway or any other giveaways in the future Make sure to go down below and follow me and my beautiful babies Instagram look at how cute Go follow those this video notification shout out goes to Emily Davis shout out to you Thank you for the cute comment on our YouTube channel We love every single one of you guys and we will see you in the next video model squad, but until then Peace


of the startup world is by whatever parameters, you look at the parameter of those who've succeeded to become unicorns, you look at the parameter of number of startups, you look at the ecosystem, the fact that the Entrepreneur Magazine can have an annual event and such a huge attendance because a lot of us are either already entrepreneurs or aspiring to be entrepreneurs. In India, we had eight unicorns in 2008, the most in a single calendar year. We moved to the 57th rank and Bangalore is among the top three cities. And India as a country is among the third as a country. So, we are the first largest ecosystem that gives you that it is fertile ground to be able to put the seeds. You know, about what, 25, 30 years ago, when I started and I became an entrepreneur, I left my job with the GE to start my own company. Startup was not a sexy word that it is today. Entrepreneurship was not an accepted term. Like one of my aunties, if many of you from Delhi will understand Punjabi, one of my aunties came and said, Putar Ki Oya, Kad Dita, that they sack you from your job, that is why you're saying you're setting up your own company. And any effort to explain to her that I have aspirations to do, to use my knowledge to leverage the opportunity and create something different fell on deaf ears. There was some very tough family dinner time conversations that I remember I had at that point of time and they were able to influence my parents who after the dinner turn around and said, Let's talk about it. Are you doing the right thing? Your employers are still saying don't leave because I was serving my notice period. And are you doing the right thing? Is it the right thing for you to listen to your employers and stay? So, a lot has changed in this ecosystem to where we are today. So, what is the enabler of this ecosystem? One of the big enablers of this is the incubators and the accelerators. Last year they grew by 11%. But what is important is that these incubators and accelerators by type et cetera come from both the corporate world. A lot of companies are initiating an incubator or an accelerator where they showcase their technology so that the startups start using their technologies and when the seed becomes a plant, the plant becomes a tree, and the tree becomes a huge fruit bearing tree, it is their technology that is being used. There is a vision for, there, it is for profit promotion by them to create these accelerators and incubators. And then there is a government supported aspect of it, where the government wants to create a level playing field and a ground where people can participate. Again, when I started my journey, this did not exist. Mentorship did not exist. You could hire consultants who are not really mentors who never set up an industry or a company or a startup of their own and you have to pay good money to get any advice. So this is a sea change of what has happened in this environment. So a lot of the startups, and I'll talk a little about it, are working on AI, IoT, artificial intelligence, virtual reality, a whole lot of things that are buzzwords just literally four, five years ago. And what is interesting is that approximately 38% of these accelerators are not in larger towns, they are in small towns and cities. So both the government and corporates play a crucial role acting as a catalyst to initiate that reaction. So I've taken three examples just to give you a little flavor. A corporate accelerator, the NetApp accelerator run by one of the corporates, they bring out what their focus areas are, what they offer to the accelerator. There are some grants, there are some equity participations, there are markets, there are inputs, there's technology, all of that. And they are very successful models that corporates run or private individuals or private entrepreneurs run because running an accelerator is an activity by itself. And then there is a state government incubator, the T-HUB. And I've had a chance of visiting this multiple times and it is quite fascinating as you go into the T-HUB, the sheer energy that you see there. And the first time I visited that, I came out saying, I don't even, can't even realize that this is where the government participates.  But my favorite example is the next one, which I think is one of the most fascinating steps taken by the central government of what is BIRAC. For biology and biotech and the other capabilities where they actually incubate. I am now an investor in about four companies that were incubated by BIRAC or they gave some initial grants. Four as a percentage, I have 70 plus investments, so four as a percentage is not very large, but in terms of the excitement of what these companies are doing, there is, for instance, a company that is creating human skin using 3D printing. So God forbid you get burnt and you need grafting of a skin in the world of tomorrow, they'll be able to use a 3D printer to generate that skin, which can be put on you and the early signs of that are very good. There's a company that is creating the human liver in a laboratory without the human being done and my first reaction was, why do I need a liver in a petri dish, what will I do with it? Still I realized that all pharmaceutical companies have to test their medicines to check how they impact the liver and they can then not do human tests and they could do it in a petri dish. So it's quite fascinating, from the outside one would not have thought what the possibilities are and BIRAC has identified those companies given initial grants and today I'm proud to say I work very closely with them, they are a part of the Indian Angel Network fund on which I'm one of the promoters and we are looking at biotech as the capability. So the government continues to play a role in what it is and I think this is the best example that I can tell you, $160 billion as an industry, the industry I represent, I'm fortunate to have been the past chairman of NASCARM. In my estimation, this $160 billion would not have happened in the time that it has happened, but for STPI, which is a catalyst for this, 30 years ago, I was very pleasantly surprised to learn that it was the 30th anniversary year, it was formed to promote the software experts and they played multiple roles, I mean I could go on and on, but I can tell you within the BPO industry what we did as American Express, which is what initiated the BPO industry, what we did as GE, what is Genpac today, which is the largest BPO company in the whole world out of India, or what the industry which directly employs about 1.6, 1.7 million people and indirectly employs another 4 million would not have happened, but for the support. The single window clearance that you see there is a critical part of it, their ability to bring state of art technology at a cost effective price to a startup is a key part of it, and they now have 57 centers of which 49 are in tier two, tier three cities, they have incubation centers, and they have BPO promotion schemes, but it is the evolution of this concept. STPI came into being because the industry went to the government and said there were some needs that were not fulfilled, we were a foreign exchange deficient economy at that point of time, and we proposed to the government we'll be able to create jobs and earn foreign exchange if the following was done, this entity was formed by the ministry, the IT ministry, and that is a perfect, BIRAC was formed by the health ministry, STPI was formed by the IT ministry, so it promoted those aspects, and that is what brings us to the world of today to say where we are and what are the challenges. All of us sitting in Delhi, I can't see you, but I see nobody wearing masks as yet, but I can tell you five years later, if we did nothing and we sat in this room, we'd all probably be wearing masks if it all we continue to stay in Delhi. I know at least 10 expats who are heading large multinationals representing their companies in India who have since left the country because they did not want their children to grow up in the polluted environment of Delhi. That is a serious aspect, and if you look at it, we are cities rank among the most polluted in the world, and the solution is to, hey, be able to fix it, to be able to find solutions either to clean the air or to reduce the aspects, and the vehicle pollution, which I'm just picking as a point where we, as a startup ecosystem, can work on to where it can be done. So to leave it just to the government would be wrong, it is up to us or where we take it and what we can do with it. And the innovation and legislation are a big aspect that will work on it. The government has taken the initiatives. There are already companies, but what I want to dwell on is what the startup is, and sitting within the startup ecosystem, in the last six months, the number of companies that I have met who've come up with the concept, it is not about making the vehicle, it is not about making the lithium-ion batteries only, it is about models that will make it happen. I met a startup that has a charging unit that is more cost-effective than what is imported today, that is more efficient than what is done, takes lesser time to install, and they could install it in petrol pumps, they could install it in parking lots, and their entire model is to bring that capability, and they manufacture in India and their technology, they have patented it, it's quite unique. I met another startup that is on electric scooters, I said, so you make scooters, he said, no, we don't make them, we just get them made by somebody, we have a design, said, so why would I, why would I want to invest in you? Said, here are stories, I said, yeah, I'm willing to hear. They have an entire model where they will put their scooters outside metro stations, you come out of the metro instead of hiring a cab or a auto, you can go there, flash your mobile phone, a barcode on it, it will unlock the scooter, you can take it to your office where there'll be another parking lot, you put it there, flash your mobile phone, and they will take into account how many kilometers they've run, and they'll charge you by the kilometer, you don't need to wait for autos, you don't need to do anything, and it is electric vehicle, it is eco-friendly, they do the charging, they do the maintenance, so that got me pretty excited, and then they told me how they financed it, they go to people saying each unit cost whatever, 36,000, 50,000 rupees, and they give a return to an individual investor, and they have people who come in and invested to say, okay, I'll buy 15 of these vehicles, and I'll pay you the money, and you give me a monthly return, and they then get their return by the kilometer, it's a fascinating business model.  Entrepreneurs just don't create wealth and jobs, they also play a part in solving the problem. If you solve the problem, yes, revenue will happen, and yes, profits will happen, so will investors, so will the general well-being, and that is the greatest aspect. It is not about the EV, it is about the business models around it that have yet to be found. Amitabh Khan spoke about some of the startups and gave their present business models. Four years ago, those startups did not exist, nobody knew of those business models, and what I'm telling you is nobody knows what will happen in the EV world or what can be done. The budget has brought a lot to it, we've cut the GST rate, we've said that if you take a loan to buy an EV, your interest will be deductible, and that will create an impetus for this to happen. There are other incentives in terms of the FAME II scheme, the tax rebates, the import duties for various things, so it is not just the import substitution, it is the entire business model that will help what it is. So to my mind, these are very exciting times for startups, I just use EV as an example, we have significant capabilities to address the biggest issues, pollutions, and within pollution, EV as one of the potential solutions is just an example. With the government resources behind us, which is what the budget has shown, which is what the present government has shown, it creates a huge opportunity. The question is, are we as entrepreneurs, are we as a startup community up to it? And as you guys make those companies, allow me to invest, together we'll make magic happen. Thank you very much.


We talk about buying a house which is of course a very big investment but it's also a place to raise a family to create memories or just kick back and relax. Now as a South African home loan provider, Netbank will help you with the home loan process and of course secure a bond at a competitive interest rate. Buying your home usually means getting a home loan. Today we have our financial expert LaPoy from Netbank Home Loans to take us through the process in detail and of doing the home loan application. Now LaPoy, when it comes to thinking of a home loan, it's overwhelming. Where does one start? So Zoe, I always think start as far as where would you like to live. Once you understand where you want to live then it becomes easier for you to determine whether you can afford it or not. Apart from affordability, I would also suggest that viewers would look at their credit profile with the Bureau. So Netbank has an instant bond indicator that does both an affordability and a credit check online and it gives you a real-time response as far as how much you would qualify for. I think that's the best starting point. I think once you know how much you qualify is then when you can start looking at the different types of homes you want but in order to apply for your home loan what are some of the various steps one needs to take? Once you've found that perfect home and you've signed the offer to purchase with the seller then all you need to do is to go online, fill in the online application form with your personal details as well as the property that you're buying and then it also allows you to upload certain documents. So specifically things like your payslip or your salary advice, the offer to purchase as well as sometimes any bank statements that you might require. And is there any pitfalls we need to look out for when applying for a home loan? If you have done the appropriate research so you've done your credit profile check, you've checked your affordability, you've drawn up your expenses every month then you can avoid some of the pitfalls around not being able to manage that home loan installment quite well. After moving into the home some of the things that might trip you up if you aren't prepared are things like the additional costs associated with living in your own home, things like rates and taxes every month or you know maintenance of a home that sometimes when you're renting doesn't really come into play. Those are some of the things that viewers need to be mindful of and budget accordingly for. How important is it to have a deposit when you go and apply for a home loan? I'm so glad you asked Zoe, I think it's quite important because it also shows your willingness as a buyer to the bank that you are part of this transaction with them. So saving for a deposit is quite important. I would consider looking at saving every month before you actually look for your dream home so that when it comes time to applying for your bond the bank can see that you have gone the extra mile to actually save for that home. Thank you so much for that great advice and you can also go and ask your bank or home loan sales consultant to assist when trying to get your home loan so that they can help you through the entire process and make life a little easier. Now visit www.privateproperty.co.za for more details and helpful resources. Whether it's your first apartment in the city, a trendy suburban townhouse or that dream family home on a gulf estate, there's a home for everyone on private property.


Simmons of Excelsior full attention of Calabar and Walker of Jamaica College. So Archer in lane number four from Excelsior he has the fastest time heading into the final champs 52.59 seconds. What is his state of mind like? Santamoy Brown in lane number three of Kingston College has the third fastest time heading into the final 34.01. Is that an interesting career at the high school level? Santamoy Brown of KC when he first arrived a 400 800 champion in class three and has transitioned to the 400 herders as his favorite event in the last few seasons was in the champs final last year as well. Sent off. Brown has started off well in the purple and white of Kingston College in lane number three Archer there are their bolts pretty much even at the moment between these two as they head down the back straight yep Archer so almost slipped here Brown has covered him Brown with the lead slightly ahead of Archer again Archer didn't clear that hurdle well and Brown now with a slight advantage Archer trying to close on the outside trying to get into the thick of things as well as Simmons but it's between these two Brown and Archer and Archer reaches that hurdle first again didn't clear it well but he manages to stretch that lead over Brown there is Archer and he's pressing away from Santamoy Brown Devonte Archer wins 51.58 seconds for Devonte Archer of Excelsior. Is that 47 39 speed that made the difference down the stretch Devonte Archer he's been putting it together so well this season and he comes here at the corporate area meet and delivers a telling blow always knew that Santamoy Brown would provide some amount of a challenge Santamoy Brown is such a difficult competitor


We're here at the World Telecommunication Development Conference in Dubai in the United Arab Emirates and I've got the great pleasure of being with Mirumeta Ben Yamina who is Minister of Communications Transport and Tourism Development for Kiribati. Minister, thank you very much indeed for being with us today. Thank you. I'd like to start off by talking about Kiribati and the climate change issues that are affecting you. All right. As you all know, the islands of Kiribati are very low-lying eagles and I think the first lot to encounter any impact of climate change, especially sea level rise and coastal erosion. That's the fact. And what impact is this going to have on telecoms and ICTs? I think we cannot escape from the fact that we badly need ICT infrastructure development because we are at the mercy of this impact of climate change. One example would be communicating with our people throughout, especially from island to island. And we should be able to tell them early warnings of what's going on so that people are at least prepared to make the decisions to try and protect themselves from the harsh conditions of climate change. And how serious and what time scale are we looking at in terms of this threat? Well, according to the international body of which is disseminating information on climate change throughout the world, it is taking place. And there are evidence all over the place that people are gradually being affected in terms of their own survival and livestock and their homes. And so I think we cannot say that it may take two or three weeks. No, it will happen. And the fact is it will happen over 50, 60 years. There are already evidence of that in Kiribati. And what does it mean to be living under this kind of a threat? It means we have to be prepared, resilient. We have to adapt ourselves to whatever impact that are forthcoming. And we are grateful of the aid that we have received through bilateral and other UN agencies that are providing us with the options of adapt and be able to face what consequences are coming. And how has RTU been helping Kiribati to address these issues? RTU has helped Kiribati in many ways. I must say, first of all, in the way how we should develop our ICT infrastructure. And I guess we have also received assistance in terms of regulation and policy regarding ICT development. We would want to see more of these forthcoming and especially to help us prepare ourselves and to try and establish all the necessary infrastructures on the out-towards. If that's the case, ITU to prepare us, we will be grateful. But we will continue to become a full ITU member as long as we can. And we are thankful that they have also provided us with funding telecentres on some of the out-towards, which is great. So is there a situation that you envisage where you would have to carry on business from elsewhere, so perhaps even e-government? That might take place eventually. But again, this will be years and years to come. But we need the ITU support in this whatever approach that we should be given so that we are prepared. And this may take place in the next two or three generations to come. I do hope the ITU now is the same ITU in the next two or three generations. You know what I mean? Giving the same support and perhaps help us in any other way. What do you hope to take away from this conference? It's every four years. What do you hope will be some of the outcomes? I haven't really looked at it because this is our first time to attend. But it is important that we are not left out of the program or whatever the development progress is going on. So it is equally important for us to keep on attending. And what is more important is the ITU to see our needs, the needs of the small island states. And this morning I was told that there was a special session for that in terms of broadband, the benefits of broadband for sustainable development addressing the needs of small island states. And I think we are going in the right direction, the right approach. So I would want to see more of the involvement of ITU addressing the needs of small island development states, in particular the infrastructure challenges, remoteness in terms of accessing these and so forth. So I think whatever is discussed here is important to us all, not only to developed countries, but we need to be part of the partnership that we always talk about. Minister, thank you very much indeed for being with us today. Thank you for the interview. Thank you. And thank you for watching.


The book has all these little gems and they're just tucked into this tiny little perfect package and that's part of it being atomic. Tiny little minuscule atoms combine and it can change everything right? Keep watching to find out what book I'm talking about today. It's Kyla Denigno and you're watching Kyla's reading day. So today we're talking about the book Atomic Habits. I don't have the book here in front of me. I usually like to show it to you guys, hold it, play around with it. I couldn't get it. The book is flying off the shelves but I was lucky enough to catch it on an audiobook download from the Columbus Public Library and I listened to it. So Atomic Habits is the name of the book. The author of the book is James Clear. The genre of the book is non-fiction. The theme of the book is self-improvement psychology business okay? So I'm three hours into reading the book and I have a light bulb moment. Kyla, you've been doing this one thing. It's been detrimental to you. You need to stop. You haven't wanted to stop. Here it is plain. Plainly you're listening to and you got to make the change and that thing is working in bed. I was always the person who said I never want a TV in my bedroom. I hear that it affects your relationship. You don't talk as much to your spouse. All this stuff okay? Well that has not happened for me. Having a TV in the bedroom means trying to multitask and really being unproductive at everything. So I would go to bed. I have my lap tray. I sit there and working and I'm trying to watch TV and then I'm like okay I can't focus on that. Let's turn to the food network. Okay and so I either find myself missing the show and we're winding it while I'm trying to edit a video or I'm editing the video and then I have to re-watch the entire episode again on TV because I missed that. Completely unproductive. Sit up until 1.30 in the morning. Exhausted when the kiddos up at 7 in the morning to get ready for school and eat breakfast and all this stuff. We would take the kid out to daycare, come back home. I would sit on the couch with the news. Why? I just started to enjoy TV. I had never been a really big TV watcher but then I started to really enjoy it and so I had it on everywhere and so I started reading the book and I was like I'm not doing again. That night I decided I wasn't taking it upstairs. The next day we took the kid to daycare. I came back home and I was like okay where am I going to work? Well I have this. This is a whole desk. I have typewriters to remind me that I'm supposed to be working. This is my working space. This is not just my YouTube corner. This is literally your office. Why aren't you using it? We also have this awesome living room space that isn't being used during the day because the kiddos at school, my husband's upstairs working in his office and this space is just here. It's a beautiful table that we bought for Thanksgiving. Chairs. I'm sitting here. I can do my book outlines. I can write my notes. I can be on the computer when I need to. So the next day the kiddo goes to bed and I'm like oh you know I want to edit ahead right? Oh I'm changing my habits. Let's work ahead. Let's go upstairs. I could not walk upstairs with my laptop and my tray just that quick right? Atomic habits. They will make you think about the things you're doing and you will make a change just that quick and I was like no you said you were never taking it upstairs again. You are never doing it and that's it period. James Clear says one purpose for one room. When he was working in an apartment you didn't have that luxury. You were kind of working wherever you had to be. I've certainly done that. We came from a two-bed room apartment living room space was office co-working this and that feeding all the stuff but we moved into a house in September 2022. I literally have the space to designate areas for work. I literally have the space to designate areas for sleep. So I'm reading a book and then I get to my quote. The quote that meant a big deal to me and I wrote it down. So the mark of whether you're made for a task is not whether you love it but whether you can handle the pain that comes along with the task better than other people. That was such a breath of fresh air to me because I am a podcaster. I love podcasting. I love talking to guests. I love finding guests. I love researching them. I love recording. I love editing and people come to me and they say hey Kyla can you talk to me about podcasting?  I'm like no I'm like no it's oh it's fine it's fine here here use a worksheet that I have. Oh I can't do it. It's too much work. I can handle the pain of it because this is a task that I am willing to do and I always thought well everyone gets there right? Everyone does it and that's what James Clear talks about in the book like in order to get into a flow state you have to enjoy what you're doing.  And this book Atomic Habits made me feel really grateful for my life. I just passed the one-year anniversary of being fired from a job and that was very very hard for me. I was like okay I went to law school I'm not practicing as a lawyer I'm not even working in a law firm anymore who am I? Well I can be whoever I want to be but I have to have the habits, the atomic teeny weeny weeny weeny tiny small habits in place. It made me really happy to read the book and be like okay some of my habits work some of them don't. So since you're already watching go on and subscribe to the channel. I talk about non-fiction books like Atomic Habits, I talk about fiction novels, I talk about children's books, I talk about books. I love books okay? So go on and subscribe and keep watching as I talk about Atomic Habits. There are four things that James Clear says are really important for making a habit and establishing a new habit. It was make it obvious, make it attractive, make it easy and make it satisfying. Okay and so for me the big focus was make it easy. So I've been telling myself that I was going to exercise.  It's not easy. I haven't done it. I was never going to do it. So now I told myself I told my husband we're going to take the kid out to bake here. We get back we're going to pull the car on the driveway. I'm going to go into the garage while it's empty jump rope for a minute and I've done it. I've done it for the last five days okay? So here's some clips of me jumping rope. All kinds of different outfits.   But for me make it easy meant don't even have to worry about moving the car. Do not even pull the car into the garage. It's completely empty. It's cold outside. It's raining. That has nothing to do with you. Jump rope. Then go get the car and move it in okay? So that has really helped me as well. Like the book has all these little gems and they're just tucked into this tiny little perfect package and that's part of it being a comic. Tiny little minuscule atoms combine and it can change everything right? So one of the challenges in the book James Clear asked you to write down your daily habits and then mark whether they are good, bad or neutral. So I'm listening to the book. I'm like I'm not going to do that. I don't care. My habits are fine. I get my work done. But I'm also a rule follower. I'm a lawyer and I was like okay we're gonna do it. So I did it and seeing it on paper most of my day is work focused or neutral. But the things that were bad were really bad. I don't know and I don't know when it switched but thank god I was able to switch it back. So atomic habits. Would I reread this book? Yes. I am going to put atomic habits in the special list of books that I will reread every single year along with the book Big Magic. I absolutely read that every year. In fact it's time for me to pick that back up. So thank you for hanging out with me today as I discuss the book Atomic Habits. Again I'm going to list it here on the screen. I could not get a copy of the book. The library is all sold out and now that I've read it I might buy it. Okay so get in the comments. Talk to me. Have you ever read the book Atomic Habits? Do you believe that your habits need to change?  Okay so I will see you next time on Kyla's Reading Day.


In the previous class, I have introduced you to numerical methods of solving differential equations. In this particular lecture, we will continue that string and we will do a few examples on psi lab and to illustrate the basic properties of most of the commonly used numerical methods. The examples which I will choose today are examples which we have done before. We have tried to analyze them using some intuition as well as linear analysis in some cases. Today, let us just first revise what we have done in the previous class. In the previous class, we learnt a few numerical methods like Euler method. So, let us just quickly go through this and then begin with our examples. Euler method or is also known as forward Euler method is the first method which is taught to any student studying numerical methods for integration of differential equations. It is a very simple method. In fact, if you have got a differential equation x dot is equal to x comma t this x comma looks a bit big. So, I will just write it down again x of t. In that case, we evaluate the function using this algorithm at discrete time steps spaced by h. So, we have got x k plus 1 is equal to x k. This is x x k plus h into f into x k and that is the function f evaluated at the point x k and t k. Forward Euler method or simply Euler method is having order 1 that is if your numerical solution can be expressed simply as a first order polynomial in t then Euler method will give you exact answers. Typically of course, you know that our responses are not a first order polynomials in t. We have in fact, come across sinusoids and exponential functions which are not first order polynomials. So, Euler method is likely to have some error whenever you integrate most common differential equations. It is also an explicit method in the sense to evaluate x k plus 1. You really do not have to solve any differential equations. You have to just write the if you know the previous value of x k. In one shot, you can get x k plus 1. So, you get essentially an explicit algebraic equation wherein x k plus 1 is written explicitly in terms of x k. Forward Euler is also a single step method. So, you just require one previous the value at the previous time step in order to evaluate the value at the next time step. If you look at a special class of systems that are linear systems, then one thing we can attempt to do since we know the solution of linear equations, we can compare what a numerical method gives us and what is the actual solution of a linear system. Now, we cannot do this in general for non-linear system because you cannot write down the response of a non-linear system in terms of simple functions. In contrast, linear systems for example, x dot is equal to A x. We know it is stable when A is less than 0. If A is a real number, then if A is less than 0, then it is a stable.  So, obviously the Euler method may give these two conditions in fact are not equivalent. If you look at for example, f h is greater than 2 by A, in that case even though A is less than 0, you may still have Euler method showing the system to be unstable. So, Euler method when you integrate the system may give you wrong answers. Even not only wrong answers, it may give a qualitatively wrong result. It may show a stable system to be an unstable one. So, this is basically one problem with Euler method that you will not get accurate answers and you may even get qualitatively wrong answers. Look at backward Euler method. Backward Euler method, the only difference is that the function evaluation is done at the new value of x. It is in fact done at the point which you want to evaluate. So, although backward Euler is also order 1, this is something you can prove. I am not prove this in this class. It is implicit. It is an implicit method in the sense to evaluate x k plus 1. You have to solve this equation x k plus 1 appears within this function. Therefore, you need a numerical method at every time step in order to get x k plus 1. So, this is if for example, for if x is f is a non-linear function, you will have to use some numerical method to solve algebraic equations like Newton-Raphson or Gauss-Seedle. Backward Euler also is a single step method. For a linear system, backward Euler is stable if this condition is satisfied. That is 1 upon the absolute value of 1 minus a h less than 1. This condition also is not equivalent to this condition. In fact, an interesting point which I will show you sometime later in one of the examples is that you can choose values of h. So, that an unstable system is shown as a stable system if you use backward Euler method. So, backward Euler method also can and sometimes depending on the choice of the value of h give you qualitatively wrong answers as far as stability is concerned of a linear system of this kind. Trapezoidal method on the other hand is an order 2 method. The basic algorithm is this. It takes the average of the function evaluation here at the previous point and the new point. It is an implicit method again and it is a single step method. It requires only the previous value of x in order to get the new value of x. For linear systems, trapezoidal rule says that the system is stable if this is less than 1, whereas the actual system is stable when a less than 0. But interestingly, this is a very interesting fact which I leave to you to work out. These two conditions are equivalent in the sense that if a less than 0, it also means that absolute value of 1 plus h by 2 divided by absolute value of 1 minus h by 2 is less than 1. So, that is an interesting thing. Trapezoidal rule whenever you are simulating linear systems will never give you a wrong result as far as stability is concerned. So, that is something interesting and useful. But of course, one important point is implicit methods like backward Euler and forward Euler are difficult to solve because especially when you come to non-linear systems because you will need to use a numerical method to compute x k plus 1 from x k. It is an iterative method at every time step itself you will have to iterate and solve a numerical or solve a non-linear algebraic equation. When you are talking of linear systems, of course, it would mean typically a matrix inversion or a solution of a linear system of equations. We also discussed in the previous class not trapezoidal. This method which I am talking of is rangekutta method. Rangekutta method of fourth order has the following evaluations. You have to evaluate k 1 at these points. I have done this in the previous lecture. You evaluate k 1 first, that is the function evaluation. k 2 is a function evaluation at a slightly different point depending on which depends on the first evaluation. So, f of x k plus h by 2 into k 1, k 1 is evaluated here. Similarly, you have got the evaluation of k 3 which requires the value of k 2 and k 4 which requires the value of k 3. So, what you see here is this is small error here. What you see here is the final value of x k is a combination of these. Of course, rangekutta requires a lot many more evaluations and it can be proven that this is a fourth order method. The one I have shown you is a fourth order method. It is of course, single step. The reason is that we require only the value of x k in order to compute all these functions k 1, k 2, k 3, k 4 and it is an implicit method. At least this one which I have shown you is an explicit sorry it is an explicit method. I am sorry rangekutta fourth order which I have shown you here is an explicit method. Explicit in the sense that all these functions evaluations are single shot. They are explicit evaluations of functions. You do not have to solve any non-linear algebraic equations to evaluate any value here. Of course, when you are trying to numerically integrate linear higher order systems, again let me point out here that the reason why I am studying the behavior of numerical methods using linear systems is not because linear systems cannot be solved. We saw in the lectures, previous lectures that linear systems in fact are amenable to exact solutions. You can write down the solution in terms of very well known functions. So, the only reason why I am using linear systems, a numerical integration of linear systems, the study of that is because I would like to know how the numerical methods behave. Now, if you have got a linear higher order system like x dot is equal to a x a is a matrix x is a vector. It is stable, you know it is stable if real of the any of the all eigen values should have the real part less than 0. Then we say it is stable, but if I numerically integrate this higher order linear system, in that case what you have is x k plus 1 is a vector is 1 plus a h times x k, which is again a vector this is a matrix. Now, this system is stable this is a discrete time system, it is stable if any the eigen values of 1 plus a h is less than 1. This is not something I am explicitly proving, but it is not difficult to prove you can use the same model analysis or linear systems analysis eigen values and eigen vectors to analyze discrete time systems as well. Now, if the absolute value of the eigen value of 1 plus a h is less than 1, then it is a stable system that is what Euler method says, the original system is stable if all eigen values have the real part less than 0. So, for a higher order linear system Euler method will give result that if 1 plus lambda h the modulus of lambda h is less than 1, then it is stable. So, where lambda is the eigen value of a. So, let me just repeat if the eigen values of 1 plus a h have the modulus or absolute value less than 1, then Euler method will show that the system is stable. It may be actually a stable system and Euler method may show to be unstable system if this becomes greater than 1. So, it really depends on the value of h here. You can show that mu is nothing but 1 plus lambda h, where lambda are the eigen values of a. So, the condition this condition can be rewritten as the condition for this discrete time system to be stable is 1 plus lambda h the absolute value should be less than 1. So, again let me just clarify what I am trying to say I am trying to say that even if the original system is stable if this condition is not satisfied then your numerical method will give you a qualitatively wrong answer about stability. So, you have to be careful when you do the simulation that you do not get wrong answers. Now, moving on we will do a few examples in psi lab it would be wonderful if you have got psi lab or mat lab or even if you program in C or Fortran and verify what I am trying to say here. What I will do is I will kind of make an algorithm for doing the numerical integration first of a system x dot is equal to minus 5 x. This is a stable system. So, I will first numerically integrate it using Euler method actually if Euler method is used remember that it will you know it is likely to give correct results one of the intuitive ways of choosing your h is that look at this eigen value this eigen value will probably will give in fact a solution of this kind. So, we will just try to verify whether we get that solution or not. In fact, you have to choose a h carefully if you look at this particular you know function it will decay almost to 0 in 5 times the time constant of the system. The time constant of the system is in this particular case 1 upon 5 that is 0.2 the time constant of the system. So, we expect that it will decay this particular thing will decay if I give a non equilibrium initial condition you will go to 0 which is the equilibrium in this case in about 1 about 1 second. So, I can just intuitively think that if I want to get a nice at least a nice picture of what is going to happen I should choose h at least you know something like 0.01 or so we will just try out with various values of h and see what we get. I done this in the previous class also, but it was somewhat in a hurried fashion let us do it a bit calmly in this particular lecture. So, this is the basic psi lab window I have already invoked psi lab what we will do is first of all let us look at the solution of. So, we will do a numerical integration using Euler method first of all we will clear this variable x we will choose a time constant let us say of time step of 0.01. Let us simulate or numerically integrate this system till three seconds because we know that it is probably to decay within that time. So, we have got a number of discrete time steps at which x has to be evaluated. So, let us assume the initial condition on x is 1. In fact, I should have written this x of 0, but psi lab does not allow 0 as an index. So, that is why I have called it 1. So, this is the value in fact at time t is equal to 0. So, I have given initial condition of 1 remember x dot is equal to minus 5 x has got an equilibrium value of 0. So, we are giving an initial condition which is not the equilibrium value. So, you will see a transient the value of a is 5 as I discussed some time back. So, what I will do is I will evaluate the behavior of the system using Euler method. So, x of i is equal to 1 plus a into h into x of i minus 1 this is what Euler method will do. So, if you recall we have got x k plus 1 minus x k upon h is equal to minus 5 of x k. So, this will effectively give us x k plus 1 is equal to 1 minus 5 h into x k.  So, what I will do is I will evaluate this and plot it. So, the way to do it is. So, we evaluate this it is giving a growing function. Now, why is this happening we will just look back perhaps I have done I have made some error here h is equal to 0.01 that is I am not saved it. So, I need to save it first I have saved it. So, I have given the value of h is equal to 0.01 which I changed from the earlier value and now I will run it again. So, let me just run this again. So, I am doing this of course, simulation for 3 seconds. So, now what we are getting is what we kind of expect exponentially decaying e raise to minus 5 t kind of response the initial value is 1 in case you are not able to see it clearly and this time period is from 0 to 3 seconds. So, what I will do next is I will change the time step to 0.1 and just to make it a bit easier to make see we save it plot it as a red. So, if you look at this you see that by changing the time step to 0.1 we have actually slightly the behavior is slightly change you see this red if you look at this red response it is slightly different. So, if I go ahead and let me let me make this 0.5 look at the response well you look what has happened your response is absolutely different from what is expected. So, if you choose a time step like 0.5 in which is in fact greater than 2 times 1 upon 5 I had mentioned that if h is greater than 2 by a the absolute value of 2 by a in that case your response is altogether different you really have a situation where you are getting an unstable response is not of course, the correct response we say the numerical integration is blowing up. So, though the actual system is stable the numerical method gives you a wrong answer. So, obviously, you should choose your time step carefully especially if you are using Euler method. Let us now go on and try to simulate the same system using backward Euler method. So, let us just have the same thing this is backward Euler method h is equal to 0.01 t final is 3 number of steps of course, is the depends on the duration as well as h the initial value of x is 1 a is minus 5. So, I just save this. So, as per backward Euler method you will have x of k or x of i is equal to x i minus 1 divided by 1 minus a h you can easily derive it from the basic formula for discretizing or you know using backward Euler method. Now, this particular sentence has been commented.  So, let us just numerically integrate and see what we get this is with a time step I believe of. So, we are getting the correct answer of course, the time step is 0.01 and a is minus 0.5. Now, I can make this also 5 we have done that before let us see what happens well the solution which you get is this the interesting thing is that of course, there is an a kind of error you know the earlier with a lower time step we got this response. Now, we are getting this slightly erroneous, but of course, it is still stable. So, what one small point which we should remember is that backward Euler a stable system will in fact, be shown as a stable system. So, that is no worry, but a very interesting thing is what if I got an unstable system a is equal to 5 and I choose my time step more than 2 times 1 upon 5 and interesting thing which you will see here is that this system which ought to be unstable you know after all a is equal to plus 5 this is the real system is unstable e raise to 5 t into x of 0 that is the solution what you will find is backward Euler when you use backward Euler it is actually showing the system to be a stable one that is very interesting. So, it really this something you should keep in mind that backward Euler in fact, is gives you can also give you wrong qualitative answers if your time steps are large this of course, not true if you choose a time step to be small. So, for example, if I choose 0.1 backward Euler of course, should give you the correct answer correct in the sense that it is an unstable system. So, you see that unstable system. So, it just grows up and if it is not visible on your screen this is almost 1.2 into e raise to 9. So, 1.2 into 10 raise to 9. So, that really means that it is really blown out in 3 seconds itself that is not very surprising. So, this is about backward Euler what if I do the same system simulation using trapezoidal rule trapezoidal rule has an interesting property that whether you are stable if you have a stable system it will show be shown as stable. So, if for example, this is for 0.01 just to say I will just check whether I have done the right thing I have to comment this statement and enable this statement which is essentially implementing trapezoidal rule this is x of k is equal to x of this x of i is equal to x of i minus 1 divided by 1 minus h by 2 into 1 plus a h by 2. So, you can actually work out this that you get this particular algorithm if you apply trapezoidal rule to x dot is equal to a x. So, if you have got a is equal to minus 5 and h is equal to 0.1 as we saw just now I save this run it remember although I have called this file backward Euler still it is doing trapezoidal it is using trapezoidal method of integration now. So, what if I increase the time step. So, I am using trapezoidal rule just remember that. So, for example, 5 this was unstable for example, Euler method when we try to integrate it with h is equal to 5 it was unstable what does trapezoidal rule is also stable of course, the errors in the response. So, if I actually go on increasing h the errors increase, but as we shall see I will just do something more sillier and make h is equal to 1 second even if I make h is equal to 1 second although I am going to get an error in the response it is not unstable this is my new response. Similarly, if a is equal to plus 5 if I choose h is equal to 1 this is an unstable system we will just do it on a fresh plot this is an unstable system and trapezoidal rule is showing it to be an unstable system. So, really trapezoidal rule you know at least in a linear system it kind of preserves the stability information. So, if I of course, if I make this time step smaller the answer is of course, closer to the right answer. So, you see this increasing here like this. So, to summarize whatever I have shown so far is that when we are trying to numerically integrate a system x dot is equal to minus 5 x. If you choose Euler method if you choose a h to be large in fact, if it is greater than the absolute value of 2 by a is equal to minus 5 in that case you in fact, get a wrong qualitative picture about the stability of course, as you go on increasing the time step a numerical methods accuracy versions. So, it is not unexpected that you will get better results if you choose h smaller, but the important point which I again emphasize is that Euler method may give you a qualitatively wrong result also it may show a stable system in fact, to be unstable if h is larger this does not happen with backward Euler or with trapezoidal rule. In fact, a stable system is always shown to be a stable system whether one uses backward Euler or trapezoidal rule, but interestingly with backward Euler method and essentially unstable system can be shown as a stable system if your h is very large. So, if h is greater than point 2 upon a the modulus of it and unstable system for example, a is equal to plus 5 is shown to be stable one whereas trapezoidal rule preserves the stability information of a linear system. So, that is one thing a very interesting point which we can we must note now the basic you know the three method which I have shown you to get accurate solutions in fact, for example, x dot is equal to minus 5 x it makes sense in fact, to use methods with time steps of 0.01 or so you will get a reasonably accurate picture you know it is another thing about you know if you choose h is equal to 0.5 etcetera not only you are going to get an inaccurate picture, but in case of Euler method you will get even a qualitatively wrong picture. So, you will use a it is a good idea if you have got a system with a time constant of 0.2 or so as in this case to use a time step of 0.01 and so on, but you know one you know for example, if your situations where a system is marginally stable you know you have got a system in which if the real part of the Eigen value of the continuous time system is 0 or near 0 Euler method is absolutely hopeless you will find that it does not give you a good you know it gives you a qualitatively wrong picture. What I will do now is try to simulate one of the systems which we have simulated before try to understand qualitatively before and the swing equations of a power system. So, the swing equations of a single machine infinite bus system are delta dot is equal to omega minus omega naught and the derivative of omega minus omega naught which is nothing but omega dot itself, because omega naught is a constant which is the frequency of the infinite bus is equal to omega b by 2 h p m minus k sin delta k is in fact we assume it to be a constant this is the internal voltage of the synchronous machine the infinite bus voltage divided by the reactance. So, if I got a single machine connected to an infinite bus e 2 angle 0 and this is e 1 well this is a synchronous machine which is modeled as an internal. So, this is our synchronous machine. So, this is e 1 angle delta this is the internal voltage this is the rotor angle x dash is a transient reactance. So, this these are the equations swing equations this is a non-linear system of equations. So, if I am trying to solve this system of equations by Euler method what is the kind of response I will get. We saw for small disturbances we have done the analysis of the system and we saw that in fact it is for the equilibrium corresponding to which is less than 90 degrees this one for small disturbances around the equilibrium we will get essentially an oscillatory response. So, we know that this if you are at this equilibrium you will get an oscillatory response. If I give of course, a very large disturbance from this then the system may even go unstable the synchronous machine may fall out of step or lose synchronism. So, we know the kind of responses which can be expected at least for small disturbances and qualitatively of course, we know that for large disturbances the system may go unstable. So, with this this is a non-linear system by the way this is not a linear system because your delta appears within the sine function. So, what I will do is try to integrate the swing equations by Euler's method. So, you just take let us assume P m is 1 h is 3 at this capital H is of course, the inertia constant k let us assume is e 1 e 2 divided by 0.4 this is omega and omega base we assume is the same as this frequency of the infinite bus. Let us take a time step of 0.01 the final time is 3 the number of steps of course, can be calculated from t final n h.  Now, remember one important point is that although delta is at the equilibrium value of the value of delta is actually its equilibrium value the value of omega is not its equilibrium value. So, as a result this we are not actually at the equilibrium because both states have to be at the equilibrium value if we are not to have any transient. So, by adding this plus 1 we have in fact, given an initial condition which is not the equilibrium value of the states. So, we are going to get a transient. So, I have given you a initial condition which is not equal to the equilibrium value if I had given you the equilibrium value the system would simply not move it would just remain where it is. So, if I apply Euler method you have got delta is equal to delta of i is equal to delta of i minus 1 plus h this is in fact, simply omega minus. So, we are just numerically integrating the system by Euler method omega i is equal to omega i minus 1 plus h times omega base divided by 2 times the inertia constant into P m minus k sine delta this easy to evaluate if I know what the initial values of delta and omega are I can find out the new values. Since, we are not at equilibrium since I have given an initial condition which is not the equilibrium condition of course, if I had made the 0 this 1 is 0 we would be at equilibrium. So, what I will do is I will just simulate the system. So, what I will do is again. So, I am evaluating this system using Euler method. So, if you look at the behavior of a system it looks as if it is an growing oscillation. In fact, if you look at we have done the small disturbance stability of this particular system and we know that the response is in fact, not a growing oscillation, but a steady oscillation if this disturbance is small. So, obviously, there is a problem in the method let us just try reducing the time step. What is the problem in reducing the time step? There is no problem you will have to wait longer for the simulation to get over and sometimes that becomes a bit that may become a limiting factor for doing any kind of study if you make the time step too small. So, now you are getting you know a different answer that shows that you have to actually reduce the time step. So, I will just redo this in fact, by using the smaller time constant time step we are in fact, getting a better answer, but you see that this is still growing it is not it is not you know you this value is slightly less than the value you have here. So, what we see is that if you have got you know marginally damped systems Euler method in fact, does not seem to be very nice because whatever I reduce it if I reduce the time step I can do that even further. Now, it is going to be painful because you will have to wait for a longer time now it is still it will take some time for this to come up. So, you see that there is slight difference in the what I got with the previous time step and the time step you are having now. One thing we can simulate for longer and longer time with this time step there is point triple 0 1, but the point you can actually try it out whatever time step you take you will find this eventually starts growing if you try to simulate beyond 3 seconds say 20 seconds even with this time step you will find that it is growing. So, Euler forward Euler method not very nice to simulate systems with which are marginally damped or have poor damping. So, let us just rerun the system using trapezoidal rule. I am using exactly the same system with the time step of point 0 1 and one problem comes here. Now, if I am going to try to do integrate this method using trapezoidal rule what is going to be the problem the problem which you see here of course, is that the statements in the program seem to have become much larger. One of the things you will notice is that delta i will not be easily obtained from the previous value that is delta i minus 1. So, let us just pay our attention on what we get if we try to discretize swing equations using trapezoidal rule. So, you have got delta dot is equal to omega minus omega naught.   So, this is the discretization which you will have to use and that is what is really been programmed there.  So, if I if I given you delta k and omega k how do I get omega k plus 1 and delta k plus 1 you will have to use an iterative method. So, that is what exactly I have done in the program what I have done is try to use within every time step within every time step solve this using Gauss-Seidel method. So, what we do is get plug in this value get plug in this value of delta k and omega k you will plug it in here here here and here you plug it in omega k plus 1 is something you do not know. So, let us guess that it is equal to omega k in the first instance and this is delta k.   So, that is exactly what I am doing this here the program is a bit complicated and probably you will find it difficult to follow the steps, but I will just indicate what I am doing. So, for what I do here is I essentially in these steps I am in fact using Gauss-Seidel method using Gauss-Seidel method I have taken a maximum number of iterations per time step as 10 with the hope that of course, that will converge. So, in fact I am checking out the convergence criterion as well. So, I will just let me read go through the whole program slowly this is the check on the convergence. So, I will just run this of course, one small point will give the same disturbances before in the sense that the this was one in the previous. So, I have given a small disturbance we are away from the equilibrium save this. So, this is basically what you get we actually evaluated this we redo it this is not the what we get with trapezoidal rule. Now, we will get what we see. So, this is the response and actually this sinusoid does not grow with time. So, this is a this is a true in fact really reasonably accurate picture of how delta varies if the system is given a small disturbance. So, even if I increase the time step remember here this is this will not really be affected. So, I just change my time step to 0.1 10 times more save this rerun it because of course I have chosen a larger time step there is slight you know kind of pokey pokey solution here this particular sinusoid has got rough edges. But on the whole it does not give you a wrong picture of stability it is a undamped sinusoid delta is an undamped sinusoid. So, this is basically what you get with trapezoidal rule. If I increase the disturbance I increase the disturbance in that case you may even go unstable. So, I am now I am going to give a large disturbance relatively wrong you see that because of this larger disturbance the delta oscillations have become larger and one interesting thing that this is no longer sinusoidal looking you see this top part looks much fatter than the bottom part. So, obviously with the if you make the disturbance larger your behavior starts moving into the non-linear zone and you do not get the sinusoidal response as predicted by linear analysis. So, if I actually make this disturbance even larger you are going to get the system loses synchronism you see that this angle just goes on increasing with time. So, if I go on increasing the you know deviation from the equilibrium we go into non-linear behavior and this is of course, captured quite well by this numerical integration method. So, this is basically what I wish to tell you about using trapezoidal rule for small disturbances it gives you the correct response if I use backward Euler method for the same problem backward Euler method is also an implicit method remember. So, I have to use it an iterative solution to get this iterative method to get the values of the states at every time step. So, let us just do this again with backward Euler. So, so an interesting thing that I have just to observe here is that when I use backward Euler on a marginally stable system which is just having zero damping Euler backward Euler shows it as if it is a stable system as a damped oscillation. So, that is an interesting point about backward Euler it really changes the picture means especially if your system is marginally damped it is got very little damping. So, it is showing you the system to be a stable system whereas, actually it is an undamped oscillation. So, the system is a showing as if it is dying down. So, that is one interesting observation about backward Euler method of course, if I use backward Euler method with a very small time step you are going to get reasonably accurate solution. So, I just numerically integrated with a small smaller time step well we are coming the answer is slightly different and we are coming closer to the solution we have in mind. So, of course, reducing time step is one of the ways you can get accurate solution or a more accurate solution. So, with this I end my discussion about the numerical integration of swing equations. My discussion about numerical integration method would be quite incomplete if I do not consider what are known as stiff systems. We have learnt about stiff systems. Stiff systems are systems in which the Eigen values of the of linear systems especially are far apart in the sense that they are small as there is large Eigen values which translates to the fact that the system has got both fast and slow transients. And if you are actually interested only in the slow transients then you really need need not model it in great the fast transients in detail. In fact, I showed to you in one of the previous lectures that when you are trying to analyze a system with slow and fast transients the fast transient differential equations corresponding to the fast transients or the states associated with the fast transients can be converted to algebraic equations and we really do not lose outmatch as far as the information on the slow transients is concerned. A similar issue will kind of crops up when you have got even a numerically integrating stiff system. So, if you have got a stiff system you have got large and small Eigen values and the question comes what is going to be the choice of time step. So, if you have got a stiff system and you have got fast transients you will have to choose a time step which is compatible with the fastest transients of their system in order to accurately get the response. So, if you have got a system with slow as well as fast transients inherent then you have to choose a time step to be compatible with the fastest transients. So, that your fast transients are accurately obtained, but consider a situation where you are not really interested in the fast transients. So, how will you choose a time step? In Euler method it is very clear that if 1 plus lambda h the absolute value is greater than 1 then the numerical integration method is going to blow up. So, even if your fast transients are stable if I choose my time step which is not compatible with this particular condition then your numerical integration as I said will blow up, but what if I am interested only in the slow transients. Unfortunately with Euler method will be forced to take a time step which is small very very small which is compatible with the fast transients. So, that is one of the problems. So, even if you are interested in the slow transients you have to choose very small time steps. So, that is one of the problems which you will encounter when you try to use Euler method to integrate a stiff system. So, let us again look at a stiff system we have done this example before this is I L 2, this is I L 1, this is V c we know that this particular system has both fast and slow transients. So, this is 10 milli Henry, this is 1 Henry. In fact, the state associated with the slow transient is this and the states associated with the fast transient are these from the participation factors which we have evaluated in a couple of lectures before. So, if you have got this particular system this contains slower and faster transients. In fact, how do you know that it contains slow and fast transients. In fact, if you know the A matrix you have got x dot is equal to A x where x is equal to I L 1, I L 2 and V c then this A is equal to minus 10, 0, minus 100, 0, 0, 1. We have done this in a previous class. So, I am not reworking out this this is minus 10000 and 0. So, this is your A the Eigen values of the system are approximately minus 5 plus or minus j lambda 1 and lambda 2. So, they are two actually complex conjugate pair. So, this is a stable system, but one of the Eigen values is very small and the other one is very large. This is a example of a stiff system. If I try to numerically integrate this system, stiff system using Euler method I will have to choose my time step. So, that 1 plus lambda 1 h is less than 1. So, h should be such such that this is satisfied. In fact, it should be satisfied for all the three 1, 2 and 3 all the three Eigen values. So, one thing you will notice that if you want to satisfy it for all the three Eigen values then h will have to be very small. In fact, it becomes completely unstable in case you choose your h say as 0.001 or so. So, I will just do this numerical integration using Euler method and then we will conclude this particular lecture. So, let us just do this one integration. So, this is your system this is the A matrix I chose a very small time step. I will just simulate it for a short while because I know it is really not going to work out with Euler method even with this small time step. Even with this small time step we have a problem with Euler method. So, the first thing I will do is integrate it using Euler method. This is a remember higher order system. So, instead of 1 plus small case a h you have got identity matrix plus a into h into x plus h into u. So, I am implementing Euler method for this system of course, there is an input here. So, I have used Euler method I have generalized it to a situation where you have got an input as well. So, if I numerically integrate this system oops there is a error here. So, I will just check out what is the problem and redo it. So, the problem probably is there is some error here. So, we will what we will do it do is redo this example again in the next lecture. We will just debug the error which is there in the program and rerun this example in the next class. So, in the next class we will continue with this particular example see how it is in fact a very interesting and important thing we are going to learn in the next lecture that for stiff systems of this kind you have to use specific numerical methods even you know explore if you want to capture. In fact, both the fast and slow transition you may even have to explore things like variable time steps and so on. So, let us conclude this lecture at this point we will redo this example which did not work out in this particular lecture again in the next lecture.