Hi Mans, 1. I don’t think I am assuming lambdas are always positive. I don’t think there is anything in the modeling that assumes a positive price. The only assumption is that the cost curve of each unit be convex. As I said previously, this means that the slopes of the segments must be increasing as you go from most negative to most positive injection.
2. There are two possible scenarios for a facility that can both consume and produce power (possibly simultaneously?) and I’m not sure which one you are trying to model. You can either model the load and generation separately, or you can model a single unit that represents the net injection. In the first case, each must have its own convex cost function. In the second case, you need to create a single convex cost function of the net injection. It sounds like you are trying to do the latter. In this case, building the appropriate cost curve is a bit more subtle. I’ll attempt to explain how I would go about it. Assuming you have a marginal benefit function (e.g. load bids) for the load and a marginal cost function (e.g. gen offers) for the generator, start by finding the price at which your generation and load can match, for a net output of zero. This is the slope of the segment of your cost curve that passes through the origin. The range of net injections that corresponds to this price will define the quantities of the endpoints of this segment. Then, decrease price from that point until you reach a new demand block (now included) or a new generation block (now excluded). The price at which you encounter a new block is the slope of your next segment (toward the left) and the size of the block determines the horizontal length of that segment. Continue decreasing the price to build up the left part of the cost curve. Then raise the price from your starting point in a similar fashion to build up the right portion of the cost curve. 3. All I’m saying is that our modeling defines a negative cost equal to the negative of the benefits, so our cost minimization problem is mathematically equivalent to a welfare maximization problem. Best, Ray > On Feb 5, 2015, at 6:45 PM, mohd <[email protected]> wrote: > > Dear Ray, > > Thanks for the clarification. Note my comments and questions in sequence: > > 1) When i derived it as injections in the objective function, the picture > became more clear to me. The only thing to note is that you are assuming > lambda to be always positive. In some markets with subsidies, the price could > go negative as in ERCOT with the current level of wind penetration. One > question that might arise is how to deal with negative price in modelling! > This means a positive cost again. If so, how can i make the shift such that > the curve still maintains its convexity. > > 2) I have a facility that has both a load as well as its internal generation. > My derivation shows convexity which is the main assumption for optimization. > However, i was asking how to model the load when it has both positive and > negative injections, especially during the shift from load to generation. > What i did is that since my load bids around 500 MW for a low price close to > zero, i ended the point as (-500,0), then i made a straight line (flat) till > point (0,0). After that, i started the generation side from (0,0) and made my > next point, all using the piece wise cost curve and using the break points > listed. I hope my approach is correct and i am sorry for not explaining > myself well in the first e-mail. > > 3) I got the point but i wanted to make sure it is not related to the > modelling assumption for dispatchable load. I am used to other terms such as > worth, cost, and social welfare but thanks for the information. > > I hope that you can comment on my points, especially point-2. Thanks and take > care. > > Kindly, > Mans > > From: [email protected] <mailto:[email protected]> > Subject: Re: Dealing with dispatchable loads > Date: Wed, 4 Feb 2015 13:47:41 -0500 > To: [email protected] <mailto:[email protected]> > > 1) The marginal cost is positive. Multiplied by a negative quantity results > in a negative total cost. Increasing the load (negative injection) decreases > the overall objective function. So the load will be fully dispatched, unless > the LMP drops below it’s marginal cost. > > 2) I’m not sure I completely understand what you’ve done, but you certainly > model this as a generator with negative PMIN and positive PMAX, with a > piecewise linear cost function that passes through the origin. It must be > convex, with increasing slopes for the segments as you go from most negative > to most positive injection. > > 3) The negative total “cost” associated with the dispatchable load is > actually just the negative of the total benefit derived by the load from it’s > consumption. That’s how we define this cost. So, minimizing the sum of this > (dispatchable load) cost and the generation cost is the same as maximizing > the total benefit to consumers minus the total cost to producers, that is > maximizing the net benefits to society, also called social welfare. > > Hope this answers your questions, > > Ray > > > > On Feb 3, 2015, at 2:42 PM, mohd <[email protected] > <mailto:[email protected]>> wrote: > > Dear all, > > I would like to understand several points about modeling dispatchable load in > matpower. I have read section 6.4.2 and i would like to understand how the > dispatchable load works in the context of its modelling. > > 1) If we have negative power injection with a negative cost. My understanding > is that it will show at the end as a net added cost since we are multiplying > two negatives?! In this case, i assume, if we want to dispatch a load, it has > to have a cost that is cheaper than the current cost of operating the most > expensive generator or at least cheaper than the set LMP price at its > location. I would like to ensure proper understanding or more clarification > about that. If my understanding is wrong, then i have an issue of why not > running all dispatchable load since they will reduce the total cost of the > system. They are negative! > > > 2) If i model a demand-responding facility that has both a load and > generation. Based on the price, it will bid either negative or positive > injection. Positive mean net export to the system and negative means net > import from the system. What is the best way of modelling that? What i did > is that, i had my own profit maximization algorithm for the facility that > provides the net injected power at each price, so i can create my bid > function. From that, i have started from the highest expected price and > calculated the associated cost by multiplying my power by the price. So, for > each change in power, i calculated a change in cost as explained, then i > started from the highest price to calculate the cumulative power and > cumulative cost to create the total cost function for the load. In this case, > how should this be handled when the net injection is positive? I do believe > that i should not use a negative cost anymore. I need someone to comment on > my method to ensure i am not making assumptions that are different from > matpower assumptions for modeling. My curve for the negative injection (load) > is convex so far but the positive injection is still confusing to me using > dispatchable load assumption. > > 3) The last paragraph in that section states that " it should be noted that, > with the definition of dispatchable loads as negative generators, if the > negative cost corresponds to a benefit for consumption, minimizing the cost > f(x) of generation is equivalent to maximizing social welfare". What do that > statement mean? does it mean minimizing negative cost means maximizing total > benefit which social welfare or is it just using the definition of social > welfare which is worth minus cost and minimizing cost means maximizing SW. > > Thanks for your time and effort in trying to answer my questions and > clarifying the points raised. > > Kindly, > Mans
