Including the Consumer Function. 1.0 Constant demand as consumer utility function

Transcription

1 Incluin the Consumer Function What we i in the previous notes was solve the cost minimization problem. In these notes, we want to (a) see what such a solution means in the context of solvin a social surplus maximization problem an (b) exten the formulation to inclue eman responsiveness.. Constant eman as consumer utility function In the formulation of our cost minimization problem, we minimize the cost of supply subject to the requirement that the total supply equale the fixe system eman. Here, our only ecision variables were eneration levels, i.e., emans were not ecision variables but rather constants. This implies that eman is insensitive to price, i.e., eman is inelastic. This means that the

2 consumer is not solvin the followin problem: ax U(x)-px () where U is the consumer s utility function (benefit from consumin x), epenent on x which is eman an p which is price, but rather is solvin the followin problem: ax U(x) () That is, the consumer is maximizin utility, which is a function of eman, but the consumer is totally inorin the price when oin so. This is the way most resiential consumers use electricity. The consumer therefore etermines how much eman they will consume via a local ecision-makin problem (e.., What o I nee to o toay ), an the social optimum can be solve treatin the consumer eman as a constant. This means that the consumer eman function is a vertical line, as shown in Fi..

3 price Fi. : Deman function for constant eman Since the consumer eman is a constant, the consumer utility is also a constant. As a result, the social optimum is obtaine via solution of a maximization problem that only inclues enerator cost functions (multiplie by -) in the objective function. This is: ax: ( C k ( q k )) () k Alternatively, the social optimum is obtaine via solution of a minimization problem that only inclues enerator cost functions in the objective function. This is what we i in our LOF, an it is what utility companies have one for years.

4 in: C k ( q k ) () k Now, however, we want to account for the possibility that the consumer will watch their price an ajust their eman as a function of that price. In this case, we must inclue the consumers utilities in the objective function, leain to the objective function use in the formulation for our present problem, which is: ( U ( x ) C ( q )) ax: k k k k (5) k In both cases, we are maximizin the social surplus, the ifference between the consumers utilities an the suppliers costs.. LOF with Consumer Utility As in previous notes on LOF, we brin in network constraints, but this time we will o so with an objective function that maximizes social surplus. In aition, as with our LOF, we will use a piecewise linear approximation of the cost curves with only piece per curve. Thus, each eneration

5 unit an each consumer is represente in the objective function by a constant times the W output for that unit or the W consumption by that consumer. So here is the formal statement of our problem: min Subject to: k s k k + { enerator buses} k { loa buses} s k k (6) ' (7) ( D A) (8) (9), max,max { enerator buses} { loa buses} k k, max, k () 5 k k, max, k () 6 where k k k, k,... N () 7 We want to maximize social surplus as efine by U k - C k, but this is the same as minimizin C k - U k. We make this chane because the L available to us in atlab is a minimizin L. So these are the DC power flow equations to represent the network. However, we must inclue all noal injections, N an all anles N in this set of equations. These are the equation to et the line flows. Aain, we nee to inclue all anles N in this set of equations. These are the limits on the line flows. Notice that there is only one circuit ratin, but it must be enforce as a limit if the flow is in one irection or in the other. 5 These are the limits on the linear cost curve variables. 6 The limits on the linear eman curve variables. 5

6 We ientify the ecision vector as: n bm b n n x ; n n s s s s c () We are now in a position to state the LOF more compactly. ax x c T Subject to: () eq b eq x A, max min x x x (5) where the equality constraints in the A eq matrix equation moel the line flow equations an DC power flow equations. 7 This equation relates the variables use in the cost curves ( kj ) to the variables use in the DC power flow equations ( k ). 6

7 ) ( + A D (6) ' + (7) an the inequality constraints are iven by:,max,max,max,max,max max,max, bm b n n n bm b n n bm ma b (8) Some particular notes about the above problem statement: The upper riht-han m n submatrix of A eq is D A. The lower riht-han n n submatrix of A eq is. 7

8 The riht-han-sie of the equality constraint equation, b eq, is all zeroes because we now have variables for the eman which means it must be inclue in the A eq matrix instea of bein a fixe constant (an therefore represente in the b eq vector).. Example: Unconstraine transmission We illustrate usin an example that is similar to the example use in the LOF notes, which is a combination of previous examples. These are The example use in the notes calle Linear rorammin Approach Usin iecewise Linear Cost Curves where we optimize a unit system, where all units were connecte to the same bus an supplyin a loa at that bus. In this example, we use sements to approximate each cost curve. 8

9 The example in the notes calle The ower Flow Equations where we ha units connecte to ifferent buses in a bus network supplyin loa at ifferent buses. The one-line iaram for the example system is iven in Fi.. y -j y -j y -j y -j y -j Fi. : One line iaram for example system We will use the same ata for the unit costcurves as we i in the LOF notes. These were 9

10 K ( ) s ( ) s ( ) s K K where the eneration variables are in pu an the coefficients are s 7 $/pu-hr s $/pu-hr s 5 $/pu-hr The constraints are 5< < 7.5< <5 5< <8 We will not a the constant factor to the objective function. We will use the followin linearize utility functions for eman: D D ( ) s ( ) s where the loa variables are in per-unit an the coefficients are s - $/pu-hr s - $/pu-hr

11 The constraints are < < < < Objective function: Let s explicitly write out the solution vector. 5 x So, usin these coefficients, the objective function is:

12 ( ) [ ] T x c x Z Equality constraints: The equality constraints are iven in eqs. (6) an (7), repeate here for convenience: ) ( + A D (6) ' + (7) We nee to buil all of these equality constraints into a matrix form of A eq xb eq. We bein by notin imensions. Columns: Since the solution vector x is x, A eq must have columns in orer to pre-multiply x.

13 Rows: Since there are 5 branches, eq. (6) will contribute 5 rows to A eq. Since there are buses, eq. (7) will contribute rows to A eq. So A eq will have total of 9 rows. Therefore, the imensions of A eq will be 9. We bein with the line flow equations, eq. (6). From the notes on ower flow equations, we can recall the D an A matrix. The D matrix is exactly the same as before, which is: D An the noe-arc incience matrix, A, is:

14 - - A The DxA prouce require by eq. () is then iven by: - - A D So base on eq. () an the solution vector, we can see that these elements will occupy the upper riht han corner of A eq. So that will take care of the last columns in the first 5 rows. ut what about the first columns? These are the elements in the line flow equations that multiply the variables,,,,,,,,, 5. Since we o not use the eneration or eman variables within the line flow

15 equations, the first 5 columns of these top 5 rows will be zeros. The last 5 columns in these top 5 rows will also be zeros, except the one element in each of these rows that multiply the corresponin line flow variable, an that element will be -. Finally, with respect to these top 5 equations, eq. (6) inicates that the rihthan-sie will be for each of them. Thus, we can now write own all elements in the first 5 rows of our matrix, as follows: 5

16 5 eq x A Now we nee to write the last equations. These are the DC power flow equations corresponin to eq. (7). Aain, we must remember that the solution vector contains all anles, an therefore the DC power flow matrix nees to be a x. This aumente DC power flow matrix is iven below: 6

17 ' So base on eq. (7) an the solution vector, we can see that this matrix will occupy the lower riht han sie of the A eq matrix. So that will take care of the last columns in the bottom rows. The resultin matrix appears as: 5 eq x A 7

18 Once aain, we nee to consier the first eiht columns. Columns 6- correspon to the line flow variables, which o not appear in the DC power flow equations, so these will be zero. 5 eq x A The first three columns multiply the eneration variables,, an, an columns an 5 multiply the loa variables an. However, the DC power flow equations, eq. (7), require the neative of the injections for all buses, an the injections are the 8

19 eneration minus the loa, i.e., k - k. So we want to moel k + k on the left-hansie in the last rows. This will be one by placin a - an + in the appropriate place. 5 eq x A Inequality constraints: The inequality constraints are simple, as iven below. Notice that the -5 to 5 constraints on line flows imply we are moelin no transmission constraints. 9

20 Solution by atlab: The coe for solvin this linear proram usin atlab is iven below: %uil objective function vector. c[ ]'; %uil A matrix for inequality constraints Ax<b. A[]; %uil b, the riht-han-sie of inequality constraints. b[]; %uil Aeq matrix for equality constraints.

21 Aeq[ - -; - - ; - - ; - - ; - - ; ; - - -; ;]; %uil riht-han sie of equality constraint. It will be vector of zeros beqzeros(9,); %uil upper an lower bouns on ecision variables. L[ pi -pi -pi -pi]'; U[ pi pi pi pi]'; [X,FVAL,EXITFLAG,OUTUT,LADA]LINROG(c,A,b,Aeq,beq,L,U); %'X', X,FVAL,'eqaulity', LADA.eqlin, 'upper', LADA.upper, 'lower', LADA.lower % % Compute the ollars pai to each participant: ollarsc.*x; %Write out the ollars pai to each participant ollars The solution vector x is iven below. The limits on the variables are also repeate here so that it is easy to see which ones are at their limit.

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23 One can easily check to see that the power is conserve at the buses. Objective function value: The objective function that atlab provies (FVAL) is Z-.8 $/hr. This is neative of the social surplus (atlab requires all problems to be minimization problems, so we ha to minimize the neative of the social surplus in orer to maximize social surplus). So the social surplus (Total Utility of Loa less Total Cost of Supply) is $.8. Not too much! This is because the emaners are valuin the enery at just a little above cost. If we chane the utility function coefficients to -5 an -, from - an -, respectively, the social surplus woul chane to $9/hr. If we chane utility function coefficients to - an -9, respectively, the social surplus woul be -$9/hr, inicatin the cost of supply is more than the utility of

24 consumption, an the only reason any power is bein consume is the lower boun constraints we have place on eneration an eman. Larane multipliers: Now let s investiate Larane multipliers for this case, assumin infinite capacity lines. These Larane multipliers (the same as the ual variables), are iven in Table. Table : Larane multipliers for infinite transmission capacity Equality constraints Lower bouns Upper bouns Equation Value* Variable value variable value

25 Larane multipliers on the last equality constraints are very interestin, since they ive the improvement in the objective function if we increase the riht-han-sie of the corresponin equation by unit. These are the noal prices, iven in $/per unit-hr. The numbers are all $/per unithr; if we ivie this by the power base ( VA), we et $./W-hr. This is also the coefficient of the eman at bus,. Now let s consier the Larane multipliers: Lower bouns: an are non-zero, inicatin they are at their lower bouns, as confirme by ecision vector on p.. Upper bouns: an are non-zero, inicatin they are at their upper bouns, as confirme by ecision vector on p.. Not constraine (reulatin): Only has Larane multipliers for both lower an upper bouns, inicatin it is not at either boun (this variable is reulatin ). 5

26 Connection! There is only ONE unconstraine variable,, an it is also the variable that is settin the noal prices ($./W-hr) throuhout the network! A look at the coefficients will show why: s $/pu-hr s - $/pu-hr s 5 $/pu-hr s - $/pu-hr s 7 $/pu-hr Think of the alorithm like this: It first sets eneration an loa at lower limits (there is no choice about this much supply an eman). One variable must come off its lower boun in orer to provie power balance. Since sum of loa lower bouns is, an sum of en lower bouns is.7, one or more of the ens must come off their lower bouns by.6 in orer to provie a feasible solution. This en will be the least expensive one(s). In this case, it is G an G. (G ets pushe to its limit in this step) Then it takes a W of supply an a W of eman from the en/loa pair that is not at 6

27 upper bouns an provies the most positive surplus. This will be the en with the least cost an the loa with the reatest utility, as lon as the surplus is positive. In our example, the first en/loa pair taken, after finin a feasible solution, are G/D. As soon as either the en or the loa of the max-surplus en/loa pair reaches its upper limit, it will replace that en or loa with the one that yiels the next larest surplus. In our case, G reaches its upper limit first, an it tries to replace it with G. ut the G/D pair has coefficients that result in a neative surplus! So the maximum surplus is foun when G reaches its upper limit. You shoul be able to see that the alorithm will always terminate with just one en or loa reulatin, an that en or loa will set the noal price throuhout the network (for the unconstraine transmission case). 7

28 Settlement: The cost (for suppliers) an the utility (for the emaners) of the market equilibrium are compute by multiplyin the cost coefficient (in the vector c) by the amount of W bouht or sol (in the vector X). In atlab, this can be achieve by usin the vectorize multiplication function c.*x. The result of oin this is in Table. Table : Unconstraine case Cost or utility ($/hr) K ( ) s 65.5 K ( ) s 86.5 K ( ) s 57. D ( ) s -. D ( ) s -. However, this is not the settlement. The settlement is the ollars actually pai by the emaners an to the suppliers. The settlement epens on the noal prices, as iven in Table. The cost & utility is also 8

29 provie in Table so that the supplier an consumer benefit may be etermine. Table : Unconstraine case col col col col col5 k or k Cost or utility λ k ($/mwhr) λ k * k or enefit col- col ($/hr) λ k * k Observe the followin:. The sum of col is neative. This is because we are efinin eneration costs as positive an eman utility as neative. So we have more utility than cost, a esirable situation.. Col5 ives benefit, which shoul sum to have a sin opposite to that of col 9

30 since col is the neative of social surplus (or social benefit).. So the social surplus is the same as the total social benefit, an this is the same as the objective function.. Example: Constraine transmission We will constrain the transmission on branch. Reference to the ol solution of Fi., repeate here for convenience, inicates that the flow on branch is.65. So we will constrain that flow to be.6.

31 .5pu.5pu pu pu.pu The atlab coe for this is iven below. %uil objective function vector. c[ ]'; %uil A matrix for inequality constraints Ax<b. A[]; %uil b, the riht-han-sie of inequality constraints. b[]; %uil Aeq matrix for equality constraints. Aeq[ - -; - - ; - - ; - - ; - - ; ; - - -; ;]; %uil riht-han sie of equality constraint. It will be vector of zeros beqzeros(9,); %uil upper an lower bouns on ecision variables. L[ pi -pi -pi -pi]'; U[ pi pi pi pi]'; [X,FVAL,EXITFLAG,OUTUT,LADA]LINROG(c,A,b,Aeq,beq,L,U);

32 %'X', X,FVAL,'eqaulity', LADA.eqlin, 'upper', LADA.upper, 'lower', LADA.lower % % Compute the ollars pai to each participant: ollarsc.*x; %Write out the ollars pai to each participant ollars The new an the ol ecision vectors are provie below, toether with the limits. New solution Ol solution Limits

33 The new solution is provie in Fi pu.pu.867pu.5pu.8pu Fi.

34 Objective function value: The objective function that atlab provie (FVAL) in the unconstraine case was Z-.8 $/hr (social surplus of $.8/hr). Now in the constraine case it is Z-$.75/hr (social surplus of $.75/hr). The social surplus has ecrease, confirmin the principle that ain new constraints can never result in an improvement in the objective function. Larane multipliers: The Larane multipliers (the same as the ual variables), are iven in Table.

35 Table : Larane multipliers for constraine transmission capacity Equality constraints Lower bouns Upper bouns Equation Value* Variable value variable value Some observations:.in the unconstraine case, all four noal prices were $/Whr, now, in the constraine case, only bus is $/Whr (set by ), which is a reulatin (not at a limit) unit. An all of the remainin noal prices are ifferent. 5

36 . is also reulatin, an therefore the bus noal price is set by the bi which was $.7/hr..uses an have loa or eneration at a limit. us has at its lower limit, an bus has at its upper limit. So neither of these buses are reulatin. Notice that the noal prices at these buses are ifferent from the cost or utility function coefficient at the bus: us has utility function coefficient of whereas its L is.7 us has cost function coefficient of 5 whereas its L is 9.. This shows that buses with reulatin units or emans set their own price, whereas non-reulatin buses have prices set by other buses in the network..if there were no binin transmission constraints (effectively an infinite transmission capacity situation), then the prices at buses an woul be set by one other bus in the network. ut with a 6

37 binin transmission constraint (i.e., presence of conestion), then the prices will be set by the units neee to supply an aitional W at the bus AND maintain flow within the limit. As we have seen before, this will necessarily involve more than one unit. Settlement: The cost (for suppliers) an the utility (for the emaners) of the market equilibrium are compute by multiplyin the cost coefficient (in the vector c) by the amount of W bouht or sol (in the vector X). In atlab, this can be achieve by usin the vectorize multiplication function c.*x. The result of oin this is in Table. This result is also compare to that of the unconstaine case to illustrate. 7

38 Table : Constraine case Constraine Case ($/hr) Unconstraine Case ($/hr) K s ( ) K ( ) s K ( ) s D ( ) s D ( ) s However, this is not the settlement. The settlement is the ollars actually pai by the emaners an to the suppliers. The settlement epens on the noal prices, as compute in Table. The cost & utility is also provie in Table so that the supplier an consumer benefit may be etermine. 8

39 Table : Unconstraine case col col col col col5 k or k Cost or utility λ k ($/mwhr) λ k * k or enefit col- col ($/hr) λ k * k I think that the col shoul be Why it is not I o not know. Some error here.rounoff? ut note the value of the col5. This is the amount of benefit. It is NOT the same as the objective. Why? ecause the conestion rents must be pai as well, in this case, it is clear that they are $.5. 9