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日期:2019-05-27 10:25

DEPARTMENT OF ENGINEERING SCIENCE

ENGSCI 391: Computing Assignment Part II – 2015 (10 marks)

Figure 1 shows a simple network representation of the New Zealand electricity transmission

grid. This is an approximation of the full grid which has about 250 nodes

at which electricity is priced in the wholesale electricity market1.

Figure 1: New Zealand transmission grid model.

You must formulate and solve a model of the linear program that dispatches power

from generators at nodes of the grid in Figure 1 to consumers in this grid. The basic

form of this model is as follows.

Here fk is the power flow in MW in line k in the direction shown, and xg is the power

generated in MW by generator g. Generator g has capacity Gg MW. The flow in

each line k can be positive or negative but must lie between Kk and Kk. The net

1You can find a map of the full New Zealand grid here.

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flow into each node is "

fk where I (n) indexes arcs with heads

at node n and O (n) indexes arcs with tails at node n. The first set of constraints

holds at each node and says that the production of electricity at each node and the

net flow in must add up to the demand dn for electric power in node n. Note that

L (n) indexes the generators at node n. The objective is the total hourly cost of

generating power at the prices φg ($/MWh) offered by each generator.

This exercise will use the data in Tables 1 and 2 (all data are in MW or $/MWh).

These very approximately represent the system data in a typical hour in June 2008

assuming full HVDC capacity2

.

Generator Node Capacity Offer Price

Table 1: Generator locations, capacities (MW), and offer prices ($/MWh).

Node Demand

Table 2: Demand at the nodes and the line capacities (MW).

2Actually most of the line capacities are guesses as they represent maximum transfers between

regions rather than the capacities of actual lines.

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1. Write some Matlab code that will build a constraint matrix for this problem.

The code should take as input vectors for φ, G, K and d, and produce matrices/vectors

A, c, and b, so that SD is in standard computational form, as

below:

The code should be able to recover the original optimal variables from the

solution to SDCF. The code should build A in blocks. So adding upper bound

constraints on variables can be done by adding a block of rows to a matrix

using:

newA=[oldA,zeros(m,n);eye(n,n),eye(n,n)];

2. Using the data in Tables 1 and 2 solve SD using your code assuming all Kk

and Gg are infinite (i.e. ignore these constraints). Check that the solution

makes sense and explain in one sentence why it is the solution you obtain.

3. Using the data in Tables 1 and 2 solve SD using your code assuming all Kk are

infinite (i.e. ignore these constraints). Check that the solution makes sense

and explain in one sentence why it is the solution you obtain.

4. Include the real Kk values and solve the problem. Present the optimal solution

in a table that shows:

(a) the generation of each generator;

(b) the flow in each transmission line;

(c) the total cost of generation;

(d) the energy price at each node (how much extra one extra MWh at node

n would cost).

5. The flows in the transmission lines must meet some loop flow constraints dictated

by Kirchhoff’s laws for DC load flow. These are as follows:

fHN + fNW = fHNP + fNPW,

fDB + fBC = 0.4 × fDC.

Add these to your model and re-solve it. Display:

(a) the generation of each generator;

(b) the flow in each transmission line;

(c) the total cost of generation;

(d) the energy price at each node.

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6. Keep the constraints added in question 5. It is known that up to 10% of power

flowing on the (HVDC) line between B and W (in either direction) is lost as

heat in transmission. In fact each MW in the first 500 MW loses 5% of the

power sent in transmission, and each MW of power flow above this loses 15%

of what is sent. (This means that if 600MW is sent then 40MW is lost so

only 560MW arrives at the other end.) Carefully reformulate the constraints

involving this line to model these losses and solve the problem again using

your code. Display:

(a) the generation of each generator;

(b) the flow in each transmission line;

(c) the total cost of generation;

(d) the energy price at each node.

What to hand in:

Hand in a listing of the Matlab code that builds the problem, and the answers to

questions 1 through 6.

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