Optimization in modern power systems

Lecture 6: Reviewing Week 1

Spyros Chatzivasileiadis

The Goals for Today!

• Review of Week 1: Questions

• Questions and Clarifications on Assignments 1 and 2

• Feedback about the course so far

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Review of Week 1

By the end of Week 1 you must be able to:

1 Formulate an optimization from a given problem description and solve it

2 Formulate the Economic Dispatch problem and solve it

3 Formulate the DC-OPF problem and solve it

4 Formulate the Bus Susceptance Matrix, and formulate and explain thePTDFs

5 Draw the Merit Order curve, and identify the marginal generators

6 Formulate the Lagrangian of a constrained optimization problem

7 Derive the KKT conditions

8 Derive the LMPs from the DC-OPF problem

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Reviewing Week 1 in Groups!

• For 10 minutes discuss with theperson sitting next to you about:

• Three main points we discussedlast week

• One (at least one!) topic orconcept that is not so clear toyou and you would like to hearagain about it

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Some take-aways from Week 1

• (The list is not complete!)

• The AC-OPF is a non-linear non-convex problem

• The DC-OPF is a linear convex problem

• Note: The DC-OPF can also be formulated as a quadratic problem, ifinstead of a linear objective function, we include quadratic costs in theobjective function. This is still convex.

• The Economic Dispatch assumes a copperplate network: noconsideration of power flows, assuming that there are no congestions.

• The Economic Dispatch is also a convex problem.

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Some take-aways from Week 1 (cont.)

• Locational Marginal prices in DC-OPF

• LMPs are equal on all nodes if there is no congestion• LMPs are di↵erent on di↵erent nodes if there is congestion

• Calculate the LMPs

• Standard Power Flow equations: the lagrangian multipliers ⌫

i

of theequality constraints

• PTDFs: LMPi

= ⌫ +P

lm2L

PTDF

lm,i

· �lm

, where ⌫ is lagrangianmultiplier of the equality constraint, and �

lm

is the Lagrangianmultiplier of the congested line l � m. If line l � m is not congested,then �

lm

= 0

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DC-OPF discussion points

• sin � ⇡ �

•� is in rad!

• B · ✓ = P

• B is in p.u.•✓ is in rad, ) dimensionless

• P must be in p.u.

• Bus Admittance Matrix B in DC-OPF

•b

ij

= 1xij

) positive

• all o↵-diagonal elements are non-positive (zero or negative)• all diagonal elements are positive• AC-OPF: This di↵ers from the case where z

ij

= r

ij

+ jx

ij

. In thatcase, it is y

ij

= g

ij

+ jb

ij

with b

ij

is negative.

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PTDF: An Example

48 Chapter 4. Mathematical Modeling CM Schemes

elasticity.

4.2.4 Modeling of the Transmission System

DC Power Flow Approximations

The above sections outlined the models used for the representation ofthe demand (loads) as well as the supply side (generators). In order toestablish a marketplace between both entities the power produced mustbe transported via the transmission grid, causing a certain power flowover the transmission lines. Figure 4.5 presents an example network asconsidered within the scope of the simulator.

~

~~

~

node 1

node 2

node 3

node 4

node 5

G1

G2

G3

G4

L1

L2

L3 L4

L5

K1

K2

0.00

64

0.0281

0.0304

0.0297

0.02

97

Zone 1

Zone 2

Zone 3

Figure 4.5: Power System Description

As shown, generators and loads are connected by transmission lines,which ‘carry’ a certain flow. For modeling the transmission lines anddetermining the flow, the so-called telegraph equations can be used.However, a simpler and more commonly used representation is the �-line model, which can be directly derived from the telegraph equations.The �-model is characterized by the series resistance, the series reac-tance as well as the shunt susceptance and the shunt conductance ofthe transmission line. Using this model in conjunction with representa-tions of generators and loads, the active and reactive power flows on thelines can be determined. Unfortunately, this approach of computing the

4.3. Summary 51

displayed in figure 4.5. Node 1 is assumed to be the slack bus.6 Therows in the table correspond to the nodes k, the columns correspondto all transmission lines l in the network. The computation of such aPTDF-matrix is for instance described in [23].

1-2 1-3 2-5 3-4 4-51 0 0 0 0 02 -0.9432 -0.0568 0.0568 -0.0568 -0.05683 -0.2496 -0.7504 -0.2496 0.2496 0.24964 -0.5133 -0.4867 -0.5133 -0.4867 0.51335 -0.6732 -0.3268 -0.6732 -0.3268 -0.3268

Table 4.1: Matrix of power transfer distribution factors for the networkpresented in figure 4.5

Using the matrix in table 4.1, the line flow equation (4.3) can be rewrit-ten as follows:

Pl =X

k

P (k, l)(Pk) (4.8)

Consistent with equation (4.4), Pk determines the sum of injections andwithdrawals at node k, Pl denotes the flow on line l and P (k, l) is theelement in the k

th row and l

th column. Initially, the advantage of re-formulating the line flow equations through PTDFs might not appearobvious. It is clear that there are no benefits from a physical modelingviewpoint. Nonetheless, the use of PTDFs simplifies the latter formula-tion of the optimization models for the di�erent congestion managementschemes. With PTDFs it is possible to state the problem clearly distin-guishing power production and its subsequent transmission as shown inequations (4.11) to (4.16). The argument is detailed in the correspond-ing section 4.4.

4.3 Summary

Above the economic and technical modeling prerequisites for power sys-tems have been described. It was outlined that one crucial objective for aliberalized marketplace for electricity is to maximize social welfare. Un-fortunately, this objective is constrained by physical as well as economic

6Note, that the location of the slack bus will not change the line flows obtained.

Figures taken from: T.Krause, Evaluating Congestion Management Schemes in Liberalized Elec-

tricity Markets Applying Agent-based Computational Economics, PhD Thesis, ETH Zurich, 2007.

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KKTs for Constrained Optimization

• Minimize Lagrange function:

L = f0(x) +mX

i=1

�

i

f

i

(x) +pX

i=1

⌫

i

h

i

(x)

• The Karush-Kuhn-Tacker first order or necessary optimality conditions:

@L

@x

= 0 ) @f(x)

@x

+mX

i=1

�

i

@f

i

(x)

@x

+pX

i=1

⌫

i

@h

i

(x)

@x

= 0

@L

@�

i

= 0 ) f

i

(x) 0 for i = 1, . . . ,m

@L

@⌫

i

= 0 ) h

i

(x) 0 for i = 1, . . . , p

�

i

f

i

(x) = 0 for i+ 1, . . . ,m

�

i

� 0

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Costrained Optimization: Example

minx1,x2

(x1 � 3)2 + (x2 � 2)2

subject to:

2x1 + x2 = 8

x1 + x2 7

x1 � 0.25x22 0

• Write down the KKT conditions for this problem.

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Constrained Optimization: Graphical Solution

Example:

minx1,x2

(x1 � 3)2 + (x2 � 2)2

subject to:

2x1 + x2 = 8

x1 + x2 7

x1 � 0.25x22 0

x1 � 0

x2 � 0

Figure taken from: Gabriela Hug, Lecture slides for class 18-879 M: Optimization in Energy Net-works, Carnegie Mellon University, USA, 2015.

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Convex Optimization

• The optimization problem

minx

f(x)

f

i

(x) 0 for i = 1, . . . , p

h

i

(x) = 0 for i = 1, . . . ,m

is convex if:

• the objective function f(x) is convex

• the inequality constraints f

i

(x) are convex

• the equality constraints h

i

(x) are linear

If the problem is convex, there is a single optimum, which is also the

global optimum!

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Some Questions

1 What are the di↵erences in the formulation of the DC-OPF based on thestandard power flow equations and based on the PTDFs?

• What is the meaning of PTDFs?• How do we calculate the PTDFs?

2 How do we determine the minimum of a function in unconstrainedoptimization?

3 What is a convex, a concave, and a non-convex function? Give anexample for each.

4 Are there guarantees that we can find the global optimum of a convex, aconcave, or a non-convex function? Explain for each case.

5 What is the sign of the Lagrangian multipliers for the equality andinequality constraints. Explain.

6 Draw a graphical solution of 2-dimensional optimization problem.

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Points you would like to discuss?

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Questions about Assignment 1 or 2?

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Feedback on the course so far?

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