Alejandro Garcés Ruiz - Mathematical Programming for Power Systems Operation

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Explore the theoretical foundations and real-world power system applications of convex programming In
, Professor Alejandro Garces delivers a comprehensive overview of power system operations models with a focus on convex optimization models and their implementation in Python. Divided into two parts, the book begins with a theoretical analysis of convex optimization models before moving on to related applications in power systems operations.
The author eschews concepts of topology and functional analysis found in more mathematically oriented books in favor of a more natural approach. Using this perspective, he presents recent applications of convex optimization in power system operations problems.
Mathematical Programming for Power System Operation with Applications in Python A thorough introduction to power system operation, including economic and environmental dispatch, optimal power flow, and hosting capacity Comprehensive explorations of the mathematical background of power system operation, including quadratic forms and norms and the basic theory of optimization Practical discussions of convex functions and convex sets, including affine and linear spaces, politopes, balls, and ellipsoids In-depth examinations of convex optimization, including global optimums, and first and second order conditions Perfect for undergraduate students with some knowledge in power systems analysis, generation, or distribution,
is also an ideal resource for graduate students and engineers practicing in the area of power system optimization.

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Some properties of the supreme and the infimum are presented below:

27 28 29 - фото 35(2.7)

28 29 210 Moreover the last case implie - фото 36(2.8)

29 210 Moreover the last case implies that - фото 37(2.9)

210 Moreover the last case implies that 211 That is to say the value - фото 38(2.10)

Moreover, the last case implies that:

211 That is to say the value of x that minimizes the function f x α - фото 39(2.11)

That is to say, the value of x that minimizes the function f ( x ) + α is the same value that minimizes f ( x ); for this reason, it is typical to neglect the constant α in practical problems.

Example 2.3

Table 2.1shows some examples of maximum, minimum, supreme, and infimum.

картинка 40

Table 2.1. Bounds of some ordered sets.

Set sup max inf min
Ω 1= {1, 2, 3, 4} 4 4 1 1
Ω 2= { xкартинка 41: 1 ≤ x ≤ 2} 2 2 1 1
Ω 3= { xкартинка 42: 3 < x ≤ 8} 8 8 3 -
Ω 4= { xкартинка 43: 2 ≤ x < 9} 9 - 2 2
Ω 5= { xкартинка 44: 4 < x < 7} 7 - 4 -

2.2 Norms

In many practical problems, we may be interested in measuring the objects in a set, either as an objective function or as a way of analyzing solutions. A norm is a geometric concept that allows us to make this measurement. The most common norm is the Euclidean distance given by ( Equation 2.12)

212 However this function is not the only way to measure a distance In - фото 45(2.12)

However, this function is not the only way to measure a distance. In general, we can define a norm as a function ‖⋅‖:Ω→R that fulfills the following conditions:

Mathematical Programming for Power Systems Operation - изображение 46(2.13)

Mathematical Programming for Power Systems Operation - изображение 47(2.14)

Mathematical Programming for Power Systems Operation - изображение 48(2.15)

Mathematical Programming for Power Systems Operation - изображение 49(2.16)

The first two conditions indicate that a norm must return a positive value, except when the input is the vector 0→ The third condition indicates that it is scalable; for example, the norm must be twice the original vector’s norm if we multiply all the vector entries by 2. The last condition, known as the triangle inequality, is a generalization of the triangles’ property (therein lies its name). The sum of any two sides’ lengths is greater (or equal) to the remaining side’s length. This property is intuitive for the Euclidean norm, but surprisingly it is general for many other functions, such as ( Equation 2.17):

Mathematical Programming for Power Systems Operation - изображение 50(2.17)

This function is known as p-norm, where p ≥ 1. Three of the most common examples of p-norms in Rn nhave a well-defined representation, as presented below:

Mathematical Programming for Power Systems Operation - изображение 51(2.18)

Mathematical Programming for Power Systems Operation - изображение 52(2.19)

Mathematical Programming for Power Systems Operation - изображение 53(2.20)

The Euclidean distance is equivalent to a 2-norm whereas 1-norm, also known as Manhattan distance, consists in measuring the distance along axes at right angles (see Figure 2.2b), and infinity-norm or uniform norm, takes the maximum distance along axes as shown in Figure 2.2c). In general, ‖ x ‖ 1≤ ‖ x ‖ 2≤ ‖ x ‖ ∞. All of these norms are suitable ways to measure vectors in the space.

Mathematical Programming for Power Systems Operation - изображение 54

Figure 2.2 Three ways to measure the vector Mathematical Programming for Power Systems Operation - изображение 552-norm or Euclidean norm, b) 1-norm or Manhattan distance, c) infinity-norm or uniform norm.

We can use a norm to define a set given by all the points at a distance less or equal to a given value r , as given in ( Equation 2.21).

221 This set is known as a ball of radius r Figure 23 shows the shape of - фото 56(2.21)

This set is known as a ball of radius r . Figure 2.3 shows the shape of unit balls (i.e., balls of radius 1), generated by each of the three previously mentioned norms.

Figure 23 Comparison among unit balls defined by norm2 norm1 and norm - фото 57

Figure 2.3 Comparison among unit balls defined by norm-2, norm-1, and norm-∞

Notice that a ball is not necessarily round, at least with this definition. All balls share a common geometric property known as convexity that is studied in Chapter 3.

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