Matching Policies

From energy conservation follows, that the power flowing into the power system has to either flow out or be consumed in the system. The generation and load at vertex iare denoted byg_i andl_i, respectively. The mismatch, defined as sum over all generation and load m = P

(g_i+l_i), is typically nonzero. To ensure the conservation of energy, the load and the generation have to be modified by some method so that they match, i.e.

Pg^b_i = P

l^b_i. There are unlimited degrees of freedom to modify the load and generation, accordingly. We introduce two methods, that we refer to as “matching policies”.

The minimum dissipation policy, described in Section 5.1.1.1, minimizes the sum over the quadratic flows. It is a benchmark as this minimizing property is known a priory.

In Section 5.1.1.2, we introduce the global factor policy, which distributes the mismatch over all vertices relative to their load or generation and can be treated analytically.

Various other policies are possible. In real systems, the employment of capacities is usually assigned based on the costs of the generators and transport [107]. The implementation of such a policy would incorporate a modelling of different classes of generators with their cost functions, their spatial distribution, and rules on economic decision making. This is beyond the scope of this work. However, as the cost of transmission are also considered, the economic dispatch can be expected not to deviate strongly from the minimum dissipation policy.

In the following, the in- and out-flows before the policy is applied are referred to as

“unmatched”. The matched in- and out-flows, that fullfill the conservation of energy condition, are denoted by s^b_j =l^b_j+g^b_j.

5.1.1.1. Minimum Dissipation

A policy that minimizes the aggregated flows in the network is of great advantage. It defines a lower bound that can be used for comparison. Trivially, to minimize the flows, the load at vertex i should be satisfied by the generation at the same vertex, as this elicits no flow in the network. Thus, we define the unmatched in- and out-flow at vertex i as si =gi+li and the matched in- and out-flows for all vertices by

~s^b =~s+~s^c, (5.8)

with s^c_i the correction at each vertex so thatP

is^b_i = 0. The minimal flow has to meet some constraints. The flow needs to satisfy Equation (5.2). For positive s_j, referred to as sources, the constraint 0 ≤s^b_j ≤s_j has to be met. For sinks, with s_j ≤0, it is the constraint s_j ≤s^b_j ≤0, in order not to introduce artificial generation or load. Further,

~

s^c has to satisfyP

is_i =m =−P

is^c_i.

Another difficulty is that the flowf_ij along the link i→j can be positive or negative, depending on the direction. Using the absolute value in the objective function would

5.1. DC Power Flow Approximation

introduce a rather inconvenient nonlinearity. Minimizing the quadratic flow f_ij² solves this problem, leading to a constrained nonlinear optimization problem, also referred to as nonlinear programming, which is not easy to solve. Gertz and Wright [55] propose a method to solve problems of the form

min

~

x ~xQ~x (5.9)

with the constraintsA~x=~bandx^l_i ≤xi ≤x^u_i, where x^l_i andx^u_i are the respective upper and lower bounds. To convert our problem into this form, we define the vector ~x as

~x= f~

~s^c

, (5.10)

wheref~is a vector of length M³ denoting the flows on the links. The in- and out-flow correction ~s^c for each vertex is of length N. Since only the flows should be minimized, the matrix Q has the form

Q=

1_M 0 0 0N

, (5.11)

where 1_M denotes a diagonalM ×M unit matrix and 0_N anN ×N zero matrix.

The constraints have to be of the formA~x=~b. As shown in Section A.2.4, the in/out flows ~s^b are related to the flow on the linksf~_ij by the incidence matrixK,~s^b =Kf~_ij. Using this relation, Equation (5.8) can be written as

~

s=~s^b −~s^c=Kf~_ij −~s^c=A⁰~x⁰ (5.12) with

A⁰ = K IN

. (5.13)

The constraint P

js^c_j = −m can be incorporated into the linear constrain equation A~x=~b via

A=

K I_N 0 1 . . . 1

. (5.14)

and

~b= ~s

−m

, (5.15)

Additional conditions for all sources are 0≤s^b_j ≤sj and for sinks sj ≤s^b_j ≤0.

So far, Equation (5.2), P~ = B⁰~δ, is not implemented explicitly. As argued in Section A.2.4, the matrix B⁰ is formally equal to the admittance matrix Yres describing

3M denotes the number of edges.

a resistor network. The flow obeying Kirchhoff’s laws is known to minimize the dissipation in a resistor network [19]. The dissipationp^d_ij at the linki→j is the product of the current fij and the voltage difference (ui−uj) at the resistor Rij. With Ohm’s law, we find

p^d_ij =f_ij(u_i−u_j) = R_ijf_ij² . (5.16) The dissipation of the whole network is defined as the sum of the energy dissipated at all links

p_d= X

ek∈E

R_ekf_e²

k (5.17)

with E the set of edges and e_k denoting an edge i → j, so that e.g. R_ek = R_ij. For R_ij = 1, Equation (5.17) is equal to the objective function in Equation (5.9). As shown in Section A.2.4, Kirchhoff’s current law can be written as ~s^b =Kf~_ij, where K is the incidence matrix. By Thomson’s Principle⁴ [73], we know that if the current law is met and the flow is minimal, Kirchhoff’s potential law is also fullfilled. Because of the formal equality of B⁰ and Y_res and of the conservation of energy, P~ = B⁰~δ (see Equation (5.2)) has to be satisfied forB_ij = 1. ArbitraryB_ij can easily be implemented by extending the vector~xin Equation (5.10) with a vector containing the voltage phases

~δ. Further, the constraint X K^T~δ −f~ = 0, where X is a diagonal matrix with the susceptances on the diagonal, is added to the matrixAin Equation (5.14) together with further zeros in Q in Equation (5.9). This is not done here, since we use only uniform susceptances B_ij = 1 in this work. Equation (5.5) is consequently fullfilled because the flow is minimal and the additional constraints would only cause a higher complexity for numerical evaluation that is thus avoided.

5.1.1.2. Global Factor Policy

A very simple approach is to divide mismatches with equal shares over all vertices.

For shortages, all vertices have to shed a fraction of the load and the same is true for surpluses concerning generation. Sinks and sources at each vertex are then multiplied with a common factor in order to match generation and load,

g_i^b = (1−α⁰)g_i l_i^b = (1−β⁰)l_i . (5.18) To ensure that form= P

isi = P

igi+li >0 only the generation is reduced, we require that 0≤α⁰ ≤1 withβ⁰ = 0. α⁰ = 0 and 0≤β⁰ ≤1 causes loads to be shedded only for negative m.

4The currents in a resistor network that fullfill Kirchhoff’s circuit laws, are the unique flows that minimize the dissipation [73].