4.6 Problem approximation
4.6.3 Control Policies and intraday
In this section, we propose two different methods, to parametrize the control policies (4.13) and, thus, obtain a tractable reformulation of the planning problem (2).
Two-stage approximation
This solution method approximates the planning problem 2 by a two-stage robust optimization problem [125] that uses Ξts as the uncertainty set. Thus, instead of having multi-stage control
policies parametrized by uj = π
uj(zw, r) and D = ¯D + πD(zw, r), a two-stage control policy is considered. This implies that the causality requirements are relaxed and it is assumed that the uncertainty is revealed for the whole prediction horizon at once. More precisely, the first stage variables are represented by the power flexibility,γ, and the baseline profile, ¯D, which are decided before the realization of the random parameters (zw, r). On the other hand, the second-stage
decisions are represented byuj and D and they can be chosen after the uncertainty has been fully
revealed.
As an example, if we focus on the input realization for a single resource,uj, considering a two-stage
structure allows the optimizer to select different input trajectories, denoted asuj,(k), for each pos-
sible realization of the uncertainty contained in Ξts. A similar approach could also be considered
to parametrize the intraday control policy, πD. However, as underlined in [118], considering this
additional degree of flexibility in the problem formulation actually comes at a cost. In fact, due to the relaxation of the causality requirement, the solution of the approximated problem can poten- tially overfit the uncertainty realizations giving rise to unrealistic behaviours. As a consequence, we decided herein to separate the intraday control policy from the planning problem and to optimize it independently. In particular, we considered the offline causal policy firstly introduced in [118] which is reported in the following.
The tracking request, r, received by the TSO might exhibit considerable bias in either direction over short periods of time. When dealing with energy-constrained resources, such as BESSs or buildings, these energy biases could have a detrimental effect on the resources causing a reduction
4.6 Problem approximation 67
in the offerable flexibility, γ. Thus, the focus of the intraday policy is to reduce such biases by modifying the future baseline. As a consequence, rather than depending on both uncertain parameters, (zw,r), the policy is just a function of r, i.e., π
D(zw, r) = πD(r).
We first define the residual tracking signal,a, as the sum of the received tracking signal, r and the intraday transaction, τ :
a := r + τ
where τ is the normalized intraday transaction corresponding to the normalized tracking request, r. The total intraday transaction is then given by γτ .
The intraday policy is then given by:
πtsD(r) := τ τi+∆intra = argmin τi+∆intra
| ˜a[0→i]+ τi+∆intra+ Er˜a[i→i+∆intra−1]+ ri+∆intra | s.t. a˜[0→i] = ˜a[0→i−1]+ τi+ ri τ0= 0, ˜a[0→−1]= 0 ∀i = 0, ..., N − 1. (4.26)
where ∆intra denotes the number of steps, in the chosen sampling time, for which the baseline is
fixed according to the specific market rules (e.g. 45 minutes in current Swiss regulations).
Essentially, the objective of the intraday control policy is to reduce the energy content of the residual tracking signal, a. Thus, the policy is obtained by minimizing for each time instant, i, the expected absolute value, in ∆intra sampling times, of the cumulative sum (energy content) of
a. This is achieved through the following steps: 1) measure the current energy content, ˜a[0→i], 2)
choose a control action,τi+∆intra, such that the expected energy content of the signal in the future is minimized. At timei the uncertainty, and for the next ∆intra, the tracking signal has not yet been
realized. As a consequence, the policy minimizes the energy content in expectation. In particular, the expected value can be estimated at each step using scenarios of the tracking signal,r.
Once the PF linearization and the intraday policy are fixed, the planning problem 2 is approximated by the two-stage optimization problem resulting in the following:
Problem 3 (Two-stage approximation). minimize γ, ¯D,uj,(k) c T energyD − c¯ Trewardγ s.t. (Resource constraints) uj,(k)∈ Uj(ˆxj, ˆdj) ∀j = 1, . . . , N cont (4.27)
(Linearized PF) zend,(k)= Aczˆzcont,(k)+ Aw,(k)ˆz zw+ ¯zend (4.28)
(Grid constraints) zend,(k)∈ Z (4.29)
(Apparent power controllable res.) zcont,(k)= ΛU(k) (4.30)
(Power tracking) Γzcont,(k)= ¯D + γπts
D(r(k)) + γr(k) (4.31)
(Power flexibility) γ ≥ 0 (4.32)
(Scenarios) ∀(zw,(k), r(k)) ∈ Ξts. (4.33)
where, as already underlined, an implicit parametrization of the second-stage variables,uj is defined
by having separate trajectories,uj,(k), for each considered scenario. Please also notice how the pre-
determined intraday policy in (4.31), which was determined for the normalized tracking signal, r, is scaled by the capacityγ in order to obtain the total intraday transaction.
The resulting optimization problem is a linear program and can, therefore, be solved efficiently using available optimization software.
Multi-stage approximation
A second approximation scheme is proposed in this section. Its main advantage with respect to the two-stage approximation lies in its capability to retain the multi-stage structure of the original problem. In particular, to reduce the infite-dimensional decision space into a finite dimensional one, we consider affine feedback policies which offer a nice trade-off between performance and computational properties [6].
Thus, we define uj := Mj
rr + Mjww + mj and D := ¯D + Krr + Kww where, in order to ensure
causality, we impose the following constraints on the policies (for bothr and w):
M[k,l]j = 0 for l > k j = {1, . . . , Ncont}
K[k,l] = 0 for l > k − ∆intra
(4.34)
where, once again, ∆intra is imposed by the current market regulations.
The uncertainty set considered in this solution method isΞms, and the reformulated problem takes