Constructing Empirically-Based System Pattern Transition Matr

and Its Applicability

3.2.4.1 Empirically-Based System Pattern Transition

Matrix Construction

From historically observed system pattern data, an empirically-based system pattern transition matrix can be constructed for each particular hour. Suppose the current time is Day D at Hour H and historically observed system pattern data since the beginning of Day D − K are available for constructing the empirically-based system pattern transition matrix. In addition, assume that J is the total number of observed system patterns from the historically observed data. Then the system pattern transition matrix from Hour H

ΓH+1|H =          γ₁₁H+1|H γ₁₂H+1|H · · · γ_1JH+1|H γ₂₁H+1|H γ₂₂H+1|H · · · γ_2JH+1|H .. . ... γ_jjH+1|H0 ... γ_{J 1}H+1|H γ_{J 2}H+1|H · · · γ_{J J}H+1|H          (3.26)

where γ_jjH+1|H0 denotes the sample probability of the system pattern transition from pat- tern j at Hour H to pattern j0 at Hour H + 1. The component γ_jjH+1|H0 is calculated by dividing the total frequency of the historically observed system pattern transition from

pattern j at Hour H to pattern j0 at Hour H + 1 by the total frequency of the histor-

ically observed system pattern j at Hour H.4 _{The transition matrix has the following}

well-known property:

ΓH+2|H = ΓH+1|H× ΓH+2|H+1, ..., ΓH+h|H = ΓH+1|H × · · · × ΓH+h|H+h−1 (3.27)

Thus, the system pattern transition matrix from Hour H to H +h can be easily calculated by means of matrix multiplications.

Net load patterns can be different by weekday and weekend, seasons, and months (or combinations of these factors). For better goodness of fit, the historically observed system pattern data can be segmented based on these factors and their combinations. The segmented empirically-based system pattern transition matrix can be constructed from the corresponding segmented data. For example, the system pattern transition matrix from Hour 12 to Hour 13 during a weekday can be constructed from the corresponding historical data. As the number of segmentations increases, however, the sample size of each specific system pattern transition matrix decreases. Thus, a sufficiently large amount of historically observed system pattern data is necessary for more segmentation.

3.2.4.2 Applicability to Status Forecasting of System Variables

System patterns are derived from solutions of DC-OPF problems subject to the phys- ical power system constraints. Therefore, historically observed system pattern data con- tain reduced forms of information for the power generation levels and the line congestion. For each generating unit i, we can define sets corresponding to i) a minimum power level system pattern set, SPi,−1_{, ii) a marginal power level system pattern set, SP}i,0_,

and iii) a maximum power level system pattern set, SPi,+1_{, as follows:}

SPi,−1 = {∪SPj, j ∈ J : gi = −1} (3.28)

SPi,0 = {∪SPj, j ∈ J : gi = 0} (3.29)

SPi,+1 = {∪SPj, j ∈ J : gi = 1} (3.30)

Assume that the system pattern at Hour H is j. Then, for each generator i, the sample probability of minimum capacity generation at Hour H + h conditional on the given SPj at Hour H, P ri,−1_H+h|H, the sample probability of marginal generation at Hour

H + h conditional on the given SPj at Hour H, P r

i,0

H+h|H, and the sample probability of

maximum capacity generation at Hour H + h conditional on the given SPj at Hour H,

P ri,+1_H+h|H, can be expressed as follows:

P r_H+h|Hi,−1 = X j0_∈SPi,−1 γ_jjH+h|H0 (3.31) P r_H+h|Hi,0 = X j0_∈SPi,0 γ_jjH+h|H0 (3.32) P r_H+h|Hi,+1 = X j0_∈SPi,+1 γ_jjH+h|H0 (3.33)

The status of each transmission line can also be forecasted from this matrix. For each transmission line τ ∈ T , we can define sets by congestion status: i) a negative congestion system pattern set, SPτ,−1_{, ii) a non-congestion system pattern set, SP}τ,0_{, and iii) a} positive congestion system pattern set, SPτ,+1_{. These sets are defined as follows:}

SPτ,−1 = {∪SPj, j ∈ J : lτ = −1} (3.34)

SPτ,0 = {∪SPj, j ∈ J : lτ = 0} (3.35)

SPτ,+1 = {∪SPj, j ∈ J : lτ = +1} (3.36)

Then the sample probability of negative congestion at Hour H + h conditional on the given SPj at Hour H, P r_H+h|Hτ,−1 , the sample probability of no congestion at Hour H + h conditional on the given SPj at Hour H, P r_H+h|Hτ,0 , and the sample probability of positive

congestion at Hour H + h conditional on the given SPj at Hour H, P r_H+h|Hτ,+1 , can be

denoted as follows: P rτ,−1_H+h|H = X j0_∈SPτ,−1 γ_jjH+h|H0 (3.37) P rτ,0_H+h|H = X j0_∈SPτ,0 γ_jjH+h|H0 (3.38) P rτ,+1_H+h|H = X j0_∈SPτ,+1 γ_jjH+h|H0 (3.39)

Thus this approach can forecast the status of the system variables by means of the corresponding calculated probabilities although it cannot provide the specific forecasting values of the system variables.

3.2.5 Applicability of Extended System Pattern Method to

Load Scenario Reduction

Centrally-managed wholesale power markets in the U.S. are structured as forward markets in advance of real-time operations. To make informed decisions in the forward markets for electric energy and reserve, market managers would ideally like to be able to forecast system variables under the set of all possible load scenarios [70]. However, the set of all possible load scenarios is too large to consider in practice. To reduce computational

complexities and time requirements, forecasting models with large number of scenarios are often approximated by models with relatively small number of scenarios [44].

The system pattern method can be applied to load scenario reduction processes in two ways. First, it can reduce all net load variations into a limited number of scenarios.

As pointed out in Section3.2.2, although the total number of system patterns in a huge

power system is expected to be a large number, the total number of feasible system patterns would be less than the total number of system patterns and the total number of historically observed system patterns would be even less than the total number of feasible system patterns. Thus, the historically observed system patterns can be small enough to handle in practice, even in a huge power system.

Second, the system pattern method can provide reasonable criteria for the classi- fication of load scenarios. Each SPR corresponds to a unique combination of binding constraints in the power system. Any severe net load volatility cannot affect power sys- tem constraints, such as the status of the generating units and the transmission line congestion, if the load fluctuations are bounded within the same SPR. Thus any net load volatility within the same SPR is manageable under the same operating condition (or binding constraints). Under this circumstance, we can conjecture that the LMP volatil- ity is relatively lower when the net load fluctuates within the same system pattern, while the LMP volatility is relatively higher when the net load fluctuates across the system patterns, because LMPs are piecewise linear functions of net load, hence bigger net load fluctuations will typically result in bigger LMP deviations.