International Journal of Electrical Power & Energy Systems
Calculation and decomposition of spot price using interior point nonlinear optimisation methods
Introduction
One of the key issues in the emerging electricity market is the spot pricing for both active power and reactive power. Recently, with the growing interests in determining the cost of ancillary services which are necessary for maintaining the reliable, secure and economical electricity supply, the spot pricing of ancillary services should be considered in the pricing model as well. Schweppe et al. [1] introduced the concept of Spot Price into power systems and summarised the ideas in their classic book. Starting from the economic theory, they point out that the optimal price for electricity can be described as:where C is the total cost of providing power to customers. ρi is the spot price of node i; Pdi is active load of node i.
Baughman and Siddiqi [2] develop this model by introducing reactive power pricing and reveal that λp and λq, Lagrangian multipliers corresponding to node power balance equations in Optimal Power Flow, represent the marginal costs of node power injections and they have the same economic meanings as active and reactive power spot prices, respectively. Dandachi et al. [3] takes account of the reactive power production cost by introducing Mvar cost curves which are the counterpart of the standard MW incremental cost curve. Successive LP method is applied to solve this nonlinear reactive power optimisation problem. El-Keib and Ma [4] introduce real power loss component into reactive power spot pricing derived from the Q-sub-problem whose objective function is real loss minimisation and the influence of reactive power on the local voltage level appears in the pricing formulas. The spinning reserve pricing problem is addressed in Ref. [5], and implemented by the reliability differentiated pricing model. Zobia and Ilic [6] focus on the ancillary service of energy imbalance compensation. Congestion alleviation cost, as the basic idea of security pricing, was advanced in Ref. [7], [8], and Kaye and Wu [8] extended this approach to consider pricing in the presence of contingencies and determine the optimal societal consumption and optimal prices. Alvarado et al. [9] consider the formal quantification of system security by computing the outage cost associated with specific operating points, as well as the influence of actions on this cost. Considering most ancillary services and incorporating constraints on power quality and environment, an advanced pricing prototype is recently introduced in Ref. [10], which combines the dynamic equations for load-frequency control with the static equations of constrained OPF. However, it is still uncertain how to solve this large scale stochastic optimal control problem attributed by the introduction of differential equations. In Refs. [11], [12], the prices for real and reactive power are decomposed into two components which represent the sum of generation and losses and system congestion. Rivier and Perez-Ariaga [13] demonstrate the physical meanings and numerical properties of spot price, especially focus on the interpretation of ‘slack bus’ and ‘system lambda’. Li and David [14] try to calculate the wheeling marginal cost of reactive power.
For state-of-the art spot pricing methods, as the comments of Ref. [12] mentioned, congestion would appear to be a ‘go’ ‘no go’ gauge, that means congestion cost will appear only if congestion happens. The spot prices change too harsh to be predicted and responded at congestion points. A more relaxed approach with the smooth increasing congestion cost when the operation constraints approach the normal limiting values will be more practical.
This paper reveals that λp,λq, the Lagrangian multipliers of node power balance functions in optimal power flow, can play an important role in spot pricing of electricity. They not only have the similar economic meanings as spot price (shadow price), but also can be further decomposed into different components reflecting the effects of various ancillary services. The derivation provided is based on the concepts of economy theory, classic economic dispatching and optimal power flow instead of abstract mathematics theory. Thus this makes it more readily to be understood by system operators. After summarising this optimal spot pricing model as a nonlinear optimisation problem, the primary-dual interior point algorithm is employed to solve it, which permits the efficient handling of large sets of equality and inequality constraints within the problem solution and can avoid the go ‘no go’ problem. Case studies on IEEE 30-bus systems are employed to gain the insight into the proposed method.
Section snippets
The proposed formulation
The spot pricing model (1) can be generalised as an extended optimal power flow problem
(i) Objectives:
Both active and reactive power production costs are considered in the objective function and simulated by quadratic curves.
(ii) Constraints:
Real power balance:Reactive power balance:Generator capacity limits:
Test result
The IEEE 30-bus system is used as an example. It has six generators and four ULTC transformers. A loading condition of 324.29 MW and 114.19 MVA is assumed in the study. The units of all quantities are given in per unit. The quadratic cost data are given in Table 1. All bus voltage upper limits are 1.05, lower limits are 0.95. The unit of spot price for active power and reactive power are €/MWh and €/Mvarh. Define:
Normal operation conditions
Fig. 2 reveals that active power spot price is dominated by system lambda and network loss compensation term on normal operation conditions. The congestion and security terms are relatively small. This conclusion is the same as Ref. [4]. Reactive power spot price at all buses are dominated by reactive power generation cost and real power loss compensation term, as shown in Fig. 3. Compared with active power, reactive power spot price is much more volatile. The security prices for both active
Congestion conditions
Line constraints force the use of a higher loss path to satisfy the demand requirements, and may also require the reallocation of generation which would increase the total cost (Fig. 4). Fig. 5, Fig. 6 show the variations of active power and reactive power spot prices when the power flow limits for line 3 are approached (The power flow limit is 68 MW, the optimisation result is 67.999 MW). Security prices are increased for all buses, especially for buses 5 and 7, as they are most sensitive to the
Voltage violation
Voltage profiles are determined mainly by reactive power supply. Therefore, voltage constraints as well as the generation and consumption pattern of reactive power by the customers have the greatest impact on the spot prices of reactive power. Tightening the voltage lower limits of bus 7 (increase it from 0.95 to 0.99), the reactive power security prices are increased rapidly, because it is dominated by voltage constraints at this situation. The phenomenon can be interpreted by Eq. (28). To
Alleviating “go” no go gauge
In order to illustrate how the PDIP alleviates go no go gauge, Fig. 8 shows the response of active power spot price of bus 5 due to active power flow limits on line 3. Under normal operation conditions, the line flow limit of line 3 is set to 100 MW and the active power flow on line 3 is 72 MW. When the line flow limit is decreasing from 100 to 74 MW, the spot price of bus 5 keeps almost the same value. This is because the security price is negligible. Once it is approaching to 72 MW, the security
Conclusions
This paper reveals that λp, λq, the Lagrangian multipliers of node power balance functions in optimal power flow, can play an important role in spot pricing of electricity. They not only have the similar economic meanings as spot price (shadow price), but also can be further decomposed into different components reflecting the effects of various ancillary services. After summarising this optimal spot pricing model as a Nonlinear optimisation problem, the primary-dual interior point algorithm is
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