A stochastic, contingency-based security-constrained optimal power flow for the procurement of energy and distributed reserve

Highlights

• Co-optimization of energy and reserves, including system contingency requirements

• Complete AC power flow formulation with static and dynamic security constraints

• Lagrange multipliers determine various day-ahead and spot market commodity prices.

• Comparison with traditional method shows improvements in system security and costs.

Abstract

It is widely agreed that optimal procurement of reserves, with explicit consideration of system contingencies, can improve reliability and economic efficiency in power systems. With increasing penetration of uncertain generation resources, this optimal allocation is becoming even more crucial. Herein, a problem formulation is developed to solve the day-ahead energy and reserve market allocation and pricing problem that explicitly considers the redispatch set required by the occurrence of contingencies and the corresponding optimal power flow, static and dynamic security constraints. Costs and benefits, including those arising from eventual demand deviation and contingency-originated redispatch and shedding, are weighted by the contingency probabilities, resulting in a scheme that contracts the optimal amount of resources in a stochastic day-ahead procurement setting. Furthermore, the usual assumption that the day-ahead contracted quantities correspond to some base case dispatch is removed, resulting in an optimal procurement as opposed to an optimal dispatch. Inherent in the formulation are mechanisms for rescheduling and pricing dispatch deviations arising from realized demand fluctuations and contingencies. Because the formulation involves a single, one stage, comprehensive mathematical program, the Lagrange multipliers obtained at the solution are consistent with shadow prices and can be used to clear the day-ahead and spot markets of the different commodities involved.

Introduction

This work combines several standard problems found in systems operation and planning in a single mathematical programming framework. The advantages of this formulation are found in greater clarity with respect to the underlying problem to be solved, and for ease of extraction of sensitivity information from the solution. The problems herein considered are

• The optimal power flow problem with a full AC nonlinear network model and constraints;

• The N − 1 contingency security problem with both static (post-contingency voltage and MVA limits) and dynamic (generator ramp rate limits; voltage angle difference limits; post-contingency load pickup governed by participation factors) constraints;

• The problem of procuring an adequate supply of both active and reactive power and corresponding geographically adequate distributed reserves in a day-ahead market scenario in light of the uncertainty of the actual realized demand and the occurrence of specific contingencies, while taking into account the costs and constraints on the corresponding post-contingency flows;

• The problem of setting the price for the day-ahead contracts for power and reserve; and

• A consistent mechanism for re-dispatching and pricing the next day under a specific realization of the set of all uncertain quantities involved.

Each of these problems is usually tackled separately, in a sequential process that revises the original dispatch produced by an optimal power flow solver to accommodate the additional restrictions. However, the sequential nature of typical practice does not ensure that these are introduced in a way that preserves optimality for the overall problem, nor allows for the original LBMPs to be used correctly for pricing both active and reactive power and reserve, or for understanding the price of security. The approach employed here tries to accommodate as many of the issues involved as possible in a single, consistent mathematical program, avoiding the use of proxies of the constraints. The specific novelty in this work lies in 1) the decoupling of the concept of day-ahead programmed dispatch and day-ahead contracted quantity, resulting in an optimal day-ahead hedge for the system operator; 2) a single stage, comprehensive problem formulation for energy and reserve allocation that is appropriate for extraction of sensitivity information important to microeconomics, namely, meaningful location-based shadow prices. The resulting problem is formidable to solve but it exhibits a structure that is amenable to decomposition and coordination approaches to its solution, making a parallel implementation possible and desirable.

Secure operation of generation and transmission systems addresses a plethora of issues. It involves planning so that the system can survive the occurrence of certain kinds of events, most notably so-called “contingencies”, in which a piece of equipment goes offline suddenly. But it also involves planning so that the system can continue to perform if the operating conditions expected at the decision-making moment do not materialize exactly, i.e. if there is uncertainty in the prediction of load, climate, wind or river flow. Of these two types of issues, perhaps the first results in more acute concerns, because a sudden realization of a contingency disturbs the state of the system before much can be done by the operators.

Several events occur in different time frames after a contingency. First, new bus voltages can be reached in a matter of seconds as the transient governed by automatic reactive controls takes place. If the controls steer the voltage towards a stable equilibrium, it still remains to be seen if the overall voltage profile that is reached is appropriate. In a longer time scale involving tens of seconds, frequency controls steer generators to balance the active power and make up for lost generation or load. Under-frequency relays may trigger network reconfiguration events in extreme cases at this stage. In a time frame of a few minutes, area exchange controls balance deviations from scheduled transactions, and operator-originated redispatches start to take place. In some cases, an automatic redispatch is initiated right after the contingency in order to improve the security and economy of the initial post-contingency operating point.

A key planning decision is the amount and location of spinning reserve that must be set aside for eventual use in case of a contingency. The required redispatches might not be feasible otherwise. Thus, correctly solving the planning problem requires addressing the issue of geographically appropriate reserve allocation. Furthermore, correct pricing of this commodity requires that it be explicitly included in the formulation.

A taxonomy of system states with respect to security is offered in [1]. The normal state is that of “secure”, when no operating limits are violated and no limits would be violated in the event of a contingency. Secure operation requires planning with respect to credible contingencies in order to both position the current state accordingly and to plan for corrective rescheduling strategies in the event that one of them does occur. There are many approaches to solving this problem, depending on the formulation, the simplifications, the available tools, and on the numerical method used. Some are only approximate in light of the simplifications, e.g. DC flow instead of AC flow, and require further examination before claiming that the solution is engineering-feasible. Others do not produce accurate pricing information due to the nature of the solution method employed, or the use of proxy constraints instead of precise models of the physical limitations. One key criterion is whether the approach is 1) direct, 2) base flow data modification-based or 3) base flow with added self-contained constraints. The first approach is used, for example, in [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] and involves actual simultaneous formulation of the post-contingency flows with additional constraints that bound the deviations of the injections in the post-contingency flows from those in the base case. These are the only coupling constraints; voltage security and rating limits are imposed directly on the post-contingency flows. Clearly, as more contingencies are considered the problem's size becomes formidable and it is tempting to exploit the problem structure with a decomposition framework, typically a price coordination scheme such as Benders' decomposition or Lagrangian relaxation, among others.

The second idea relies on modification of the original problem data for the base case OPF so as not to violate limits in a post-contingency state. A typical example is to artificially reduce the rating in a transmission line or the maximum generation capacity in a given unit to alleviate a congestion problem that would occur in a post-contingency state. This is amenable to sequential modification of a base case OPF after a given OPF solution is analyzed and found to be insecure with respect to contingencies. However, the order in which contingencies are studied might be important in determining the final secure dispatch, which raises the possibility of not finding the true optimum.

The third idea adds more constraints to the base case OPF to force the resulting solution to be secure. Like the second approach, it is amenable to sequential introduction of constraints into the base OPF, dictated by an analysis of the security of a given solution. These new constraints may typically be linearizations of the constraints that were violated in a post-contingency flow, and are thus proxies that may not be entirely accurate.

We now discuss some of the ingredients of the overall problem and how they have been dealt with over the years. Every now and then, reference will be made to specific Matpower implementation conventions and algorithms. This stems from the fact that this software package's generalized optimal power flow capabilities have been taken advantage of in order to code the prototype implementation. A detailed description of its capabilities and algorithms can be found in [13], [14].

Survival of a contingency implies a state trajectory that does not exceed system ratings or operating limits and which reaches an equilibrium that does not violate any limits. Then, the system can be steered towards a more economical and secure operating point with the resources available. The initial response is automatic, as voltage, frequency and automatic generation controls respond to errors. Assuming that no dynamic instability occurs, the final resting point is easy to predict when the ramp rates, participation factors, scheduled area interchanges and voltage setpoints are known. It takes a load flow with a particular form of distributed slack to solve this [15]. In this work, a direct approach (as explained earlier) is employed, meaning that all of the post-contingency situations are modeled by specific load flows that join the overall problem formulation. Once the variables defining those flows are incorporated, they become available to impose coupling constraints such as ramp rates on them, as well as normal voltage, generator capability and transmission capacity limits for the post-contingency solution. This is different from continuation load flow approaches to voltage security such as [16].

If post-contingency load shedding is a possibility, then such loads are modeled as price-responsive loads with their first block priced at the same level as the value of lost load. This is consistent with a welfare maximization problem formulation.

Most previous works model the generation limits using box bounds on the active and reactive output. True generator capability curves, however, come from the intersection of several curves, each arising from physical limits being reached in a specific component of the generator [17]. A trapezoidal approximation to these curves is employed in the underlying Matpower [13] OPF formulation which is closer to true generator capability curves.

In market-based scheduling settings, offers for both generation and curtailable loads are usually structured in blocks at a given price, not as a polynomial cost. Block-based costs resulting in convex piece-wise linear cost functions are dealt with by internally adding new linear constraints and variables using the capability of the generalized OPF solver in Matpower; this is transparent to the user. The specific method employed defines one cost variable y_i for each generator or load with block-based costs, which is added to the problem's cost functional, and then constraints of the form

$$y_i\ge m_j{p}_i+b_j,j=1\dots \#\text{of}\text{cost}\text{segments}$$

are formulated, resulting in a convex feasible region for (y_i,p_i). The minimization process drives (y_i,p_i) against the boundary, which is exactly the cost curve; see [13], [18] for further details. Of course, Matpower also allows polynomial costs and these two representations can both be present in a given problem.

The generalized formulation employed in Matpower allows the specification of price-responsive loads as negative injections. For welfare maximization, the negative of the benefit function can be specified; market bids are assumed otherwise (Fig. 1). Thus, a load demand as in Fig. 2 can be converted to an injection offer. Because a load's reactive consumption cannot be dispatched, price-responsive injections with negative active power are assumed to exhibit a constant power factor. This models the behavior of such loads more accurately and is a standard feature in Matpower.

Load shedding can be modeled by specifying a demand curve whose first block's price corresponds to that of the value of lost load. This approach is appropriate for maximization of social welfare, where the value of lost load should be taken into account. If the actual value of lost load should not be allowed to set the nodal prices at the solution, an alternative approach is to use whatever price caps are in effect in the market. This models load shedding in a setting in which the consumers are not compensated.

It should be noted, however, that true load shedding is a non-convex problem; normally, if the first block in a load's demand curve is made price-responsive to model load shedding, this does not mean that there is an ability to dispatch it half-way through; in a normal OPF setting the solution algorithm might try to split the block. This would require an adjustment to the post-contingency flow in order to shed the whole block.

Secure dispatch and post-contingency rescheduling require that resources be available for redispatch if needed. Traditional security rules include the N − 1 spinning reserve criterion for each control area, in which the amount of reserve must be enough to cover the loss of the largest generation unit. Other rules specify 10 and 30 minute reserve as a percentage of the load being served. Non-integrated market approaches such as [19], [20] require pre-specified amounts of reserve to be met, usually divided by zones. However, the true locational aspect of reserve has not always been addressed. The reserve resources must have an appropriate geographic distribution to be able to harness their energy should it be needed if a contingency occurs. Works such as [5], [6], [7] address exactly this issue, as opposed to, for example [21], [22], [23], in which an integrated market is optimized but the reserve amounts to be met are specified in a zone by zone basis, not a contingency analysis-originated basis. The direct formulation approach used there, without simplification of network constraints, is helpful for obtaining solutions that need no further adjustment. The approach suggested in [5] simply provides a solution from which it is feasible to transition to any of the post-contingency states considered; the raison d'être for reserve is implicit in the dispatch itself. In [6], [7], the concept of reserve amount and reserve contract is introduced, so that reserve markets can be designed, and a full AC flow setting is employed. This work expands [7] to consider both upward and downward excursions as “reserve”, albeit of a different kind, as well as reactive reserve. This makes it easier to integrate the approach to a day-ahead market-based scheduling framework in which there must be a real time follow-through. Other efforts have included [24], [25], [26], [27] with a linear flow formulation.

Security-constrained OPF models that rely on explicit formulation of post-contingency flows can be thought of as multi-scenario planning models with coupling constraints. These constraints are there to model transition-related limits, ramp rates in particular. This suggests a tree structure for the problem, the branches representing both transitions and coupling constraints. This approach has been suggested explicitly in the setting of unit commitment algorithms [28] but is certainly inherent in other “direct” treatments of the security-constrained OPF. In fact, this approach can be generalized further by allowing several tree structures in a single problem. This way, more than one probability-weighted “base case” can be considered, each with corresponding contingency-originated transitions and constraints, also probability-weighted. The probability weightings used here can be computed from individual equipment failure rates and line outage probabilities based also on weather prediction, as well as historic data.

The proposed scheme can be useful to model high-load and low-load predictions in addition to the central 24 hour-ahead load prediction. Of course, this adds to the dimension of the problem. An example of such a tree is shown in Fig. 3, which considers two base cases, with two contingencies considered for each. Here, an additional refinement has been introduced in that the transition to a post-contingency state can be modeled in two stages if necessary, the first being the immediate post-contingency state of the system, after voltage controls have acted, but before AGC has had a chance to correct frequency and area interchange; and a second and final stage in which economic redispatch is assumed to have taken place.

In the proposed scheme, the cost of operation for every scenario is weighted by its probability of occurrence, making the problem one of constrained stochastic optimization. This makes economic sense as it solves for the least expected cost of procurement.

One could certainly consider (N − 2)-type contingencies sprouting from each of the terminal N − 1 contingency nodes, but it is clear that the dimensionality of the problem would become unmanageable with both current and envisioned computing capabilities. Even when the ramp limits are ignored and a linear (DC) network model is used as in [29], N − 2 security results in huge mathematical programs.

This approach, in which the transition direction is important, is different from that considered in [6], [7], where the redispatch amount needed to transition between any two considered scenarios is bounded to be less than the available “ramp rate”. Thus the formulation in this work is less conservative.

A related view of the problem is that of a receding-horizon optimal stochastic control problem. The N − 1 security translates to a one-stage horizon from the moment that the control actions are implemented, and the 24 hour-ahead planning translates to a 24-hour control delay. The probabilities employed in the formulation are those estimated day-ahead, which will certainly be different from those in real time, when there is little uncertainty about the load level and the weather. It is important to recognize this because the realized system state one day later is bound to be at least slightly different from the central day-ahead prediction, i.e. the base case. Thus, for completeness of the problem, any real-time or spot rescheduling mechanisms must be taken into account in the day-ahead planning. That is the reason why in this work additional costs on deviations from the contracted day-ahead quantity are employed; these must be provided by participants in the market at the same time that they offer in the day-ahead energy and reserve market.

A major feature of the proposed formulation is that the day-ahead contract quantities are not constrained to be equal to the base case dispatch. Rather, additional contracted quantity variables together with several sets of inequalities involving the incremental dispatches, reserve variables and actual base and post-contingency dispatches are employed. This offers more flexibility in selecting a day-ahead optimal contract to the independent system operator. In integrated, co-optimized markets this flexibility is actually needed in some cases to be able to reach an optimum hedge. When the contracted quantities are set to be equal to the base case dispatch, the shadow prices on energy may require modification and the system cost can increase.

The remainder of the paper is organized as follows: Section 2 poses the formulation of the day-ahead problem; Section 3 analyzes the Lagrangian and shadow prices relationships; Section 4 analyzes the real-time redispatch adjustment, Section 5 discusses a test implementation based on Matpower and Section 6 presents some preliminary numerical results using the IEEE 30-bus and 118-bus systems. Finally, conclusions and closure are offered in Section 7.