**I. INTRODUCTION: Uncertainty management in power system operation**

The Uncertainty in power systems is a persistent issue. In power systems, generally, the uncertainty is a state for all the system operation, components, and the objective environment, where has it is impossible to exactly describe the existing all state, as future outcome, or more than one possible outcome due to limited knowledge. Due to uncertainty, the power system can be exposed to potential safety issues as well as economic loss. Mitigating the uncertainty is desirable, and therefore, an important research topic. The North American Electric Reliability Corporation (NERC) has published a report stating that uncertainty must be addressed in long-term planning, calling for more robust and flexible systems [1]. During short periods, however, the uncertainty is absorbed by the power system operation. The Reliability Assessment Guidebook, for instance, requires that the uncertainty should be addressed within the assumptions, such as load variation, generation dispatch, the effect of loop flows as well as the status of transmission elements. Intuitively, it is more complex and critical to handle the uncertainty in system operation rather than in planning.

In system planning, the uncertainty prediction is typically inaccurate, often far from the actual situation, thus requiring a rough approximation of uncertainty that is acceptable. The deviation of uncertainty prediction in the planning stage can be “rescued” in the operation stage. However, the price of uncertainty in the operation stage is higher than in the planning stage. There are a variety of methods and theories on uncertainty in power system operation that have emerged requiring continuous study and improvement. In traditional power systems, the uncertainty mainly exists in the electric component outages, such as unit outage, transmission line breakdown, and breaker faults. Based on the highly controllable power generation and accurate load forecast, the system is operated by unit commitment (UC) and economic dispatch (ED) with N-1 contingency [3], which greatly reduce the uncertainty from the outage. Furthermore, to meet the increased uncertainty, traditional systems adopt deterministic dispatch or worst-case dispatch methods that have been developed by acquiring substantial capacity reserves, but which can also increase the energy cost and emissions [4]. In most cases, these methods guarantee power system security, while seeing substantial improvement over the course of their use.

II. OVERVIEW OF POWER SYSTEM OPERATION

A crucial objective in power system operation is to ensure its reliability at all times, where the power balance is the most fundamental requirement, as shown in (1).

s(t) = d(t), (1)

where s(t) and d(t) are the power supply and power demand at time t, respectively. Additional operational constraints may also be necessary in different operation cases. The second critical objective of power system operation is economic benefits, achieved by reducing operational costs. To accomplish the above two goals, power systems are generally operated in the context of timeframes: seconds-tominutes, minutes-to-hours, hours-to-days, days-to-one week and beyond, shown in Fig. 1. In the seconds-to-minutes timeframe, bulk power system reliability is almost entirely

controlled by automatic equipment and control systems, e.g., Automatic Generation Control (AGC). In the minutes-to oneweek timeframe, system operators are in charge of committing and dispatching units to balance the bulk power system. In the past, the major uncertainty in the day-ahead timeframe has been in component outages, which is typically smoothed by security constrained UC (SCUC) under N-1 contingency. In the hours-to-days timeframe, the uncertainty is reduced by security constrained ED (SCED) under N-1 contingency. The implementation framework for Independent System Operators (ISOs) is shown in Fig. 2 [3].

Now, the high penetration of large-scale renewables and the active load demand lead to the injection of large amounts of randomness into the power supply and demand side, so the power balance equation (1) should be a stochastic equation, rather than the deterministic one, shown in (2).

sd(t) + ss(t) = dd(t) + ds(t), (2)

where sd(t) and dd(t) are the deterministic supply and demand, while ss(t) and ds(t) are the stochastic parts

Therefore, the crucial problem in current operation paradigms is how to manage the stochastic components. The uncertainty in the system, fortunately, is not a chaotic mass because it can be either forecasted by a probability distribution function (PDF), or restricted in a certain interval. Based on the availability of prediction, actions such as unit commitment and dispatch, and reserve procurement and demand side resource management, should be undertaken to reduce the increased uncertainty [9]. According to this requirement, the new operation paradigm is constructed, shown in Fig. 3.

Fig. 3 shows the counterpart operation paradigm in which there are three major changes: First, meteorological and electrical data are required and must be updated by the Supervisory Control and Data Acquisition (SCADA) systems in state-of-the-art forecasting. Second, SCUC and SCED must be modified to accommodate renewable units that are uncontrollable. Finally, N-1 contingency analysis must be replaced by new SCUC and SCED dealing with uncertainty.

III. GENERAL MATHEMATICAL MODELS

The objective of operations is to ensure the safety and security of power systems, and to obtain higher economic benefits. Mathematically, this objective can be converted to an

optimization problem, where safety and security are the constraints, and economic benefit the objective function, shown in (3).

minimize f(x,ey)

s.t. hi(x,ey) ≤ 0, ∀i ∈ {1, · · · , n}, (3)

where x and ey are the vector for deterministic and stochastic variables, respectively. f(·) is the cost function, and hi(·) is the ith constraints.The stochastic elements in constraints are often tackled by stochastic programming, and robust optimization [12]. In stochastic programming, two common methods are used. One is the discretization for the stochastic variable, usually by Monte Carlo simulation. The general model in (3) can be transformed into (4). The continuous variable ey is sampled into m discrete variables, denoted by yj with probability pj respectively. Thus, the objective function is altered in expectation form, and the constraints must be matched in each scenario. Related technology is discrete sampling and scenario reduction.

The second method is to take advantage of the cumulative distribution function, where the constraints are commonly transferred into probability constraints with an acceptable probability level η, while the objective function is changed to an expectation, shown in (5).

In robust optimization, the constraints and objective function are often expressed as (6). where ε is the interval of ey. Obviously, in stochastic programming, there is a small probability that the constraints are not met, and thus are less reliable; however, they have significant economic benefits. In robust optimization, the reliability can be guaranteed definitely, with the price being worse economic benefits. Thus, in realistic operation processes, the mathematical model that is adopted depends on the trade-off between economic benefits and reliability, according to different operation requirements. The specific operation methods based on these models are illustrated in detail in the next section.

IV. UNCERTAINTY MANAGEMENT WITH OPERATION

METHODS

A. Scenario-based Stochastic Programming

The scenario-based stochastic programming is generally modeled using the steps in [13]–[16]. The discretization of stochastic components is the first step. For straightforward illustration, we regard the stochastic components, denoted by net load l(t), to be one dimension, shown in (7). It is easy tobe extended into the high dimension.

l(t) = d(t) − s(t). (7)

Based on Monte Carlo simulation, the continuous PDF for net load is sampled into N(t) discrete points with different probabilities, and each discrete point is denoted by

li(t) (i ∈ {1, · · · , N(t)}).

For the second step, a scenario tree is formed [13], [16], [17]. If the operation is concerned about the next t(t ∈ {1, · · · , T}) hours, for each period t, step one is repeated.

Thus, for the whole time horizon, the scenario tree is formed, and each trajectory is called one scenario. For example,

provided that T = 2, N(1) = 2, and N(2) = 3, the scenario tree is shown in Fig. 4.

The next step is the problem formulation [13], [14], [17]– [19]. For simplification, only the power balance is considered as the constraint.

where c0, s0 and d0 are operating cost, dispatch power, and demand at the initial time period. pi, ci , si(t), and di(t) are the probability, operating cost, dispatch power, and demand in scenario i. Thus, the objective function is composed of a deterministic part in the initial stage and a stochastic part in the subsequent stage; this is the so-called two-stage problem.

B. Look-ahead Dispatch

The principle of look-ahead dispatch is model predictive control (MPC) [25]–[29]. The MPC approach deals with the dynamic receding horizon optimization control problem [30]. Usually, MPC systems employ a stochastic model for the uncertainty. In this way, the systematic stochastic terms can be effectively compensated by the decisions of the system. In addition, proper inclusion of integration in the model can eliminate steady state error from the system outputs. The mathematical model of look-ahead dispatch is shown in (9).

minimize f(u)

s.t. xk+1 = g(xk, uk), ∀k ∈ {1, · · · , N}

h(xk, xk+1, uk) ≤ 0, ∀k ∈ {1, · · · , N}

x0 = Z(k), (9)

where k is the index of the time period, N is the whole time horizon, xk is the state variable in period k, u = [u1, · · · , uk] is the vector of control variables. In each time period, the predictions for the whole time horizon are made, expressed by the first constraint in (9). Based on the predictions, the control strategies are generated by optimization (9), but only the strategy for the current period is realized. When the time comes to the next period, a similar control is duplicated, up until the final period. In sum, there are k optimization problems solved.

C. Risk-limiting Dispatch

The risk-limiting dispatch is also a dynamic sequential operation method. The basic ideas of risk-limiting factor dispatch is to restrict the risk for the final operation objective through stages of operation [4], [9], [33]–[35]. Therefore, in risk-limiting has dispatch, there are a series of recourse decisions inserted in between the initial stages and of the final goal stages, and the final goal of stages can be either as single time point, or a time horizon. Furthermore, the global optimal scheduling strategy can be found in risk-limiting dispatch, providing enough prediction information. To specify the major features, Fig. 6 shows the operation strategies for the whole time horizon. Here the final operation goal is the power balance for real time t. At the initial stage, the operation strategy, either a UC or ED, is generated. As time goes by, at the recourse stages, the operation strategies are made with the updated prediction information that can be viewed mathematically as the conditional probability. Then, in real time, the final strategy prevents potential emergencies. The mathematical model is shown in (10).

D. Robust Optimization

Since RO only requires moderate knowledge of the uncertainty and its solution immunizes against any realization of the uncertainty set, RO applications in power system operation have been extensively studied. Due to the consideration of computational tractability, robust linear programming (RLP) has become an area of interest. Normally, rather than the basic non-adjustable RLP, the more complex Adjustable Robust Linear Optimization (ARLP) [46] is employed to reduce the degree of conservatism and to cooperate with the multistage nature of problems in power systems. ARLP includes some “here and now” as of decisions to be determined before the uncertainty reveals itself, and some “wait and see” decisions to be determined after the uncertain data are known. Usually, a twostage optimization problem is modeled. Benders decomposition is one of the major algorithms for solution methodology development.

V. CONCLUSION

Power system operation is always a battle between reliability and economic benefits, especially with influx of uncertainty resulting from high penetration of renewables generation. For the four prevailing operation methods, the weight between reliability and economic benefits are different. In scenario-based stochastic programming, the reliability method can be secured exhaustively in all selected scenarios.