Interval optimal power flow applied to distribution networks under uncertainty of loads and renewable resources
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Abstract
Optimal power flow (OPF) has been used for energy dispatching in active distribution networks. To satisfy constraints fully and achieve strict operational bounds under the uncertainties from loads and sources, this paper derives an interval optimal power flow (I-OPF) method employing affine arithmetic and interval Taylor expansion. An enhanced I-OPF method based on successive linear approximation and second-order cone programming is developed to improve solution accuracy. The proposed methods are benchmarked against Monte Carlo simulation (MCS) and stochastic OPF. Tests on a modified IEEE 33-bus system and a real 113-bus distribution network validate the effectiveness and applicability of the proposed methods.
Keywords
Active distribution network Optimal power flow Interval uncertainty Affine arithmetic Second-order cone programming1 Introduction
Optimal power flow (OPF) is one of the fundamental static power system calculations. It has wide application in electrical engineering, including scheduling of generators, loss reduction, congestion management, and expansion planning. As distributed generation (DG) and controllable loads (e.g., electric vehicles) proliferate, active network management has been introduced in distribution systems [1]. In this context, OPF is no longer limited to the domain of high voltage transmission networks and has been gradually investigated for application to distribution networks [2].
In general, all the input data of OPF are deterministic. Governed by nonlinear Kirchhoff’s laws, such deterministic optimization problems can be solved by many methods, such as successive linear/quadratic programming [3, 4], trust-region-based methods [5], the Lagrangian Newton method [6] and the interior-point method [7]. However, with increasing internal and external factors of uncertainty, such as the power demand affected by daily economic activities, power generated by renewable energy, and grid parameters obtained by approximate measurements, the input data have increasing uncertainty, which challenges conventional deterministic OPF models. The degree of uncertainty for some factors can be reduced, but for most uncontrollable factors, it is very difficult to decrease the impact of their uncertainty. Hence, OPF should be able to manage uncertainties in power flow performance.
Most conventional methodologies to address uncertainty are based on probabilistic methods that account for the variability and stochastic nature of the input data. Current OPF research on this topic can be divided into two categories, probabilistic OPF (P-OPF) and stochastic OPF (S-OPF). P-OPF [8] is a well-respected approach for characterizing the output of an implicit function whose inputs are random variables, where the cumulant method [9] and the point estimate method [10] are examples of very efficient P-OPF computation. However, the solution of P-OPF is influenced indirectly by the randomness of input variables, and only the probability distributions of control variables can be determined.
In S-OPF problems, the objective function and constraints are usually described by probability equations or inequalities, which means that the randomness of input variables can directly impact the solution [11, 12]. Thus, constraint satisfaction in an uncertain environment can be achieved. Another competitive choice for modeling S-OPF problems often considered is chance-constrained programming (CCP) [13] in which the constraints (some or all) can be violated with a pre-assigned level of probability. These are referred to as “chance constraints”. However, it is generally agreed that closed-form solutions for S-OPF are rarely available due to the different types of parameter distributions and the nonlinear nature of OPF, and thus one has to resort to approximation methods to solve instances of nontrivial size. Researchers have devised solution methods that rely mainly on discretization of the uncertain parameters. Theoretically, these scenario-based approaches to S-OPF can achieve any desired level of accuracy, but the required computational resources for scenario generation and coordinated optimization could be prohibitively expensive.
Although scenario-free approaches to chance-constrained OPF have also been developed, specific uncertainty distributions (e.g., Gaussian distribution) should usually be assumed for analytical reformulation of the chance constraints. Furthermore, to deal with the higher complexity of chance-constrained OPF, the existing approaches either assume a DC-OPF [14], a linearized OPF [15], a convex relaxation-based OPF [16], or they solve iteratively linearized instances of nonlinear OPF [13]. In addition to these approximation or relaxation approaches, accurate probability information in the form of scenarios or pre-defined distributions is always necessary, which is difficult to obtain in practice.
Robust optimization (RO) is another promising approach to model OPF involving uncertainties. It should be noticed that RO is applicable for problems with convex feasible regions. Thus, it is mainly applied to OPF with DC power flow constraints [17] or linearized power flow constraints [18]. To improve its accuracy, second-order cone programming (SOCP) is introduced by relaxing the power flow equality constraints [19] and thus the robust optimal power flow model is converted into a mixed-integer SOCP model through the robust counterpart. However, these approximation-based or relaxation-based methods may still cause a gap to the original exact power flow equations.
Interval arithmetic (IA) offers another approach to model and analyze uncertainties. It can be used to give the variation ranges of output variables and simultaneously verify the satisfaction of constraints. One of its successful applications in electrical engineering is to determine the strict solution bounds of power flow [20]. Ranges of uncertainty expressed by intervals are easily available in practice, so programming methods incorporating IA have received widespread attention. Early attempts in this area were focused on interval linear programming (ILP), [21, 22]. Since the ILP model can be transformed into several deterministic linear programming models according to the signs of the interval coefficients and variables, it can be solved in an efficient way. However, due to the nonlinear and non-convex characteristics of OPF, the best and worst optimums are not necessarily on the bounds of the intervals, resulting in an interval nonlinear programming (INLP) problem that is much more difficult to solve. To the best of our knowledge, the general idea for solving INLP problems in existing literature is to use approximate approaches via order relations of interval numbers or Taylor expansions [23, 24, 25].
To conquer the drawbacks including the “dependency problem” and the “wrapping effect” in IA, affine arithmetic (AA) is proposed in [26]. By using AA, two self-validated computation approaches for the power flow calculation and OPF with uncertainties are proposed in [27] and [28]. To satisfy constraints fully and to achieve accurate operational bounds of the solution under uncertainties, the authors in [29] further combined the linear approximation method and AA-based power flow calculation to address the uncertain reactive power optimization problem in transmission grid. It is worth noting that OPF incorporating intervals is a typical INLP problem. Although the above solution algorithms have been applied successfully in some applications, an analytical OPF model incorporating intervals like the deterministic OPF model is still unavailable.
- 1)
An interval OPF (I-OPF) model applied to distribution networks is derived to deal with uncertainties in power flow and constraint satisfaction, by employing the affine arithmetic and interval Taylor expansion.
- 2)
To improve solution accuracy, combining second-order cone relaxation and successive linear approximation, an enhanced method accounting for the equivalent error caused by high-order items is also derived.
- 3)
The proposed I-OPF model and its enhanced method are tested on a modified IEEE 33-bus distribution system and a real 113-bus system, using Monte Carlo simulation and stochastic OPF as comparative analyses.
The remainder of this paper is arranged as follows. Section 2 presents a brief background of the deterministic OPF model using the branch flows, as well as the formulation of OPF incorporating interval uncertainties. The I-OPF model is derived in Section 3. Section 4 derives in detail the enhanced method with improved accuracy. Case studies based on a modified IEEE 33-bus system and a real 113-bus system are presented in Section 5, followed by the conclusion in Section 6.
2 Problem formulation
Consider a distribution network with a set of controllable devices denoted by \({\mathcal{G}}\), including DG units, energy storage units and var compensation devices, and loads denoted by \({\mathcal{L}}\). Since the distribution network typically has a radial structure, we model it as a connected tree graph \({(\mathcal{N}},{\mathcal{E})}\), where each node \(i \in {\mathcal{N}}\) represents a bus and each link \(\text{(}i,j\text{)} \in {\mathcal{E}}\) represents a branch. The root of the tree is the feeder with a fixed voltage and flexible power injection.
2.1 Formulation of the deterministic OPF
2.2 OPF model with interval uncertainty
Due to the wrapping effect of IA, (10) cannot be fully consistent with the actual physical system described by real values. The existence of interval data also makes (10) hard to solve directly with the classical algorithms used in deterministic OPF problems. For these two problems, we need to transform (10) using some additional measures to obtain a more practical and direct description, which are introduced in the following section.
3 Formulation of I-OPF model
To transform (10) into equivalent deterministic one, we first replace interval arithmetic with AA and further introduce an interval Taylor expansion [25] to reconstruct the nonlinear constraints with intervals.
3.1 Introduction of AA and interval Taylor expansion
3.2 AA-based I-OPF model
It should be pointed out that the proposed I-OPF model in this paper is investigated for a balanced distribution network, being a single-phase formulation. Load unbalance is a typical characteristic of distribution networks, especially in low-voltage distribution networks. When I-OPF is required for an unbalanced system, a three-phase I-OPF model is required, which can be obtained by using the three-phase Distflow model derived in [32] and [33] and applying the same modeling process as described in this paper.
I-OPF depends on the special nature of a radial distribution network, namely that its power flows can be specified by a simple set of linear and quadratic equalities if voltage angles are eliminated, and constraints constructed with central values are exact. In addition to satisfying constraints fully under uncertainty, the interval extensions of state variables in the model also make the operational bounds of solution available, which is one of the advantages of I-OPF or interval programming.
4 Enhanced I-OPF model
The I-OPF model in (30) is deterministic so that it can be solved directly by using conventional nonlinear algorithms, such as the interior-point method. Note that its accuracy mainly relies on the approximation of the interval power flow equalities. When the interval uncertainties increase, the I-OPF model will be less accurate due to neglecting the higher order terms in (27). To solve this problem, an enhanced model is developed through modifications discussed in this section.
4.1 Accurate approximation
4.2 Successive linear approximation-based modification
It has been shown in [31] that, when the objective function is convex, (5) can be exactly relaxed to a second order cone constraint. Roughly speaking, such a relaxation for radial networks is exact if the power injection at each bus is not too large and the voltages are kept around their nominal values [34]. For the relaxation shown in (37), the introduction of a small constant does not affect this feature. Due to space limitations, we do not present the detailed proof process, which is available on request from the authors.
5 Case studies
This section illustrates the solutions obtained with the proposed I-OPF model and its enhanced method for two test cases. All the tests were implemented using the OPTI and SDPT3 packages in MATLAB, running on an Intel 4-Core i7-CPU 3.40 GHz personal computer, and they were based on a modified IEEE 33-bus system and a real 10 kV 113-bus distribution system.
5.1 Modified IEEE 33-bus system
Optimization results of I-OPF, enhanced I-OPF and S-OPF
Method | P (MW) | Q (Mvar) | Objective (10^{−1}) | CPU time (s) | Total time (s) | ||||
---|---|---|---|---|---|---|---|---|---|
PV1 | PV2 | ESS | PV1 | PV2 | SVC | ||||
OPF | 0.5909 | 0.8 | 0.2207 | 0.1942 | 0.2630 | 0.2 | 0.5808 | 0.096 | 3.159 |
I-OPF | 0.7120 | 0.8 | 0.2400 | 0.2340 | 0.2630 | 0.2 | 0.5954 | 0.125 | 3.835 |
Enhanced OPF | 0.7220 | 0.8 | 0.2400 | 0.2373 | 0.2630 | 0.2 | 0.6055 | 0.267 | 6.186 |
S-OPF | 0.7069 | 0.8 | 0.2400 | 0.2324 | 0.2630 | 0.2 | 0.6025 | 2.250 | 98.783 |
5.2 Real 113-bus system
1) Case 1: network loss objective with the 113-bus system. To illustrate the effect of load uncertainty on the accuracy of the I-OPF model, a tolerance sequence {±5%, ±10%, ±15%, ±20%} was successively assigned to the load demand. The voltage magnitudes were limited within [0.95, 1.05] p.u. and the active power traded between the main grid and the distribution network was constrained within [0, 5.5] MW.
The MCS results show that I-OPF and its enhanced version are both applicable to a large distribution system and they satisfy boundary constraints well, in particular, the constraint on the active power injected at the root bus. Also, we notice that there are still some errors in constraint satisfaction for I-OPF shown in Fig. 9. These errors are caused the approximation that ignores higher order information of partial deviations for state variables, which becomes noticeable for the large load variation of ±20%. However, the enhanced I-OPF modified to include higher order terms still works well.
Mean network loss, CPU time and E_{mean} of I-OPF and enhanced I-OPF
Method | Load variation | Mean loss (MW) | CPU time (s) | E_{mean} (V) (10^{−4} p.u.) | E_{mean} (I) (10^{−3} p.u.) |
---|---|---|---|---|---|
I-OPF | ± 5% | 0.1707 | 0.242 | 0.1132 | 0.0705 |
I-OPF | ± 10% | 0.1707 | 0.241 | 0.4662 | 2.9608 |
I-OPF | ± 15% | 0.1707 | 0.244 | 1.0706 | 5.9920 |
I-OPF | ± 20% | 0.1716 | 0.294 | 1.9188 | 11.0460 |
Enhanced I-OPF | ± 5% | 0.1713 | 0.635 | 0.0915 | 0.0414 |
Enhanced I-OPF | ± 10% | 0.1731 | 0.693 | 0.2618 | 1.8813 |
Enhanced I-OPF | ± 15% | 0.1762 | 0.781 | 0.5772 | 4.6759 |
Enhanced I-OPF | ± 20% | 0.1816 | 0.852 | 1.0119 | 6.9626 |
The I-OPF model and its enhanced version are both computationally efficient for the large system. Although the mean bound errors of node voltages and line currents increase as the load variation increases, the enhanced method can maintain the error level within an acceptable range. These characteristics further demonstrate the superiority of the proposed I-OPF model and its enhanced version. Note that I-OPF without modification is better for application to the large system because it is more efficient. On the other hand, enhanced I-OPF can achieve higher accuracy, especially when the load uncertainty is significant. Therefore, when the system has significant scale or the load variation level is not large, we recommend using the original I-OPF, otherwise the enhanced version is recommended.
2) Case 2: utilization ratio objective with the 113-bus system. The primary target of active network management is to improve the utilization ratio of the renewable energy sources. OPF introduced for dispatch in an active distribution network is generally operating on a 5-minute control cycle or longer. When considering the uncertainties of uncontrolled DG and loads within this control interval, some control instructions may allow overvoltage to occur. The I-OPF model can be employed to prevent overvoltage at the system level and simultaneously to make the most of DGs.
6 Conclusion
- 1)
Compared with MCS, the I-OPF model and its enhanced version perform well in bounds estimation and constraint satisfaction of state variables. Enhanced I-OPF can offer a more accurate estimate of state variables and thus meet optimization objectives more effectively than unmodified I-OPF.
- 2)
Compared with scenario-based approaches, the I-OPF model and its enhanced version are more efficient. Unmodified I-OPF is better to use for large distribution systems due to its concise form and computational efficiency, while enhanced I-OPF requires additional computing resources to satisfy constraints accurately when there are large load variations.
Notes
Acknowledgements
This work was supported by Fundamental Research Funds for the Central Universities (No. 2016XS02), and National Natural Science Foundation of China (No. 61772167).
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