Distributed Optimal Load Frequency Control Considering Nonsmooth Cost Functions
Abstract
This work addresses the distributed frequency control problem in power systems considering controllable load with a nonsmooth cost. The nonsmoothness exists widely in power systems, such as tiered price, greatly challenging the design of distributed optimal controllers. In this regard, we first formulate an optimization problem that minimizes the nonsmooth regulation cost, where both capacity limits of controllable load and tieline flow are considered. Then, a distributed controller is derived using the Clark generalized gradient. We also prove the optimality of the equilibrium of the closedloop system as well as its asymptotic stability. Simulations carried out on the IEEE 68bus system verifies the effectiveness of the proposed method.
keywords:
Nonsmooth optimization, distributed control, load frequency control, Clark generalized gradient.1
1 Introduction
With the proliferation of renewable generations, frequency control in power systems is facing a great challenge as power mismatch can fluctuate rapidly in a large amount. In this situation, the conventional centralized hierarchical control architecture may not respond fast enough due to large inertia of the traditional synchronous generators dorfler2016breaking (); wang2019distributed (). On the other hand, loadside controllable resources with fast response capabilities provide a new opportunity to frequency regulation Schweppe1980Homeostatic (). In addition, as controllable loads are usually dispersed geographically vast across the power system, a distributed architecture is more desirable for load frequency control than the centralized one.
Recently, the socalled reverse engineering methodology is proposed by combining frequency control with optimal operation problems in power systems zhang:real (); Stegink:aunifying (); Li:Connecting (); Cai:Distributed (); Changhong:Design (). Under this framework, distributed load frequency control is widely investigated Changhong:Design (); mallada2017optimal (); Distributed_I:Wang (); Kasis:Primary1 (); Distributed_II:Wang (); zhao2018distributed (); wang2018distributed (). In Changhong:Design (), an optimal load frequency control problem is formulated and a distributed controller is derived using controllable loads to realize primary frequency control. To eliminated the frequency deviation, the method is further extended in mallada2017optimal (); zhao2018distributed () to realize a secondary load frequency control. At the same time, the tieline power limit is considered. The design approach is generalized in Kasis:Primary1 (), where the specific model requirement is eliminated. It only requires that the bus dynamics satisfy a passivity condition to guarantee asymptotic stability. In Distributed_I:Wang (); Distributed_II:Wang (), the operational constraints including regulation capacity limits and tieline power limits are considered, which guarantee both steadystate and transient capacity limit constraints. In wang2018distributed (), the distributed load frequency control under timevarying and unknown power injection is investigated, which can recover the nominal frequency even under unknown disturbances. The distributed load frequency control is of course a paid service, i.e., the system operator needs to pay for the controllable load to regulate their power. In the existing literature, the cost of controllable load is assumed to be differentiable, or equivalently, the price of the controllable load is continuous. This is not true for a variety of cases, e.g., the price may have step changes when controllable load values are in different intervals. In such a situation, the regulation is inherently nonsmooth, which makes existing methods difficult to apply.
This work designs a distributed controller for the optimal load frequency control in power systems, where the regulation cost function can be nonsmooth. We relax the assumption of the objective function from being differentiable to nonsmooth. This work is partly motivated by zeng2018distributed (). However, different from it, we consider the interplay between the solving algorithm and the power system dynamics and prove the stability of the closedloop system. Another difference is that the objective function in zeng2018distributed () is strictly convex with respect to all decision variables. That is not necessary in our work, where some variables may not appear in the objective function. In such a situation, we prove the asymptotic convergence of the closedloop system as well as the optimality of equilibrium.
The rest of this paper is organized as follows. In Section II, we introduce some preliminaries and system models. Section III formulates the optimal load frequency control problem and introduces the distributed controller. In Section IV, convergence of the closedloop system and optimality of the equilibrium point are proved. We confirm the performance of the controller via simulations on IEEE 68bus system in Section V. Section VI concludes the paper.
2 Problem Description
2.1 Preliminaries and notations
2.1.1 Notations
In this paper, use () to denote the dimensional (nonnegative) Euclidean space. For a column vector (matrix ), () denotes its transpose. For vectors , denotes the inner product of . denotes the Euclidean norm of . Use 1 to denote the vector with all elements. For a matrix , stands for the entry in the th row and th column of . Use to denote the Cartesian product of the sets . Given a collection of for in a certain set , denotes the column vector with a proper dimension, and as its components.
2.1.2 Preliminaries
Let be a locally Lipschitz continuous function and denote its Clarke generalized gradient by (clarke:optimization, , Page 27). For a continuous strictly convex function , we have , where and .
Define the projection of onto a closed convex set as
(1) 
Use to denote the identity operator, i.e., , . Define . We have (bauschke2011convex, , Chapter 23.1).
A basic property of a projection is
(2) 
Moreover, we also have (facchinei2003finite, , Theorem 1.5.5)
(3) 
Define , and then is differentiable and convex with respect to (liu2013one, , Lemma 4). Moreover, we have
(4)  
(5)  
(6) 
where the inequality is due to (2). From (2.1.2), holds only when .
2.2 Network model
A power network is usually composed of multiple buses, which are connected with each other through transmission lines. It can be modeled as a graph , where is the set of buses and is the set of edges (transmission lines). Let denote the number of lines. The buses are divided into two types: generator buses, denoted by and load buses, denoted by . A generator bus contains a generator ( possibly with certain aggregate load). A load bus has only load with no generator. The graph is treated as directed with an arbitrary orientation and use or interchangeably to denote a directed edge from to . Without loss of generality, we assume the graph is connected and node is a reference node. The incidence matrix of the graph is denoted by , and we have .
We adopt a secondorder linearized model to describe the frequency dynamics of each bus. We assume that the lines are lossless and adopt the DC power flow model Distributed_I:Wang (); Changhong:Design (). For each bus , let denote the rotor angle at node at time and the frequency. ^{1}^{1}1Sometimes, we also omit for simplicity. Let denote the controllable load. Let given constant denote any change in power injection, that occurs on the generation side or the load side, or both. Define as the angle difference between bus and , and its compact form is denoted by . Then for each node , the dynamics are
(7a)  
(7b)  
(7c)  
where are inertia constants, are damping constants, and are line parameters that depend on the reactance of the line . 
The scenario is that: the system operates in a steady state at first. A certain power imbalance occurs due to variation of power injection . Then controllable load accordingly changes its output to eliminate the imbalance.
3 Problem Formulation
In this section, we first formulate the optimal load frequency problem with a nonsmooth objective function. Then, we propose a distributed controller based on the Clark generalized gradient to drive the power system to the optimal solution.
3.1 Optimization problem
The optimization problem is
(8a)  
s.t.  
(8b)  
(8c)  
(8d) 
where are constants, denoting the lower and upper bound of . are also constants, denoting the lower and upper bound of angle difference. The first constraint is the local power balance. is the virtual phase angle, which equals to at the optimal solution. Use to denote the virtual phase angle difference. In the DC power flow, we have , where is the power of line . Thus, (8d) is in fact the tieline power limit constraint. We have the following assumptions.
Assumption 1.
is strictly convex.
Assumption 2.
The Slater’s condition (boyd2004convex, , Chapter 5.2.3) of (8) holds, i.e., problem (8) is feasible provided that the constraints are affine.
Remark 1.
Assumption 1 could be further relaxed, since a nonstrictly convex function can be strictly convexified by using a nonlinear perturbation mangasarian1979nonlinear ().
Remark 2.
Problem (8) allows the cost function to be nonsmooth, which is required to be differentiable in the existing literature Changhong:Design (); mallada2017optimal (); Distributed_I:Wang (); Kasis:Primary1 (); Distributed_II:Wang (); zhao2018distributed (); wang2018distributed (). Thus, the problem (8) is more general and suitable for a variety of real problems whose regulation costs are not smooth. A typical example is the tiered price, where the price discontinuously increases with respect to the amount of controllable load. It also should be noted that the decision variable is absent in the objective function of (8). It makes the paper not a trivial application of zeng2018distributed (), i.e., the objective function is not required to be strictly convex to all the decision variables. It makes the convergence proof more challenging.
Remark 3.
In the existing literature, the controller usually involves the projection of a gradient onto a convex set. If the objective function is nonsmooth, it becomes the projection of a subdifferential set onto a convex set. In this situation, the existence of trajectories is not guaranteed zeng2018distributed (); zhou2019adaptive (), which makes existing load frequency control methods inapplicable to the nonsmooth case.
3.2 Controller Design
To help the controller design, we make a modification on the problem (8).
(9a)  
s.t.  (9b) 
where . For any feasible solution to (8), . Thus, (8) and (9) have same solutions.
Define the sets
(10) 
Then, we give the controller for each controllable load, which is denoted by OLC.
(11a)  
(11b)  
(11c)  
(11d)  
(11e)  
(11f)  
(11g)  
(11h) 
Combining with the power system dynamics, we have the closedloop system (7), (11).
Remark 4 (Load demand estimate).
In power systems, the load demand is difficult to measure. Similar to mallada2017optimal (); zhao2018distributed (); Distributed_II:Wang (), in (11b) can be substituted equivalently in following ways. For ,
For ,
In this way, the measurement of load demand is avoided. We only need to measure , which are much easier to realize. Moreover, the power loss can be treated as unknown load demand, which can be also considered by this method.
4 Optimality and Convergence
In this section, we address the optimality of the equilibrium point and the convergence of the closedloop system.
4.1 Optimality
Denote and . Let be an equilibrium of the closedloop system (7), (11). Then, there exists such that
(12a)  
(12b)  
(12c)  
(12d)  
(12e)  
(12f)  
(12g)  
(12h)  
(12i)  
(12j) 
Now, we introduce the properties of the equilibrium points.
Theorem 1.
Proof.
1) From (12b) and (12d), we have . From (12a), we have with a constant . As is a diagonal positive definite matrix, we have .
2) From (12c) and (12f)(12j), we have
(13a)  
(13b)  
(13c) 
or equivalently,
(14a)  
(14b)  
(14c) 
By the KKT condition in (ruszczynski2006nonlinear, , Theorem 3.34), (12d), (12e) and (14) coincide with the KKT optimality condition of the problem (8). Then, we have this assertion.
4.2 Convergence
Define the function
(15) 
where
(16) 
(17) 
Then, we have the following result about .
Lemma 2.
Proof.
1) By (2), we know that . From (15), (4.2) and (4.2), we know and holds only at the equilibrium point.
2) By (6), the gradient of is
(18) 
Then, there is such that the time derivative of is
(19) 
where the last item is due to the fact that, for each
(20) 
Theorem 3.
Proof.
1) From Lemma 2, we know that is bounded. By (7c), is also bounded. Since is compact, there exists a constant such that
(25) 
Define following function
(26) 
The time derivative of along the closedloop system is
(27) 
Thus, is bounded, so is . Similarly, we can also have that and are bounded.
2) By the invariance principle in (cortes2008discontinuous, , Theorem 2), we know that the trajectory converges to the largest weakly invariant subset contained in , i.e., once a trajectory enters this subset, it will never departure from it.
From , we know , i.e., . Note that if due to the strict convexity of . Thus, we have in the set . Moreover, from (24), we have , or , which implies that . Then, we have from (11b). Similarly, by (11). Up to now, we know is constant except for .
Moreover, the equality in (3) holds only when or and . Thus, for , there are four combinations:

and ;

and , ;

and ;

and .
Thus, converges to equilibrium of the closedloop system.
3) Fix any initial state and consider the trajectory of the closedloop system. As is bounded, there exists an infinite sequence of time instants such that as , for some . Using this specific equlibrium point in the definition of , we have
Here, the first equality uses the fact that is nonincreasing in while lowerbounded, and therefore must render a limit value ; the second equality uses the fact that is the infinite subsequence of ; the third equality uses the fact that is absolutely continuous in ; the fourth equality is due to the continuity of , and the last equality holds as is an equilibrium point of .
5 Case studies
5.1 Test system
In this section, the IEEE 68bus New England/New York interconnection test system Changhong:Design () is utilized to illustrate the performance of the proposed controller. The diagram of the 68bus system is given in Fig.1. We run the simulation on Matlab using the Power System Toolbox PST (). Although the linear model is used in the analysis, the simulation model is much more detailed and realistic. The generator includes a twoaxis subtransient reactance model, IEEE type DC1 exciter model, and a classical power system stabilizer model. AC (nonlinear) power flows are utilized, including nonzero line resistances. The upper bound of is the load demand value at each bus. Detailed simulation model including parameter values can be found in the data files of the toolbox.
The objective function of each controllable load is
(28) 
It can be verified that is continuous, strictly convex and nonsmooth.
5.2 Simulation results
We consider the following scenario: at s, there is a step change of p.u. load demand at buses 4, 8, 20, 37, 42, and 52 respectively. Neither the original load demand nor its change is known. The load estimate method in Remark 4 is utilized.
At first, we do not set limits to the tieline power. In this subsection, we analyze the dynamic performance of the closedloop system under the proposed controller OLC. In addition, automatic generation control (AGC) is tested in the same scenario as a benchmark. The setting of AGC is the same as that in zhao2018distributed (). The frequency dynamics under OLC and AGC are given in Fig.2. It is shown that both AGC and OLC can recover the frequency to the nominal value. They also have similar frequency nadir. Compared with AGC, the frequency under OLC has faster convergence speed.