Proceedings of the 29th Chinese Control Conference July 29-31, 2010, Beijing, China
Distributed Control for Optimal Economic Dispatch of Power Generators* R. Mudumbai, S. Dasgupta, B. Cho Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242 E-mail: [rmudumbai,dasgupta]@engineering.uiowa.edu,
[email protected] Abstract: In this paper, we present a simple, distributed algorithm that adjusts the power-frequency set-points of generators to correct for generation and load fluctuations, and asymptotically achieves the lowest-cost power allocation among generators while driving the power imbalance exponentially to zero. In this algorithm each generator executes its own control law using only the power imbalance in the network, a quantity in turn locally obtained through local monitoring of the frequency deviation on the grid. Our results only assume that the cost of each generator is a convex function. The distributed approach is motivated by the anticipated needs of the future electric grid with a large penetration of distributed small-scale alternative energy generators, and also consumers with “smart metering” capabilities. We present analytical results and numerical simulations to illustrate the performance of our algorithm. Key Words: Distributed, Energy, Smart Grids, Optimal Dispatch
1 INTRODUCTION We present a simple distributed algorithm for load frequency control of an electric grid, where each generator uses local knowledge of its own cost of generation along with measurements of frequency deviations to dynamically adjust its real power generation; we show that the algorithm automatically achieves the condition of optimal economic dispatch under mild assumptions on the cost functions of the generators. This work is motivated by the anticipated needs of the next generation electric grid which is expected to have smart consumer end-nodes [1] and a high penetration of alternative energy generators. Since the availability of alternative energy sources such as wind and solar generators is inherently intermittent in time and dispersed in geography [2], the ability of the electric grid to dynamically adjust the generation and consumption is key to achieving a high load factor and efficient energy use. In a traditional electric grid, control of generators, i.e. Automatic Generation Control (AGC) is accomplished on multiple time-scales using multiple different mechanisms [3]. Primary control is implemented in a distributed fashion at the generators, but secondary and tertiary control (corresponding to load frequency control (LFC) and economic dispatch (ED) respectively) are implemented from a centralized control station at the transmission system operator (TSO) and Load Serving Entity (LSE) [4]. While the primary control operates over a time-scale of upto 30 see [5], secondary and tertiary control operate over much longer time-scales of around 5 min and 45 min respectively. The goal of the secondary control process is to reduce the Area Control Error (ACE) to zero. The ACE is a measure of the imbalance between rated generation capacity and power consumed within the control area, and the LFC algorithm adjusts the power generation levels in order to achieve power balance within the control area. Traditionally, an ad-hoc allocation is used by the sec-
ondary controller to return ACE to zero without consideration of cost minimization; the latter function is the responsibility of the tertiary control process or economic dispatch (ED). The economic dispatch process periodically reallocates the total generation power among generators to minimize total cost; the power allocations once set by the ED algorithm may over time deviate from their optimal values because of cumulative load fluctuations and the actions of the secondary controller. ED is typically implemented as a multi-variable constrained optimization problem [6] that can be numerically solved using Lagrangian techniques such as “lambda iteration”. However, when line losses are included in the model, the ED problem becomes analytically intractable even with simplified models for the generator cost functions. Under these conditions, complex numerical optimization methods such as genetic algorithms, partical swarm optimization or Monte-Carlo methods are often employed to find the optimal solution. Our distributed algorithm offers the following advantages over the traditional centralized approaches.
*This work is supported in part by US NSF grants ECS-0622017, CCF072902, and CCF-0830747.
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1. Scalability. The centralized dispatcher requires knowledge of the cost functions of each generator which limits its scalability, especially in a power grid supplied by a large number of small distributed generators. 2. Dynamic adaptability. Since under the traditional centralized approach, the ED algorithm is implemented on long time-scales (on the order of 45 mins), it is unable to respond to load fluctuations and changes in the state of the generators occurring on faster time-scales. 3. Analytical tractability. Solving the minimum cost allocation problem requires the controller to have knowledge of not only the cost functions of each of the generators, but also an estimate of line losses and reactive power flows corresponding to each possible allocation. This makes the problem analytically intractable. The distributed approach solves the optimization problem in an iterative “online” fashion and as such, does not
require the use of approximations and simplifying assumptions. In this paper, we consider an idealized model of the economic dispatch problem, where we neglect power losses and also ignore constraints on maximum allowable reactive power flows, voltage deviations and so on. This model allows us to develop an intuitive understanding of the properties of the algorithm, and offers a starting point for investigating more realistic models. Related Work. The problem of engineering the power grid to accomodate a high penetration of alternative energy generators has recently been a very active topic of research [7]. Deregulation of the electric utilities in many countries [8] in recent decades led to research on a decentralized model of control [9], where utilities, transmission system operators (TSO) and independent power producers (IPP) cooperate and compete using market and other mechanisms. This model uses a hierarchical control scheme [10], and individual generators are still centrally controlled [11]. Fully distributed control algorithms have only recently attracted interest for power systems, usually in the context of a system with distributed generators [12]. Recent interest in small isolated power systems or microgrids [13, 14] has provided an impetus to research in distributed control. In this context, market pricing is commonly used as a basis for devising coordination mechanisms [15, 16]. To the best of our knowledge, the idea of using distributed control algorithms for optimal allocation of generation capacity or other related control applications, has not been considered before. The remainder of this paper is organized as follows. In Section 2, we present our model of the economic dispatch problem, and describe our distributed algorithm. We present analytical results that show the optimality problems of the algorithm in Section 3. We illustrate the performance of the algorithm and comapre it with traditional dispatch algorithms using numerical simulations in Section 4. Section 5 concludes.
2 PROBLEM DESCRIPTION We model the economic dispatch problem as follows. We assume that there are N generators in the system. At timestep k, the total power consumed is Pload [k], and the generator power setting for the rated system frequency is given by Pi [k], i ∈ 1 . . . N . As a result, the power imbalance in the system is given by ∆P [k] = Pload [k] −
N X
Pi [k].
(2.1)
i=1
where we neglected power losses in the system. We assume the power imbalance causes a proportional frequency deviation ∆f [k] = β∆P [k] on the grid that can be monitored continuously by each generator. This is analogous to the ACE observed by the secondary controller in a traditional LFC implementation. We assume that β remains constant for all values of Pi [k] and ∆P [k]. Note that this is an approximation that is essentially equivalent to assuming similar droop for the power-frequency curves of all the generators in the system.
Let Ji (P ) be the cost function for generator i. We take the cost functions to be fixed in time for this paper. In practice, intermittent fluctuations in wind-speed or other conditions may change the generation cost, however as long as the changes occur on a time-scale slow compared to the algorithm, it is reasonable to take the cost as constant over a large number of time-steps. The goal of the dispatch algorithm is to choose Pi [k] to PN minimize the total cost J[k] = i=1 Ji (Pi [k]) and force the power imbalance ∆P [k] to zero. The following assumptions hold for each Ji (·). Assumption 2.1. Each Ji (·) is twice differentiable and a convex function. In fact there exist βi > 0, and α > 0 such . 2 Ji (P ) that for all P , the second derivative J ′′ (P ) = d dP obeys 2 β1 P α > J ′′ (P ) > β2 . . i (P ) Define the marginal costs Ji′ (P ) = dJdP . The following assumption on the initial values of the marginal costs is justified subsequent to its statement. Assumption 2.2. Each marginal cost is positive at the initial time. This assumption is physically reasonable because we expect that at least at the intial time, additional units of generation to have positive cost and the cost to increase with the power level. Finally, we assume that there are no other constraints on the Pi [k], thus we neglect the commonly imposed maximum and minimum limits on the power of the active generators. Our model is flexible and can be modified to accomodate most of the above assumptions, however we leave that for future work. It is easy to show using Lagrangian techniques that the solution to the above optimization problem satisfies: Ji′ (Pi ) ≡
dJi (Pi ) . = constant = λ, ∀i ∈ 1 . . . N. dPi
(2.2)
Equation (2.2) has the the well known interpretation that at the minimum cost allocation of power, the marginal cost Ji′ (Pi ) of an additional unit of power is constant across all generators; we denote this optimal marginal cost by λ. 2.1 Distributed algorithm for optimal economic dispatch We now describe our distributed algorithm. This is an iterative algorithm under which at time-step k, generator i updates its rated power as follows. ( 1 , if ∆P [k] > 0, Pi [k] + α1 ∆P [k] J ′ (P ) i i Pi [k+1] = Pi [k] + α2 ∆P [k]Ji′ (Pi ), otherwise. (2.3) where α1 > 0 and α2 > 0 are paramaters controlling the rate of adaptation. The intuition behind (2.3) is explained as follows. When the power imbalance ∆P [k] is positive, then the generators make a small increase to their rated powers in inverse proportion to their marginal cost. Thus generators with low marginal costs increase their allocation more rapidly than high cost generators. Conversely when the ∆P [k] is negative, then the low cost generator reduces its power less
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rapidly compared to high cost generators. Over time, this algorithm tends to equalize the marginal cost across generators and this lead to the minimum cost solution. We now analyze the properties of the algorithm in more detail.
3 PROPERTIES OF THE DISTRIBUTED DISPATCH ALGORITHM It is more convenient to analyze this algorithm by looking at its continuous time version described as follows. The results directly extend to the discrete time version for sufficiently small αi . ( 1 , if ∆P (t) > 0, α1 ∆P (t) J ′ (P dPi (t) ) i i (3.4) = dt α2 ∆P (t)J ′ (Pi ), otherwise. i
Also, d∆P (t) ˙ = −1T P(t) dt = −α1 ∆P (t)
∆P˙ (t) < −γ∆(P (t)).
Theorem 3.1. Consider (3.4), under assumptions 2.1 and 2.2. Suppose Pload is constant and with the unique δi as above obeys N X δi < Pload . (3.6)
(3.7)
(3.8)
Hence (c) holds. Finally suppose that ∆P (0) < 0. Since ∆P = 0 is stationary trajectory, by assumption 2.1 (b) and hence (a) holds. Further, in this case d∆P (t) ˙ = −1T P(t) dt = −α2 ∆P (t)
N X
Ji′ (Pi (t)).
(3.9)
i=1
We next assert that there exists a δ > 0, such that for all t, N X
(3.5)
The underlying structural change in (3.4) on either side ∆P = 0, and the appearance of the marginal costs in the denominator of the update kernel, raises the prospect of lack existence of solution. Theorem 3.1 proves that not only does the solution to (3.4) exist but in fact this solution corresponds to ∆P (t) converging exponentially to zero. Before we prove this theorem we observe that the convexity of each cost function guarantees that if for some i, there is a δi for which Ji′ (δi ) = 0, then this δi is unique.
1
, J ′ (Pi ) i=1 i
i.e. as J ′ (Pi (t)) > J ′ (Pi (0)) > 0, by assumption 2.1 there is a, γ > 0, such that
In (3.4) all variables have physical interpretations analogous to the discrete time counterparts i.e. PNthe instantaneous power imbalance ∆P (t) ≡ Pload (t) − i=1 Pi (t) and so on. Fur. ther denote P = [P1 , P2 , . . . , PN ]T i.e. P : R → RN has elements representing the power allocations across the generators. Denote the vector corresponding to the optimal . allocation (2.2) by Popt . Further let 1 = [1, 1, . . . , 1]T denote the “all-ones” vector. Then we can write ∆P ≡ Pload − 1T P.
N X
Ji′ (Pi (t)) > δ.
(3.10)
i=1
To this end observe that if at any t0 and some i, Pi (t0 ) = δi , then for all t > t0 , Pi (t) = δi . Then because of (3.6), and the fact that under (b), ∆P (t) 6 0, the convexity of the cost functions ensures (3.10). Then (c) holds. Remark 3.1. The proof of this theorem reveals an important point. If ∆P (0) > 0 then for all t, Ji′ (Pi (t)) > 0. If ∆P (0) < 0 then for all t, Ji′ (Pi (t)) > 0. Note that Theorem 1 guarantees the algorithm reaches an equilibrium point exponentially rapidly; however it does not guarantee that the equilibrium point has minimum cost. Nonetheless in the special case where all cost functions are identical, i.e. for some F (·), and all i,
i=1
Ji (·) = F (·).
(3.11)
Then there hold: a stronger result is possible. (a) The solution to (3.4) exists. (b) For all t > 0, ∆P (t)∆P (0) > 0. (c) ∆P (t) converges exponentially to zero. Proof: First observe that when ∆P (0) = 0, clearly the solution exists and (a)-(c) hold. Now suppose ∆P (0) > 0. Then clearly, under assumption 2.2 P˙i (0) > 0 for all i ∈ {1, · · · , N }, and the solution exists and is continuous for some t ∈ [0, t1 ]. Further, ∆P (t) > 0 and Pi (t) > Pi (0) for all t ∈ [0, t1 ]. Since Ji (·) is strictly convex, J ′ (Pi (t)) > J ′ (Pi (0)) for all t ∈ [0, t1 ]. Thus, arguing as above, the solution exists unless at least at some t2 , ∆P (t2 ) = 0. Of course the solution exists even beyond such a t2 and obeys in fact Pi (t) = Pi (t2 ), for all t > t2 . Further as ∆P = 0 is a stationary trajectory, by continuity (b) holds.
Theorem 3.2. Consider (3.4), under (3.11) and the conditions of Theorem 3.1. Suppose ∆P (t) 6= 0, and for some i, j, F ′ (Pi (t)) 6= F ′ (Pj (t)). Then there holds: d(F ′ (Pi (t)) − F ′ (Pj (t))) (F ′ (Pi (t)) − F ′ (Pj (t))) < 0. dt Proof: Suppose first that for some t, ∆P (t) > 0. Then from Theorem 3.1 ∆P (0) > 0. Suppose now F ′ (Pi (t)) > F ′ (Pj (t)). From remark 3.1 and assumption 2.1, Pi (t) > Pj (t). Then the result follows by noting that d(Pi (t) − Pj (t)) 1 1 = α1 ∆P (t) − dt F ′ (Pi ) F ′ (Pj )
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< 0.
(3.12)
Now suppose ∆P (t) < 0. Then from Theorem 3.1 ∆P (0) < 0. Suppose now F ′ (Pi (t)) > F ′ (Pj (t)). From remark 3.1 and assumption 2.1, Pi (t) > Pj (t). Then the result follows by noting that d(Pi (t) − Pj (t)) = α1 ∆P (t) (F ′ (Pi ) − F ′ (Pj )) dt < 0. (3.13)
questions for future work. One key issue is extending the distributed algorithm to take advantage of smart metering infrastructure, energy storage and communication facilities that can provide more detailed information about the state of the system than just the power imbalance. Another issue is in further developing the analytical results reported here to characterize the dynamics of the algorithm under weaker assumptions and including more constraints.
REFERENCES The significance of this result is as follows. While ∆P (t) 6= 0, the algorithm will tend to drive the marginals closer. If ∆P (t), becomes zero before the marginals are equalized, then the slightest noise in the Pi or load fluctuations that enforce the condition ∆P (t) 6= 0, will again tend to derive the marginals closer to each other. Over time the practical effect of this is to equalize the marginals. Note that under (3.11), the optimal allocation is equal power allocation i.e. Pi ≡ Pload N , ∀i ∈ 1 . . . N . Thus over time this algorithm will eventually find an optimal operating point. Simulations presented in the next section attest to this fact.
[1]
[2] [3]
[4]
[5]
4 SIMULATION RESULTS We now present some numerical simulations to illustrate the performance of the algorithm. We consider an isolated micro-grid with 3 generators, all with the cost function F (P ) = 0.00482P 2 + 7.97P + 78. Fig. 1 shows the power allocations Pi (t), marginal costs Ji′ (Pi (t)) ≡ F ′ (Pi (t)) and the power imbalance ∆P (t), for a case where the generators’ initial allocations are far from optimal. The Pload (t) is nominally 800 kW, with added noise of 5 kW per iteration to represent fluctuations. The algorithm responds to instantaneous load imbalance to quickly reduce ∆P (t) to zero, however the effect of the small fluctuations serves to drive the power allocations to the costminimizing optimum over time as seen in the figure. We also note that the marginal cost of the generators are equalized over time. Fig. 2 shows the algorithm dynamics for the same system as in Fig. 1, except that in this case, the load power is assumed to be a constant throughout, while random perturbations are applied to the generator power Pi (t). This reflects the analysis in Section 3 more faithfully. As in Fig. 1, we see that the algorithm successfully drives the power allocations and marginal costs towards the optimum over time, so the effect of generator power perturbations and load fluctuations are similar as far as the dynamics of the algorithm are concerned.
[6]
[7] [8] [9]
[10]
[11]
[12] [13]
[14]
5 CONCLUSIONS We presented a simple distributed algorithm to control power the allocation to generators in an isolated electric grid. The algorithm requires each generator only to have knowledge of its own generation cost function and the overall power imbalance. We presented analytical arguments to show that the algorithm achieves minimum cost allocation over time under some assumptions on the cost functions. The preliminary work reported in this paper shows that the distributed approach has many advantages especially for alternative energy generators, and opens up many interesting
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Power Outputs 800 P
Load
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Fig. 1 Dynamics of the distributed dispatch algorithm with random load fluctuations
Power Outputs 800 P
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Fig. 2 Dynamics of the distributed dispatch algorithm with random perturbations of generator power.
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