Large population optimal demand response for thermostatically controlled inertial loads

Transcription

1 Large population optimal demand response for thermostatically controlled inertial loads Gaurav Sharma, Le Xie and P. R. Kumar Department of Electrical and Computer Engineering, Texas A & M University. {gash, le.xie, prk}@tamu.edu. Abstract In this paper, we formulate and analyze the optimal set point problem of thermostatically controlled loads in a smart microgrid environment. From an aggregator s perspective, we show that flexibility can be extracted from the thermal inertia embedded in residential customers and be leveraged for maximum utilization of variable generation such as wind and solar. Through rigorous theoretical results we show that there is a fully decentralized mechanism for setting individual household temperature bounds while (a) maximum flexibility is extracted from inertial loads to balance the stochastic generation; (b) optimal staggering of demand responses of inertial loads can be achieved for smoothing out the overall power consumption; and (c) individual customers comfort zone is taken into account. An easy-to-implement practical control setting is also introduced and tested using a simple model of wind and load distribution. I. INTRODUCTION Inertial thermal loads, such as building air-conditioners, can be pro-actively switched on and off to absorb power from intermittently available renewable energy sources such as wind. Their remaining energy need which renewables cannot provide can be supplied from nonrenewable sources. The resulting advantage is that the peak-to-average power ratio of nonrenewable supply will be lowered by the smart utilization of inertial thermal loads to absorb the fluctuations in stochastic renewables. This permits a lower operating reserve for employing renewables; see Figure. Under such a demand response strategy, the inertial thermal loads could however get synchronized. This is dangerous, since any external disturbance, such as a coordinated change in temperature settings of all the buildings, caused, for example, by many households congregating in their family rooms at Superbowl time, can cause huge surges and swings in power drawn from the nonrenewable sources. Thus it is important to stagger the demand responses of the large population of inertial loads in order to guard against such contingencies while still absorbing the fluctuations of renewable power supply. This paper examines this fundamental yet subtle question arising in using demand response to maximize the utilizability of intermittently available renewable energy sources: What is the optimal staggering of inertial thermal loads in demand response? We further show that this can be achieved in an architecture consisting of an aggregator or load serving entity that controls the set points of the loads. The aggregator does not need to know the individual temperatures of the loads, which alleviates concerns about the privacy of load information. Also, the control of the load is locally done, thus avoiding excessive communication bandwidth in actuation. The loads need to only know a signal related to the availability of renewable power. While several smart grid solutions in the form of demand response schemes have been proposed, including price control, incentive payments and dynamic load control [], [], [3] the focus in this paper is on the problem of demand response Power Availability Power Dispatched Renewable Energy Sources Time Nonrenewable Energy Sources Time + Temperature Time Iner4al Thermal Load Time varia4on in nonrenewable power needed is lowered because iner4al loads can absorb the 4me varia4ons of renewable power supply. Fig.. Inertial loads can be used to absorb variations in renewable power availability and thus reduce operating reserve required from conventional generators. in the form of direct load control [4], [5], for thermostatically controlled loads (TCLs) via a central load serving entity (LSE), which is connected to both a renewable power driven microgrid as well as the main power grid. The LSE can set the temperature limits of each of a collection of the TCLs. There are several fundamental questions that arise: To what extent can the power consumption of inertial loads, such as building A/C units, be matched to the stochastic power generation of renewable energy sources, while taking into account the comfort level settings of the occupants of the buildings? To what extent can the power needed to be drawn from nonrenewable sources be kept as constant as possible, so that the operating reserve required is minimized? This can be regarded as addressing the value proposition of renewables from the point of view of the LSEs/aggregators. How should the LSE optimally control the power to satisfy the above two objectives, the first on the demand side and the second on the nonrenewable power supply side? We will model both these aspects by incorporating them into a joint cost function. We will model the renewable power as a stochastic process. The problem then becomes one of choosing the power consumption of the inertial loads to minimize the cost criterion while satisfying load constraints. To avoid synchronization of the temperatures of all the loads [6], [7], we explicitly incorporate in the model the possibility that all the thermal loads may on occasion simultaneously change their comfort zone settings at the same time instant. We show that the optimal solution then is to stagger the demand responses of the A/C loads whenever their temperatures are getting high, but to re-synchronize them when the temperatures Comfort Range

2 are lower and thus it is safer. This maximizes the utilization of stochastic renewables while minimizing operating reserve needed. Since optimal staggering of the loads is obtained by choosing different upper temperature limits of the thermostats or different loads, the critical question is what is the optimal distribution of the thermostat upper limit settings that the LSE should use? We determine the exact explicit form of the optimal distribution in the large population limit of a large number of TCLs. We also show that this large population limit is actually quite accurate even when number of TCLs is quite small, even for ten loads. Thus, the large population limit provides a simple explicit solution to an otherwise complex demand response problem of varying finite population size. What are the privacy implications of the LSE architecture? The desirable feature of our solution is that the LSE does not need access to the instantaneous temperatures of the buildings, or to monitor their power consumptions. This alleviates privacy concerns which arise from the fact that knowledge of the power consumption allows an observer to deduce the activities of the occupants [8], [9]. In our solution, the LSE only needs to occasionally set the set-points at which an A/C is triggered. This does not require intrusive sensing of a building s temperature. Also, the LSE does not need to intrusively actuate building A/Cs either. This is done in a distributed fashion by each building on its own. What are the communication requirements entailed by a large population of TCLs? For the same reason as above, the LSE requires minimal communication. The TCLs only need to know the state of the renewable energy source. This is akin to a price signal concerning availability of power, and some such one way communication of power availability appears to be an essential component of any demand response scheme. II. THE MODEL We will conduct our analysis under very simple models of loads and renewable power variations. All the results can be generalized. Our goal is to show how this problem can be modeled and solved. More detailed models of wind, loads, etc., only changes details of the calculations, not their nature. In this work we will only consider a collection of identical buildings with A/Cs, and suppose that the renewable energy is wind. We will assume that the temperature of each building follows the differential equation ẋ = h P, where h is due to the ambient heating, and P is the power used by the air conditioner. We will suppose that the maximum power that can be drawn by the A/C is h+c, so that the steepest rate of cooling is when ẋ = c. Every building has a comfort range [Θ m(t), Θ M (t)] within which it would like the temperature to be. For simplification we translate the comfort range to make Θ m(t) =. To ensure that the designed policies are robust, we suppose that all buildings change their Θ M (t) in a coordinated fashion at the same instant of time (á la Superbowl Sunday). For simplicity we model Θ M (t) as switching between two values Θ and Θ (with Θ < Θ ) as a Markov process with mean holding times τ l and τ h, respectively in the two states. We will suppose that the wind power available is a two state Markov process, which can be generalized to have multiple states. When the wind is Not Blowing, the power generated by the wind is. When the wind is Blowing, it generates power that is sufficient to cool the collection of all buildings at a maximum rate c. The mean holding times in the two states are τ and τ respectively. We suppose that when the wind is blowing, each building uses wind power to cool itself at maximum rate, unless it is already at it s lowest permissible level in the comfort zone ( in our case), in which case it uses just enough wind power to maintain itself { at this lowest temperature. Thus c when x(t) > and wind is blowing ẋ(t) = when x(t) = and wind is blowing. When the wind is not blowing, each building operates as follows. Each building is assigned a set point ; if the temperature is less than, then the A/C is not used. However when the temperature hits then it draws just enough nonrenewable power to maintain { its temperature at. Thus if x(t) = and wind is not blowing ẋ(t) = h if x(t) < and wind is not blowing. Letting P w(t) denote the power from wind and Pg(t) the power drawn from non-renewable sources, the above -policy can be summarized as follows (see Figure ): ẋ(t) = h P w(t) Pg(t), where, { h + c when x(t) > P w(t) = h when x(t) =, and, { h when x(t) = P g(t) = when x(t) <. Wind Temperature P g (t) Blowing Not Blowing h Fig.. Behavior under -policy. When wind is blowing, houses cool at the maximum rate. When wind is not blowing, we don t allow houses to heat beyond a level. Let us now consider a collection of N such buildings labeled,,.., N, with building i employing a i -policy. We denote its temperature by x i (t), and the nonrenewable power it draws by P g,i (t). We will allow the set-point i to be chosen in the range i Θ. However, if Θ < i Θ, then there are occasions when a building s temperature x i (t) may exceed its comfort level Θ(t). We will use [(x i (t) Θ(t)) + ] to measure the instantaneous discomfort of building i at time t, where y + = max(, y). Over a long time interval, the cost of its average discomfort is lim T T T [(x i(t) Θ(t)) + ] dt. The total power drawn from non-renewable sources is N P g,i (t) at time t. We want the nonrenewable power to be i=

3 as close to constant as possible, and so we will employ a [ N cost function P g,i (t) ] dt as a means to reduce lim T T T i= peak-to-average power consumption. This ensures a low operating reserve. Wind power Nonrenewable power Load Serving Entity Min C(~z) Load Load Dividing rows of (4), we get dpz dx (x) = h c dp z dx (x). This yields p z (x) = h c pz (x)+dz, where D z is a constant. Next, substituting the above in the first row of (4), we get dp z ( dx (x) = ) p z Dz hτ cτ (x) = µp z Dz cτ (x). cτ Hence, p z (x) = Kz e µx µcτ Dz, where K z is a constant, and subsequently p z (x) = h c Kz e µx + hµτ Dz. Lastly, since p z, pz, δz and δz z form a probability distribution, and P(wind is blowing) = τ τ +τ we require, (p z + pz )(x)dx + δz + δz z =. and, p z (x)dx + δz = τ τ + τ. Solving the above two equations for D z and K z yields D z = and K z given by (3). Where C(~z) = lim T! T T NX (N) [(xi(t) (t)) + ] + i= User discomfort cost NX Pg,i(t) dt i= Load N Grid power cost M M N Comfort range varia5on Fig. 3. Overall architecture and problem. The load serving entity sets the optimal set-point values { i } for different loads, based on the wind model. We therefore consider the following model illustrated in Figure 3. The LSE needs to choose the settings = (,,..., N ) for the N buildings to minimize the cost: T { N lim γ (N) [(x i (t) Θ(t)) + ] + ( N P g,i (t) ) } dt. T T i= i= Above, γ (N) > is a relative weight by which to trade-off discomfort versus operating reserve. III. ANALYSIS OF A -POLICY We begin by analyzing the steady state probability distribution resulting from a -policy employed in a single building. For simplicity, we will assume that the cooling due to the wind power is sufficient on average to compensate for the ambient heating, i.e., cτ > hτ or µ := hτ cτ >. From Figure, it can be seen that the TCL will occupy temperature levels X = and X = each for a non-zero fraction of time on average. Denote the steady state probability masses by δ := P({X = }) and δ := P({X = }). For the intermediate temperatures x (, ), denote by p z (x) and pz (x), the steady state occupation probability densities of the TCL at temperature x when wind state is on and off respectively. Lemma : The probability density functions and mass functions are p z (x) = Kz e µx, p z (x) = c h Kz e µx, () where, K z = [ cτ + δ z = cτ K z, δ z z = cτ K z e µz, () ( + c ) ( h µ + cτ ( + c h ) ) ] e µz. (3) µ Proof: We employ the Chapman-Kolmogorov equations for probability densities and probability mass: dp z [ dx (x) dp z cτ ] [ ] = cτ p z (x) dx (x) hτ hτ p z (x) (4) ] [ ] [ ] cτ p z = () hτ p z (). [ δ z δ z z IV. ANALYSIS OF A FINITE NUMDER OF LOADS We now analyze the case when there are a finite number N of TCLs. To gain intuition consider N = 3 and = (6, 7, 8) with Θ =. Figure 4 shows typical temperature trajectories. For any pair of TCLs, (T CL i, T CL j ) following ( i, j )-policy with i < j and for any realization of wind process ω Ω, starting from identical initial conditions (or if we ignore the transients due to initial conditions), the temperature trajectories satisfy X i (t, ω) X j (t, ω). Therefore, we have the following inclusion of probability events: {X j (t, ω) = j } {X i (t, ω) = i }, for i < j. (5) Wind Temperature Blowing Not Blowing 3 Load Load Load 3 Fig. 4. Temperature trajectories under a -policy for 3 TCLs as wind power varies between blowing and not blowing. Now we consider the expected cost of a -policy. The simplest case is when we have one TCL. With relative weight γ () >, the total expected cost of the -policy is given by C () () = h P z ({X = }) + γ () E W t,θ t{[(x Θ(t)) + ] }. For brevity [ we will refer to the second term as Φ(z) : ] Φ(z) := E Θt = τ h τ h + τ l [(x Θ(t)) + ] (p z + pz )(x)dx + [(z Θ(t))+ ] δz [ ] [(x Θ ) + ] (p z + pz )(x)dx + [(z Θ ) + ] δz. For two TCLs with, the total grid power is h when X = because from (5) in Section III, X = X =. Also, when X < but X =, the grid power is h. So the total cost C () with relative weight γ () > is C () () = (h) P({X = }) + h P({X < } {x = }) + E Wt,Θ t {[(X Θ(t)) + ] + [(X Θ(t)) + ] } = h ( P({X = }) + P({X < } {X = })) + γ () (Φ( ) + Φ( )). (6)

4 Also, from (5), {X = } {X = }, and so we have P({X = }) = P({X = } {X < }) + P({X = }), i.e., P({X = } {X < }) = δ δ. (7) Similarly, for N TCLs with... N, the cost is N C (N) () := γ (N) Φ( i ) + h (N P(X N = N ) i= N + (N i) P(X N i = N i X N i+ < N i+ )). i= A critical scaling parameter is γ (N). The term T ( N i= P g,i ) dt grows like Ω(N ), however the term T Ni= [(x i (t) Θ(t)) + ] dt grows like Ω(N). Hence, we let the relative weight scale as γ (N) = γ.n. We thus obtain the following expression for normalized cost Ĉ (N) : N = γ Ĉ (N) () = C(N) () N N Φ( i) + h (P(X N = N ) N + ( i N ) P(X N i = N i X N i+ < N i+ )). (8) We thereby arrive at the following optimization problem for the case of a finite number N of TCLs: Minimize Ĉ (N) () s.t. i i+ for i =,,..., N,, N Θ. (9) This problem is however difficult to solve as the population N grows, first because it is not clear that the objective function is convex in, and second, as N increases, the number of inequalities grows linearly. We will later evaluate the solution using numerical methods. However, we will now show that in the limiting case when N is large that we can approximate it by a continuum, and thus actually obtain an explicit optimal solution! V. THE EXPLICIT SOLUTION FOR THE INFINITE POPULATION CASE We now consider the case of a continuum of TCLs, i.e., {T CL i : i [, ]}. We could use R [,] + to define a - policy. However, as we saw in Section IV it is more convenient to use a population distribution function u(z) denoting the fraction of TCLs whose set-point is below or equal to z, to describe the policy. Let U denote the admissible space of piecewise continuous increasing positive functions on [, ]. Equation (8) ( when there are N TCLs is a special case with Ni= ) u (N) H(x (x) = i ) (using Heaviside s step function N H(x) := { for x <, and, for x }.). Generalizing (8) to a continuum for u U, the cost is: C [,] (u) =γ Φ(z)u dz + h (δ Θ Θ + u (z)p({x z = z} {Xz+dz < z + dz})) where the first term is the cost due to temperature variation, and second term is that due to grid power usage. h δ Θ Θ corresponds to the case when all the TCLs are consuming grid power ( Where δ Θ Θ = P({X Θ = Θ }) as given in ()). The integral term in the grid power considers the case when a TCL with set point z is consuming grid power but a TCL with set point z + dz is not. Using (7), we obtain P({X z = z} {X z+dz < z + dz}) = ( dδz z dz )dz. Denote := dδz z [ ] Θ C [,] (u) = h δ Θ Θ + (h u (z) + u(z)(z))dz. dz. Thus Since only the second term is dependent on u(z), the optimization problem becomes: Minimize J[u] = (h u (z) + (z)u )dz () s.t. u U () u() =, u(θ ) =. () The Euler-Lagrange equation [] suggests the optimizer u of J[u] = Θ F (z, u, u )dz satisfies F u dz d F u =. That is, h u(z) dz d (z) =. Hence, u(z) = h. However this solution need not satisfy u U or the boundary condition u() =, u(θ ) =. Therefore it is not admissible for all the scenarios. Our next theorem provides a solution by an alternative approach. Theorem : The solution ( to the optimization problem is u min, ) (z) (z) = h If z < Θ (3) If z = Θ. We use the following algebraic result to prove this theorem. Lemma : Define f(z) := Φ. Then f U, that is, f(z) is a positive, piece-wise continuous increasing function of z. Proof: f(z) is continuous by definition. Also, as Φ [.Θ ], it is sufficient to show f, z [Θ, Θ ]. We can write K z in (3) as K z = (a + be µz ), where a := cτ + ( + h c ) µ and b := cτ ( + h c ) µ. It is clear that a > and b < since b = cτ ( ( cτ + hτ ) µ ) and µ = hτ cτ < hτ + cτ µ ( hτ + cτ ) >. Evaluating f(z) = Φ, upto multiplicative positive constants, (( using (6), and := δz z we get: f(z) = (z Θ ) + + ) ( a + be µz ) ) hτ cτ aµ + (z Θ ) +( a + be µz ) b z + (x Θ ) +( + c ) e µx dx. aµ acτ h Differentiating ( with respect to z, we get (for z Θ f a + be µz ) ( ( ( (z) = + (z Θ ) + + ) )). a µ hτ cτ This completes the proof as both terms are non-negative. Proof of Theorem : For the case =, the h z=θ Euler-Lagrange solution holds, since it satisfies () and (). For the other cases we use the following result from [], Theorem Since u(z) U, u(z) has bounded variation, and Φ(z) is a continuous function by definition. This, along with u() = and u(θ ) =, gives Φ(z)u + u(z)φ dz = u(θ )Φ(Θ ) u()φ() = Φ(Θ ). Thus the cost from () becomes Θ J[u] = h u (z)dz + (Θ ) u(z) dz ( ) = h u(z) h dz + (Θ ) γ Φ 4c +4 dz. (4) Lemma implies h is increasing function of z. We observe the following from (4):

5 Consider h, z [, Θ ]. Since only the first term of (4) is dependent on u(z), it is always non-negative and is when u (z) = h, and since by Lemma, h U, we see that u optimizes (). In addition, we require u (Θ ) = to satisfy (). Consider h > for some z = Θ, Then for all u(z) U ( ) ( ) with u, since Φ < u(z) Φ for all z > Θ, so we have u (z) = for all z > Θ. This completes the argument as u (z) defined in (3) satisfies the above two conditions. The optimal solution can be categorized into three cases (C- C3) as shown in Figure 5: (Θ ) h D(Θ ) C: =. This corresponds to the case when both cost components strike a balance to yield a well defined density function for all z [Θ, Θ ]. C: (Θ ) h D(Θ ) >. For this case there is a Θ (Θ, Θ ) with C3: (Θ ) h D(Θ ) =, and so the density u = for all z [Θ, Θ ]. For this case the component of the cost due to temperature deviation becomes more significant than the grid power cost component. Therefore the optimal density is for TCLs having a set-point higher than Θ. (Θ ) h D(Θ ) <. For this case there is a jump discontinuity in u (z) (i.e., a point mass in optimal density u ) at z = Θ. The cost due to grid power is more significant than the cost due to temperature deviation. The cost term due to grid power is minimized by pushing the distribution towards higher, since δ z z and are decreasing functions in z. Distribution U * (z) Density U * (z) Case Case Case 3 Case Case Case 3 Set point Fig. 5. Different cases for optimal population distribution u (z) and corresponding optimal density function u. The exact optimal distribution for the large population case (u (z)) is very useful for easily generating approximately optimal solutions for the otherwise intractable finite population case. We can simply generate N random set-points according to the distribution u (z) in (3). We will see in the simulations that this method performs impressively well even when N is not too large, say about N =. VI. SIMULATION RESULTS A. Comparison of finite and large population cases We first compare the analytic result for the asymptotic case with the numerically computed finite case optimal solution. For parameters (c, h, τ, τ, τ h, τ l ) = (.4,., 5, 5,, ). Figure 6 shows that u (N) (z) is remarkably close to u (z) even for N = 5. Figure 7 shows that the optimal cost c [,] (z) is also well approximated. Distribution Fig. 6. Cost U 5 (z) U (z) U * (z) x 6 3 Temperature Comparison between U (z) and numerically computed U (N) (z). Ĉ (N) Population size, N Fig. 7. Comparison between C and numerically computed ĈN. B. Cost of synchronization One natural question presents itself after deriving the optimal distribution: How much do we actually gain from desynchronization? The best we can achieve with any synchronized policy is by constraining = =... = N in (9). For the same parameters as before with N =, the optimally staggered policy achieves a 5% lower cost (Figure 8). Cost Fig. s C* Cost of optimal common policy Cost of optimal policy Comparison of optimal common vs optimal staggered policy. Figure (9) compares the nonrenewable power drawn and load factor as a function of time by a population N =, under the optimal finite population case -policy and the optimal Average Power common -policy. (where Load Factor := Peak Power ). One may observe that the power fluctuations are significantly more in the common -policy than in the optimal policy. Nonrenewable power Load factor Fig. 9. Optimal case Common case x Optimal case Common case Nonrenewable power time-profile and load factor comparisons. C. Effect of temperature variations in comfort range One may see that in the absence of cost of user comfort setting variation, the entire population should set Θ. We now compare the total cost when there is no threshold variation to the cost when only a percentage of the population changes their threshold to Θ, (Θ < Θ ). We choose this population iid with probability p =.5. To put these two cases into perspective, we also compared with the optimal -policy in Figure. For

6 N = TCLs and relative weight γ =., the cost increases by approximately % when we additionally take temperature variation into account. We can reduce this cost by 56% using the finite population optimal policy. Cost Cost Θ w/o variation cost Total cost of optimal policy Cost Θ with variation cost x 4 Fig.. Cost comparison of common policy ( Θ ) with optimal staggered policy. D. Approximation of solution to finite population by infinite population solution For N = we compare in Figure the -policy for the numerically computed optimal policy with the policy obtained by approximating by the infinite population policy as described earlier. Distribution Optimal infinite case distribution Approximation based finite distribution Optimal finite distribution Setpoint, Fig.. Comparison of numerically computed optimal finite case distribution, with the approximate solution from u (z). Figure compares the time average grid and temperature variation cost components between the two polices. Apparently, in terms of cost, the approximate solution is a good substitute for the optimal solution. Nonrenewable cost Variation cost Total cost Optimal grid cost Grid cost for generated Optimal temperature variation cost.8 Variation cost for generated x Optimal total cost x 4.6 Total cost for generated x 4 Fig.. Comparison between grid and variation cost components. VII. CONCLUSION & FUTURE WORK We have studied the potential for extracting flexibility from control of thermal inertial loads in a smart microgrid environment. We have formulated and analyzed the optimal set-points for individual thermal loads. Through rigorous theoretical results we have shown how to design a fully decentralized control set-point for thousands of household loads, which allows the load serving entities to maximize the utilization of stochastic wind generation and achieve a high load factor throughout the year. A set point distribution for optimal staggering of thermal inertial loads is also presented, which is fully decentralized. Numerical results suggest that the proposed method could lead to easy-to-implement control logic for load aggregators. This paper is only a first step towards rigorous assessment of flexibility embedded in demand side in smart grids []-[6]. Our results have been developed in a simplified context of wind, thermostat, and other models, but can be generalized. Future work could also include consideration of economic incentives for loads to participate in the proposed control paradigm. The proposed solution could be tested in a realistic microgrid setting such as Texas A&M campus. ACKNOWLEDGMENT This material is based upon work partially supported by NSF under Contract Nos. CPS-396, CPS-36, and CPS- 36. REFERENCES [] M. H. Albadi and E. F. 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